trigonometry assignments

Introduction to Trigonometry

Trigonometry (from Greek trigonon "triangle" + metron "measure")

Want to learn Trigonometry? Here is a quick summary. Follow the links for more, or go to Trigonometry Index

Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more!

Right-Angled Triangle

The triangle of most interest is the right-angled triangle . The right angle is shown by the little box in the corner:

Another angle is often labeled θ , and the three sides are then called:

Adjacent : adjacent (next to) the angle θ
Opposite : opposite the angle θ
and the longest side is the Hypotenuse

Why a Right-Angled Triangle?

Why is this triangle so important?

Imagine we can measure along and up but want to know the direct distance and angle:

Trigonometry can find that missing angle and distance.

Or maybe we have a distance and angle and need to "plot the dot" along and up:

Questions like these are common in engineering, computer animation and more.

And trigonometry gives the answers!

Sine, Cosine and Tangent

The main functions in trigonometry are Sine, Cosine and Tangent

They are simply one side of a right-angled triangle divided by another.

For any angle " θ ":

(Sine, Cosine and Tangent are often abbreviated to sin, cos and tan .)

Example: What is the sine of 35°?

Using this triangle (lengths are only to one decimal place):

sin(35°) = Opposite Hypotenuse = 2.8 4.9 = 0.57...

The triangle could be larger, smaller or turned around, but that angle will always have that ratio .

Calculators have sin, cos and tan to help us, so let's see how to use them:

Example: How Tall is The Tree?

We can't reach the top of the tree, so we walk away and measure an angle (using a protractor) and distance (using a laser):

We know the Hypotenuse
And we want to know the Opposite

Sine is the ratio of Opposite / Hypotenuse :

sin(45°) = Opposite Hypotenuse

Get a calculator, type in "45", then the "sin" key:

sin(45°) = 0.7071...

What does the 0.7071... mean? It is the ratio of the side lengths, so the Opposite is about 0.7071 times as long as the Hypotenuse.

We can now put 0.7071... in place of sin(45°):

0.7071... = Opposite Hypotenuse

And we also know the hypotenuse is 20 :

0.7071... = Opposite 20

To solve, first multiply both sides by 20:

20 × 0.7071... = Opposite

Opposite = 14.14m (to 2 decimals)

The tree is 14.14m tall

Try Sin Cos and Tan

Play with this for a while (move the mouse around) and get familiar with values of sine, cosine and tangent for different angles, such as 0°, 30°, 45°, 60° and 90°.

Also try 120°, 135°, 180°, 240°, 270° etc, and notice that positions can be positive or negative by the rules of Cartesian coordinates , so the sine, cosine and tangent change between positive and negative also.

So trigonometry is also about circles !

Unit Circle

What you just played with is the Unit Circle .

It is a circle with a radius of 1 with its center at 0.

Because the radius is 1, we can directly measure sine, cosine and tangent.

Here we see the sine function being made by the unit circle:

Note: you can see the nice graphs made by sine, cosine and tangent .

Degrees and Radians

Angles can be in Degrees or Radians . Here are some examples:

Repeating Pattern

Because the angle is rotating around and around the circle the Sine, Cosine and Tangent functions repeat once every full rotation (see Amplitude, Period, Phase Shift and Frequency ).

When we want to calculate the function for an angle larger than a full rotation of 360° (2 π radians) we subtract as many full rotations as needed to bring it back below 360° (2 π radians):

Example: what is the cosine of 370°?

370° is greater than 360° so let us subtract 360°

370° − 360° = 10°

cos(370°) = cos(10°) = 0.985 (to 3 decimal places)

And when the angle is less than zero, just add full rotations.

Example: what is the sine of −3 radians?

−3 is less than 0 so let us add 2 π radians

−3 + 2 π = −3 + 6.283... = 3.283... rad ians

sin(−3) = sin(3.283...) = −0.141 (to 3 decimal places)

Solving Triangles

Trigonometry is also useful for general triangles, not just right-angled ones .

It helps us in Solving Triangles . "Solving" means finding missing sides and angles.

Example: Find the Missing Angle "C"

Angle C can be found using angles of a triangle add to 180° :

So C = 180° − 76° − 34° = 70°

We can also find missing side lengths. The general rule is:

When we know any 3 of the sides or angles we can find the other 3 (except for the three angles case)

See Solving Triangles for more details.

Other Functions (Cotangent, Secant, Cosecant)

Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another:

Trigonometric and Triangle Identities

And as you get better at Trigonometry you can learn these:

Enjoy becoming a triangle (and circle) expert!

Trigonometry

Ramp up on the fundamentals of trigonometry as you master identities, polar graphic, and solving triangles.

Defining Angles

Before we dive into trigonometry, get comfortable with two ways of measuring angles.

Trigonometric Ratios

Explore the ratios of triangles to find trigonometry functions.

The Unit Circle

Create a reference tool for finding exact values of trig functions.

End of Unit 1

Complete all lessons above to reach this milestone.

0 of 3 lessons complete

Solving Right Triangles

Solve for lengths of triangles using trigonometric functions.

Solving for Angles

Find the angles of triangles using trigonometric functions.

Law of Cosines

Solve for measurements of non-right triangles.

Law of Sines

Learn a technique that works when the law of cosines doesn't.

Ambiguous Cases

See when measurements make multiple triangles possible, or none.

End of Unit 2

0 of 5 lessons complete

Trigonometry Graphs

What do the graphs of sine and cosine look like?

Other Trigonometric Graphs

Explore some unusual trigonometry graphs, including secant and cosecant.

Trigonometric Graph Problem Solving

Creatively apply your skills with trig graphs to solve some challenging puzzles.

Trigonometric Graph Symmetry

Apply symmetry as a method of simplification.

Inverse Trigonometry

Learn about and apply the inverse trigonometric functions: arcsine, arccosine, and others.

End of Unit 3

Polar Coordinates

Explore plots that use angle and radius to identify points on the plane.

Roses, Cardioids and Limaçon

Discover curves with loops and petals.

Function Transformations

Rotate, translate, and dilate polar graphs.

End of Unit 4

Reciprocal and Quotient Identities

Rewrite trigonometric expressions by dividing one trig function by another.

Triangle Identities

Go back to right triangles and relate trig functions to one another.

Sum and Difference Identities

Discover a clever method of handling addition and subtraction inside trig functions.

Double- and Half-Angle Identities

Explore scenarios where angle in a trig identity is doubled or halved.

End of Unit 5

0 of 4 lessons complete

Course description

This course will teach you all of the fundamentals of trigonometry, starting from square one: the basic idea of similar right triangles. In the first sequences in this course, you'll learn the definitions of the most common trigonometric functions from both a geometric and algebraic perspective. In this course, you'll master trigonometry by solving challenging problems and interacting with animated graphs, not by relying on rote memorization. Additionally, you'll learn how to apply trigonometry in the contexts of measuring and manipulating sound waves and creating complex, artistic designs using polar graphing. By the end of this course, you’ll be a master at solving triangles, graphing trig functions in both Cartesian and polar coordinates, and applying trigonometric identities.

Topics covered

Inverse Trig Functions
Trigonometric Identities
Wave Frequency

Prerequisites and next steps

You should know the content up through our Algebra II course.

Prerequisites

Pre-Calculus
Complex Numbers
Calculus Fundamentals

paper-free learning

Mathematical Symbols
Linear Equations
Quadratic Equations
Simultaneous Equations
Percentages

Trigonometry

Transformations
Graphs and Coordinates
Probability

Trigonometry (Mathematics Curriculum)

What is trigonometry, dictionary definition.

Where Does the Word Trigonometry Come From?

The curriculum, right triangle.

Right triangles (in American English) or right-angled triangles (in British English) are triangles containing one angle of 90° (called a right angle). In this mini-curriculum, you will learn about right triangles and their sides.

Pythagoras' Theorem

Pythagoras' theorem concerns the relationship between the lengths of the three sides of a right triangle. In this mini-curriculum, you will learn about Pythagoras' theorem and how to use it.

Sine Function

The sine function is a trigonometric function. The sine of the angle in a right triangle is equal to the ratio of the length of the opposite side to the length of the hypotenuse. In this mini-curriculum, you will learn about the sine function and how it can be used to find angles and lengths of sides in a right triangle.

Cosine Function

The cosine function is a trigonometric function. The cosine of the angle in a right triangle is equal to the ratio of the length of the adjacent side to the length of the hypotenuse. In this mini-curriculum, you will learn about the cosine function and how it can be used to find angles and lengths of sides in a right triangle.

Tangent Function

The tangent function is a trigonometric function. The tangent of the angle in a right triangle is equal to the ratio of the length of the opposite side to the length of the adjacent side. In this mini-curriculum, you will learn about the tangent function and how it can be used to find angles and lengths of sides in a right triangle.

Area of a Triangle Using Trigonometry

Even though trigonometry mostly concerns right triangles, it can be used for any triangle. In this mini-curriculum, you will learn about finding the area of a triangle using trigonometry.

This page was written by Stephen Clarke .

Trigonometry Curriculum

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1.E: Right Triangle Trigonometry Angles (Exercises)

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Page ID 3329

Michael Corral
Schoolcraft College

These are homework exercises to accompany Corral's "Elementary Trigonometry" Textmap. This is a text on elementary trigonometry, designed for students who have completed courses in high-school algebra and geometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but a more geometrical approach is taken than usual. Also, some numerical methods (e.g. the secant method for solving trigonometric equations) are discussed.

1.1 Exercises

For Exercises 1-4, find the numeric value of the indicated angle(s) for the triangle $\triangle\,ABC $.

1.1.1 Find $B $ if $A = 15^\circ $ and $C = 50^\circ $. 1.1.2 Find $C $ if $A = 110^\circ $ and $B = 31^\circ $. 1.1.3 Find $A $ and $B $ if $C = 24^\circ $, $A = \alpha $, and $B = 2\alpha $. 1.1.4 Find $A $, $B $, and $C $ if $A = \beta $ and $B = C = 4\beta $. For Exercises 5-8, find the numeric value of the indicated angle(s) for the right triangle $\triangle\,ABC $, with $C $ being the right angle. 1.1.5 Find $B $ if $A = 45^\circ $. 1.1.6 Find $A $ and $B $ if $A = \alpha $ and $B = 2\alpha $. 1.1.7 Find $A $ and $B $ if $A = \phi $ and $B = \phi^2 $. 1.1.8 Find $A $ and $B $ if $A = \theta $ and $B = 1/\theta $. 1.1.9 A car goes 24 miles due north then 7 miles due east. What is the straight distance between the car's starting point and end point? 1.1.10 One end of a rope is attached to the top of a pole 100 ft high. If the rope is 150 ft long, what is the maximum distance along the ground from the base of the pole to where the other end can be attached? You may assume that the pole is perpendicular to the ground. 1.1.11 Prove that the hypotenuse is the longest side in every right triangle. ( Hint: Is $a^2 + b^2 > a^2$?) 1.1.12 Can a right triangle have sides with lengths 2, 5, and 6? Explain your answer. 1.1.13 If the lengths $a $, $b $, and $c $ of the sides of a right triangle are positive integers, with $a^2 + b^2 = c^2 $, then they form what is called a Pythagorean triple . The triple is normally written as ($a $,$b $,$c$). For example, (3,4,5) and (5,12,13) are well-known Pythagorean triples. (a) Show that (6,8,10) is a Pythagorean triple. (b) Show that if ($a $,$b $,$c$) is a Pythagorean triple then so is ($ka $,$kb $,$kc$) for any integer $k >0 $. How would you interpret this geometrically? (c) Show that ($2mn $,$m^2 - n^2 $,$m^2 + n^2$) is a Pythagorean triple for all integers $m > n > 0 $. (d) The triple in part(c) is known as Euclid's formula for generating Pythagorean triples. Write down the first ten Pythagorean triples generated by this formula, i.e. use: $m=2 $ and $n=1$; $m=3 $ and $n=1 $, $2$; $m=4 $ and $n=1 $, $2 $, $3$; $m=5 $ and $n=1 $, $2 $, $3 $, $4 $. 1.1.14 This exercise will describe how to draw a line through any point outside a circle such that the line intersects the circle at only one point. This is called a tangent line to the circle (see the picture on the left in Figure 1.1.6), a notion which we will use throughout the text.

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On a sheet of paper draw a circle of radius 1 inch, and call the center of that circle $O $. Pick a point $P $ which is $2.5 $ inches away from $O $. Draw the circle which has $\overline{OP} $ as a diameter, as in the picture on the right in Figure 1.1.6. Let $A $ be one of the points where this circle intersects the first circle. Draw the line through $P $ and $A $. In general the tangent line through a point on a circle is perpendicular to the line joining that point to the center of the circle (why?). Use this fact to explain why the line you drew is the tangent line through $A $ and to calculate the length of $\overline{PA} $. Does it match the physical measurement of $\overline{PA}$? 1.1.15 Suppose that $\triangle\,ABC $ is a triangle with side $\overline{AB} $ a diameter of a circle with center $O $, as in the picture on the right, and suppose that the vertex $C $ lies on the circle. Now imagine that you rotate the circle $180^\circ $ around its center, so that $\triangle\,ABC $ is in a new position, as indicated by the dashed lines in the picture. Explain how this picture proves Thales' Theorem.

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1.2 Exercises

For Exercises 1-10, find the values of all six trigonometric functions of\\angles $A$ and $B $ in the right triangle $\triangle\,ABC $ in Figure 1.2.3.

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1.2.1 $a = 5 $, $b = 12 $, $c = 13$ 1.2.2 $a = 8 $, $b = 15 $, $c = 17$ 1.2.3 $a = 7 $, $b = 24 $, $c = 25$ 1.2.4 $a = 20 $, $b = 21 $, $c = 29$ 1.2.5 $a = 9 $, $b = 40 $, $c = 41$ 1.2.6 $a = 1 $, $b = 2 $, $c = \sqrt{5}$ 1.2.7 $a = 1 $, $b = 3$ 1.2.8 $a = 2 $, $b = 5$ 1.2.9 $a = 5 $, $c = 6$ 1.2.10 $b = 7 $, $c = 8$ For Exercises 11-18, find the values of the other five trigonometric functions of the acute angle $A $ given the indicated value of one of the functions. 1.2.11 $\sin\;A = \frac{3}{4}$ 1.2.12 $\cos\;A = \frac{2}{3}$ 1.2.13 $\cos\;A = \frac{2}{\sqrt{10}}$ 1.2.14 $\sin\;A = \frac{2}{4}$ 1.2.15 $\tan\;A = \frac{5}{9}$ 1.2.16 $\tan\;A = 3$ 1.2.17 $\sec\;A = \frac{7}{3}$ 1.2.18 $\csc\;A = 3$ For Exercises 19-23, write the given number as a trigonometric function of an acute angle less than $45^\circ $. 1.2.19 $\sin\;87^\circ$ 1.2.20 $\sin\;53^\circ$ 1.2.21 $\cos\;46^\circ$ 1.2.22 $\tan\;66^\circ$ 1.2.23 $\sec\;77^\circ$ For Exercises 24-28, write the given number as a trigonometric function of an acute angle greater than $45^\circ $. 1.2.24 $\sin\;1^\circ$ 1.2.25 $\cos\;13^\circ$ 1.2.26 $\tan\;26^\circ$ 1.2.27 $\cot\;10^\circ$ 1.2.28 $\csc\;43^\circ$ 1.2.29 In Example 1.7 we found the values of all six trigonometric functions of $60^\circ $ and $30^\circ $. (a) Does $\;\sin\;30^\circ ~+~ \sin\;30^\circ ~=~ \sin\;60^\circ$? (b) Does $\;\cos\;30^\circ ~+~ \cos\;30^\circ ~=~ \cos\;60^\circ$? (c) Does $\;\tan\;30^\circ ~+~ \tan\;30^\circ ~=~ \tan\;60^\circ$? (d) Does $\;2\,\sin\;30^\circ\,\cos\;30^\circ ~=~ \sin\;60^\circ$? 1.2.30 For an acute angle $A $, can $\sin\;A $ be larger than $1$? Explain your answer. 1.2.31 For an acute angle $A $, can $\cos\;A $ be larger than $1$? Explain your answer. 1.2.32 For an acute angle $A $, can $\sin\;A $ be larger than $\tan\;A$? Explain your answer. 1.2.33 If $A $ and $B $ are acute angles and $A < B $, explain why $\sin\;A < \sin\;B $. 1.2.34 If $A $ and $B $ are acute angles and $A < B $, explain why $\cos\;A > \cos\;B $. 1.2.35 Prove the Cofunction Theorem (Theorem 1.2). ( Hint: Draw a right triangle and label the angles and sides .) 1.2.36 Use Example 1.10 to find all six trigonometric functions of $15^\circ $. 1.2.37 In Figure 1.2.4, $\overline{CB} $ is a diameter of a circle with a radius of $2 $ cm and center $O $, $\triangle\,ABC $ is a right triangle, and $\overline{CD}$ has length $\sqrt{3} $ cm.

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(a) Find $\;\sin\;A $. ( Hint: Use Thales' Theorem. ) (b) Find the length of $\;\overline{AC} $. (c) Find the length of $\;\overline{AD} $. (d) Figure 1.2.4 is drawn to scale. Use a protractor to measure the angle $A $, then use your calculator to find\\the sine of that angle. Is the calculator result close to your answer from part(a)? Note: Make sure that your calculator is in degree mode. 1.2.38 In Exercise 37, verify that the area of $\triangle\,ABC $ equals $\frac{1}{2} AB \cdot CD $. Why does this make sense? 1.2.39 In Exercise 37, verify that the area of $\triangle\,ABC $ equals $\frac{1}{2} AB \cdot AC \; \sin\;A $. 1.2.40 In Exercise 37, verify that the area of $\triangle\,ABC $ equals $\frac{1}{2} (BC)^2 \;\cot\;A $.

1.3 Exercises

1.3.1 From a position $150 $ ft above the ground, an observer in a building measures angles of depression of $12^\circ $ and $34^\circ $ to the top and bottom, respectively, of a smaller building, as in the picture on the right. Use this to find the height $h $ of the smaller building.

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1.3.2 Generalize Example 1.12: A person standing $a $ ft from the base of a mountain measures an angle of elevation $\alpha $ from the ground to the top of the mountain. The person then walks $b $ ft straight back and measures an angle of elevation $\beta $ to the top of the mountain, as in the picture on the right. Assuming the ground is level, find a formula for the height $h $ of the mountain in terms of $a $, $b $, $\alpha $, and $\beta $.

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1.3.3 As the angle of elevation from the top of a tower to the sun decreases from $64^\circ $ to $49^\circ $ during the day, the length of the shadow of the tower increases by $92 $ ft along the ground. Assuming the ground is level, find the height of the tower. 1.3.4 Two banks of a river are parallel, and the distance between two points $A $ and $B $ along one bank is $500 $ ft. For a point $C $ on the opposite bank, $\angle\,BAC = 56^\circ $ and $\angle\,ABC = 41^\circ $, as in the picture on the right. What is the width $w $ of the river? ( Hint: Divide $\overline{AB} $ into two pieces. )

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1.3.5 A tower on one side of a river is directly east and north of points $A $ and $B $, respectively, on the other side of the river. The top of the tower has angles of elevation $\alpha$ and $\beta $ from $A $ and $B $, respectively, as in the picture on the right. Let $d $ be the distance between $A $ and $B $. Assuming that both sides of the river are at the same elevation, show that the height $h $ of the tower is \[\nonumber h ~=~ \frac{d}{\sqrt{(\cot\,\alpha)^2 ~+~ (\cot\,\beta)^2}} ~. \]

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1.3.6 The equatorial parallax of the moon has been observed to be approximately $57' $. Taking the radius of the earth to be $3956.6 $ miles, estimate the distance from the center of the earth to the moon. ( Hint: See Example 1.15.)

1.3.7 An observer on earth measures an angle of $31'\;7'' $ from one visible edge of the moon to the other (opposite) edge. Use this to estimate the radius of the moon. ( Hint: Use Exercise 6 and see Example 1.16. ) 1.3.8 A ball bearing sits between two metal grooves, with the top groove having an angle of $120^\circ $ and the bottom groove having an angle of $90^\circ $, as in the picture on the right. What must the diameter of the ball bearing be for the distance between the vertexes of the grooves to be half an inch? You may assume that the top vertex is directly above the bottom vertex.

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1.3.9 The machine tool diagram on the right shows a symmetric worm thread , in which a circular roller of diameter $1.5 $ inches sits. Find the amount $d $ that the top of the roller rises above the top of the thread, given the information in the diagram. ( Hint: Extend the slanted sides of the thread until they meet at a point. )

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1.3.10 Repeat Exercise 9 using $1.8 $ inches as the distance across the top of the worm thread. 1.3.11 In Exercise 9, what would the distance across the top of the worm thread have to be to make $d $ equal to $0 $ inches? 1.3.12 For $0^\circ < \theta < 90^\circ $ in the slider-crank mechanism in Example 1.18, show that \[\nonumber c ~=~ \frac{\sqrt{b^2 ~-~ r^2 \;(\sin\,\theta)^2}}{\cos\;\theta} ~\qquad\text{and}\qquad~ a ~=~ r\;\sin\;\theta ~+~ \sqrt{b^2 ~-~ r^2 \;(\sin\,\theta)^2}~\tan\;\theta ~. \] (Hint: In Figure 1.3.2 draw line segments from $B $ perpendicular to $\overline{OA} $ and $\overline{AC} $.) 1.3.13 The machine tool diagram on the right shows a symmetric die punch . In this view, the rounded tip is part of a circle of radius $r $, and the slanted sides are tangent to that circle and form an angle of $54^\circ $. The top and bottom sides of the die punch are horizontal. Use the information in the diagram to find the radius $r $.

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1.3.14 In the figure on the right, $\angle\,BAC = \theta $ and $BC = a $. Use this to find $AB $, $AC $, $AD $, $DC $, $CE $, and $DE $ in terms of $\theta $ and $a $.(Hint: What is the angle $\angle\,ACD\,$?)

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For Exercises 15-23, solve the right triangle in Figure 1.3.4 using the given information.

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1.3.15 $a = 5 $, $b = 12$ 1.3.16 $c = 6 $, $B = 35^\circ$ 1.3.17 $b = 2 $, $A = 8^\circ$ 1.3.18 $a = 2 $, $c = 7$ 1.3.19 $a = 3 $, $A = 26^\circ$ 1.3.20 $b = 1 $, $c = 2$

1.3.21 $b = 3 $, $B = 26^\circ$ 1.3.22 $a = 2 $, $B = 8^\circ$ 1.3.23 $c = 2 $, $A = 45^\circ$ 1.3.24 In Example 1.10 in Section 1.2, we found the exact values of all six trigonometric functions of $75^\circ $. For example, we showed that $\cot\;75^\circ = \frac{\sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}} $. So since $\tan\;15^\circ = \cot\;75^\circ $ by the Cofunction Theorem, this means that $\tan\;15^\circ = \frac{\sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}} $. We will now describe another method for finding the exact values of the trigonometric functions of $15^\circ $. In fact, it can be used to find the exact values for the trigonometric functions of $\frac{\theta}{2} $ when those for $\theta $ are known, for any $0^\circ < \theta < 90^\circ $. The method is illustrated in Figure 1.3.5 and is described below.

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Draw a semicircle of radius $1 $ centered at a point $O $ on a horizontal line. Let $P $ be the point on the semicircle such that $\overline{OP} $ makes an angle of $60^\circ $ with the horizontal line, as in Figure 1.3.5. Draw a line straight down from $P $ to the horizontal line at the point $Q $. Now create a second semicircle as follows: Let $A $ be the left endpoint of the first semicircle, then draw a new semicircle centered at $A $ with radius equal to $AP $. Then create a third semicircle in the same way: Let $B $ be the left endpoint of the second semicircle, then draw a new semicircle centered at $B $ with radius equal to $BP $.

This procedure can be continued indefinitely to create more semicircles. In general, it can be shown that the line segment from the center of the new semicircle to $P $ makes an angle with the horizontal line equal to half the angle from the previous semicircle's center to $P $. (a) Explain why $\angle\,PAQ=30^\circ $. ( Hint: What is the supplement of $60^\circ$?) (b) Explain why $\angle\,PBQ=15^\circ $ and $\angle\,PCQ=7.5^\circ $. (c) Use Figure 1.3.5 to find the exact values of $\sin\;15^\circ $, $\cos\;15^\circ $, and $\tan\;15^\circ $. (Hint: To start, you will need to use $\angle\,POQ = 60^\circ$ and $OP = 1 $ to find the exact lengths of $\overline{PQ} $ and $\overline{OQ} $.) (d) Use Figure 1.3.5 to calculate the exact value of $\tan\;7.5^\circ $. (e) Use the same method but with an initial angle of $\angle\,POQ = 45^\circ $ to find the exact values of $\sin\;22.5^\circ $, $\cos\;22.5^\circ $, and $\tan\;22.5^\circ $. 1.3.25 A manufacturer needs to place ten identical ball bearings against the inner side of a circular container such that each ball bearing touches two other ball bearings, as in the picture on the right. The (inner) radius of the container is $4 $ cm.

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(a) Find the common radius $r $ of the ball bearings. (b) The manufacturer needs to place a circular ring\\inside the container. What is the largest possible (outer) radius of the ring such that it is not on top\\of the ball bearings and its base is level with the\\base of the container? 1.3.26 A circle of radius $1 $ is inscribed inside a polygon with eight sides of equal length, called a regular octagon . That is, each of the eight sides is tangent to the circle, as in the picture on the right.

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(a) Calculate the area of the octagon. (b) If you were to increase the number of sides of the\\polygon, would the area inside it increase or decrease? What number would the area approach, if any? Explain. (c) Inscribe a regular octagon inside the same circle. That is, draw a regular octagon such that each of its eight vertexes touches the circle. Calculate the area of this octagon. 1.3.27 The picture on the right shows a cube whose sides are of length $a > 0 $.

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(a) Find the length of the diagonal line segment $\overline{AB} $. (b) Find the angle $\theta $ that $\overline{AB} $ makes with the base of the cube. 1.3.28 In Figure 1.3.6, suppose that $\alpha $, $\beta $, and $AD $ are known. Show that:

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(a) $BC ~=~ \dfrac{AD}{\cot\;\alpha - \cot\;\beta}$ (b) $AC ~=~ \dfrac{AD\;\cdot\;\tan\;\beta}{\tan\;\beta - \tan\;\alpha}$ (c) $BD ~=~ \dfrac{AD\;\cdot\;\sin\;\alpha}{\sin\;(\beta - \alpha)}$ (Hint: What is the measure of the angle $\angle\,ABD\;$?) 1.3.29 Persons A and B are at the beach, their eyes are $5 $ ft and $6 $ ft, respectively, above sea level. How many miles farther out is Person B's horizon than Person A's? (Note: $1 $ mile = $5280$ ft)

1.4 Exercises

For Exercises 1-10, state in which quadrant or on which axis the given angle lies. 1.4.1 $127^\circ$ 1.4.2 $-127^\circ$ 1.4.3 $313^\circ$ 1.4.4 $-313^\circ$ 1.4.5 $-90^\circ$ 1.4.6 $621^\circ$ 1.4.7 $230^\circ$ 1.4.8 $2009^\circ$ 1.4.9 $1079^\circ$ 1.4.10 $-514^\circ$ 1.4.11 In which quadrant(s) do sine and cosine have the same sign? 1.4.12 In which quadrant(s) do sine and cosine have the opposite sign? 1.4.13 In which quadrant(s) do sine and tangent have the same sign? 1.4.14 In which quadrant(s) do sine and tangent have the opposite sign? 1.4.15 In which quadrant(s) do cosine and tangent have the same sign? 1.4.16 In which quadrant(s) do cosine and tangent have the opposite sign? For Exercises 17-21, find the reference angle for the given angle. 1.4.17 $317^\circ$ 1.4.18 $63^\circ$ 1.4.19 $-126^\circ$ 1.4.20 $696^\circ$ 1.4.21 $275^\circ$ For Exercises 22-26, find the exact values of $\sin\;\theta $ and $\tan\;\theta $ when $\cos\;\theta $ has the indicated value. 1.4.22 $\cos\;\theta = \frac{1}{2}$ 1.4.23 $\cos\;\theta = -\frac{1}{2}$ 1.4.24 $\cos\;\theta = 0$ 1.4.25 $\cos\;\theta = \frac{2}{5}$ 1.4.26 $\cos\;\theta = 1$ For Exercises 27-31, find the exact values of $\cos\;\theta $ and $\tan\;\theta $ when $\sin\;\theta $ has the indicated value. 1.4.27 $\sin\;\theta = \frac{1}{2}$ 1.4.28 $\sin\;\theta = -\frac{1}{2}$ 1.4.29 $\sin\;\theta = 0$ 1.4.30 $\sin\;\theta = -\frac{2}{3}$ 1.4.31 $\sin\;\theta = 1$ For Exercises 32-36, find the exact values of $\sin\;\theta $ and $\cos\;\theta $ when $\tan\;\theta $ has the indicated value. 1.4.32 $\tan\;\theta = \frac{1}{2}$ 1.4.33 $\tan\;\theta = -\frac{1}{2}$ 1.4.34 $\tan\;\theta = 0$ 1.4.35 $\tan\;\theta = \frac{5}{12}$ 1.4.36 $\tan\;\theta = 1$ For Exercises 37-40, use Table 1.3 to answer the following questions. 1.4.37 Does $\sin\;180^\circ ~+~ \sin\;45^\circ ~=~ \sin\;225^\circ\;$? 1.4.38 Does $\tan\;300^\circ ~-~ \tan\;30^\circ ~=~ \tan\;270^\circ\;$? 1.4.39 Does $\cos\;180^\circ ~-~ \cos\;60^\circ ~=~ \cos\;120^\circ\;$? 1.4.40 Does $\cos\;240^\circ ~=~ (\cos\;120^\circ)^2 ~-~ (\sin\;120^\circ)^2 \;$? 1.4.41 Expand Table 1.3 to include all integer multiples of $15^\circ $. See Example 1.10 in Section 1.2.

1.5 Exercises

1.5.1 Let $\theta = 32^\circ $. Find the angle between $0^\circ $ and $360^\circ $ which is the (a) reflection of $\theta $ around the $x$-axis (b) reflection of $\theta $ around the $y$-axis (c) reflection of $\theta $ around the origin 1.5.2 Repeat Exercise 1 with $\theta = 248^\circ $. 1.5.3 Repeat Exercise 1 with $\theta = -248^\circ $. 1.5.4 We proved Equations 1.4-1.6 for any angle $\theta $ in QI. Mimic that proof to show that the formulas hold for $\theta $ in QII. 1.5.5 Verify Equations 1.4-1.6 for $\theta $ on the coordinate axes, i.e. for $\theta = 0^\circ $, $90^\circ $, $180^\circ $, $270^\circ $. 1.5.6 In Example 1.26 we used the formulas involving $\theta + 90^\circ $ to prove that the slopes of perpendicular lines are negative reciprocals. Show that this result can also be proved using the formulas involving $\theta - 90^\circ $. ( Hint: Only the last paragraph in that example needs to be modified. ) For Exercises 7-14, find all angles $0^\circ \le \theta < 360^\circ $ which satisfy the given equation: 1.5.7 $\sin\;\theta = 0.4226$ 1.5.8 $\sin\;\theta = 0.1909$ 1.5.9 $\cos\;\theta = 0.4226$ 1.5.10 $\sin\;\theta = 0$ 1.5.11 $\tan\;\theta = 0.7813$ 1.5.12 $\sin\;\theta = -0.6294$ 1.5.13 $\cos\;\theta = -0.9816$ 1.5.14 $\tan\;\theta = -9.514$ 1.5.15 In our proof of the Pythagorean Theorem in Section 1.2, we claimed that in a right triangle $\triangle\,ABC $ it was possible to draw a line segment $\overline{CD} $ from the right angle vertex $C $ to a point $D $ on the hypotenuse $\overline{AB} $ such that $\overline{CD} \perp \overline{AB} $. Use the picture below to prove that claim. ( Hint: Notice how $\triangle\,ABC $ is placed on the $xy$- coordinate plane. What is the slope of the hypotenuse? What would be the slope of a line perpendicular to it? ) Also, find the $(x,y) $ coordinates of the point $D $ in terms of $a $ and $b $.

$alt$

1.5.16 It can be proved without using trigonometric functions that the slopes of perpendicular lines are negative reciprocals. Let $y = m_1{}x+b_1 $ and $y = m_2{}x+b_2 $ be perpendicular lines (with nonzero slopes), as in the picture below. Use the picture to show that $m_2 = -\frac{1}{m_1} $.( Hint: Think of similar triangles and the definition of slope .)

$alt$

1.5.17 Prove Equations 1.19-1.21 by using Equations 1.10-1.12 and 1.13-1.15.

Math Article

Trigonometry

Trigonometry is one of the important branches in the history of mathematics that deals with the study of the relationship between the sides and angles of a right-angled triangle. This concept is given by the Greek mathematician Hipparchus. In this article, we are going to learn the basics of trigonometry such as trigonometry functions, ratios, trigonometry table, formulas and many solved examples.

What is Trigonometry?

Trigonometry is one of the most important branches in mathematics that finds huge application in diverse fields. The branch called “Trigonometry” basically deals with the study of the relationship between the sides and angles of the right-angle triangle. Hence, it helps to find the missing or unknown angles or sides of a right triangle using the trigonometric formulas, functions or trigonometric identities. In trigonometry, the angles can be either measured in degrees or radians. Some of the most commonly used trigonometric angles for calculations are 0°, 30°, 45°, 60° and 90°.

Trigonometry is further classified into two sub-branches. The two different types of trigonometry are:

Plane Trigonometry
Spherical Trigonometry

In this article, let us discuss the six important trigonometric functions, ratios, trigonometry table, formulas and identities which helps to find the missing angles or sides of a right triangle.

Trigonometry Ratios-Sine, Cosine, Tangent

The trigonometric ratios of a triangle are also called the trigonometric functions. Sine, cosine, and tangent are 3 important trigonometric functions and are abbreviated as sin, cos and tan. Let us see how are these ratios or functions, evaluated in case of a right-angled triangle.

Consider a right-angled triangle, where the longest side is called the hypotenuse, and the sides opposite to the hypotenuse are referred to as the adjacent and opposite sides.

Six Important Trigonometric Functions

The six important trigonometric functions (trigonometric ratios) are calculated using the below formulas and considering the above figure. It is necessary to get knowledge about the sides of the right triangle because it defines the set of important trigonometric functions.

Even and Odd Trigonometric Functions

The trigonometric function can be described as being even or odd.

Odd trigonometric functions: A trigonometric function is said to be an odd function if f(-x) = -f(x) and symmetric with respect to the origin.

Even trigonometric functions: A trigonometric function is said to be an even function, if f(-x) = f(x) and symmetric to the y-axis.

We know that

Sin (-x) = – Sin x
Cos (-x) = Cos x
Tan (-x) = -Tan x
Csc (-x) = – Csc x
Sec (-x) = Sec x
Cot (-x) = -Cot x

Therefore, cosine and secant are the even trigonometric functions, whereas sine, tangent, cosecant and cotangent are the odd trigonometric functions. If we know the even and odd trigonometric functions, it helps us to simplify the trigonometric expression when the variable inside the trigonometric function is negative.

Trigonometry Angles

The trigonometry angles which are commonly used in trigonometry problems are 0 ° , 30 ° , 45 ° , 60 ° and 90 °. The trigonometric ratios such as sine, cosine and tangent of these angles are easy to memorize. We will also show the table where all the ratios and their respective angle’s values are mentioned. To find these angles we have to draw a right-angled triangle, in which one of the acute angles will be the corresponding trigonometry angle. These angles will be defined with respect to the ratio associated with it.

For example, in a right-angled triangle,

Sin θ = Perpendicular/Hypotenuse

or θ = sin -1 (P/H)

θ = cos -1 (Base/Hypotenuse)

θ = tan -1 (Perpendicular/Base)

Trigonometry Table

Check the table for common angles which are used to solve many trigonometric problems involving trigonometric ratios.

In the same way, we can find the trigonometric ratio values for angles beyond 90 degrees, such as 180°, 270° and 360°.

Unit Circle

The concept of unit circle helps us to measure the angles of cos, sin and tan directly since the centre of the circle is located at the origin and radius is 1. Consider theta be an angle then,

Suppose the length of the perpendicular is y and of base is x. The length of the hypotenuse is equal to the radius of the unit circle, which is 1. Therefore, we can write the trigonometry ratios as;

List of Trigonometry Formulas

The Trigonometric formulas or Identities are the equations which are true in the case of Right-Angled Triangles. Some of the special trigonometric identities are given below –

Pythagorean Identities
sin²θ + cos²θ = 1
tan 2 θ + 1 = sec 2 θ
cot 2 θ + 1 = cosec 2 θ
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ – sin²θ
tan 2θ = 2 tan θ / (1 – tan²θ)
cot 2θ = (cot²θ – 1) / 2 cot θ
Sum and Difference identities-

For angles u and v, we have the following relationships:

sin(u + v) = sin(u)cos(v) + cos(u)sin(v)
cos(u + v) = cos(u)cos(v) – sin(u)sin(v)
$\begin{array}{l}tan(u+v) = \frac{tan(u)\ +\ tan(v)}{1-tan(u)\ tan(v)}\end{array} $
sin(u – v) = sin(u)cos(v) – cos(u)sin(v)
cos(u – v) = cos(u)cos(v) + sin(u)sin(v)
$\begin{array}{l}tan(u-v) = \frac{tan(u)\ -\ tan(v)}{1+tan(u)\ tan(v)}\end{array} $
If A, B and C are angles and a, b and c are the sides of a triangle, then,
a/sinA = b/sinB = c/sinC

Cosine Laws

c 2 = a 2 + b 2 – 2ab cos C
a 2 = b 2 + c 2 – 2bc cos A
b 2 = a 2 + c 2 – 2ac cos B

Trigonometry Identities

The three important trigonometric identities are:

tan² θ + 1 = sec² θ
cot ² θ + 1 = cosec² θ

Euler’s Formula for trigonometry

As per the euler’s formula,

e ix = cos x + i sin x

Where x is the angle and i is the imaginary number.

$\begin{array}{l}\sin x=\frac{e^{i x}-e^{-i x}}{2 i}\\ \cos x=\frac{e^{i x}+e^{-i x}}{2}\\ \tan x=\frac{\left(e^{i x}-e^{-i x}\right)}{i\left(e^{i x}+e^{-i x}\right)}\end{array} $

Trigonometry Basics

The three basic functions in trigonometry are sine, cosine and tangent. Based on these three functions the other three functions that are cotangent, secant and cosecant are derived.

All the trigonometrical concepts are based on these functions. Hence, to understand trigonometry further we need to learn these functions and their respective formulas at first.

If θ is the angle in a right-angled triangle, then

Cos θ = Base/Hypotenuse

Tan θ = Perpendicular/Base

Perpendicular is the side opposite to the angle θ.

The base is the adjacent side to the angle θ.

The hypotenuse is the side opposite to the right angle

The other three functions i.e. cot, sec and cosec depend on tan, cos and sin respectively, such as:

Cot θ = 1/tan θ

Sec θ = 1/cos θ

Cosec θ = 1/sin θ

Cot θ = Base/Perpendicular

Sec θ = Hypotenuse/Base

Cosec θ = Hypotenuse/Perpendicular

Trigonometry Examples

There are many real-life examples where trigonometry is used broadly.

If we have been given with height of the building and the angle formed when an object is seen from the top of the building, then the distance between object and bottom of the building can be determined by using the tangent function, such as tan of angle is equal to the ratio of the height of the building and the distance. Let us say the angle is ∝, then

Tan ∝ = Height/Distance between object & building

Distance = Height/Tan ∝

Let us assume that height is 20m and the angle formed is 45 degrees, then

Distance = 20/Tan 45°

Since, tan 45° = 1

So, Distance = 20 m

Applications of Trigonometry

Its applications are in various fields like oceanography, seismology, meteorology, physical sciences, astronomy, acoustics, navigation, electronics, etc.
It is also helpful to measure the height of the mountain, find the distance of long rivers, etc.

Video Lesson on Applications of Trigonometry

Trigonometry Problems and Solutions

Example 1 : Two friends, Rakesh and Vishal started climbing a pyramid-shaped hill. Rakesh climbs 315 m and finds that the angle of depression is 72.3 degrees from his starting point. How high is he from the ground?

Solution : Let m is the height above the ground.

To find: Value of m

To solve m, use the sine ratio.

Sin 72.3° = m/315

0.953 = m/315

m= 315 x 0.953

m=300.195 mtr

The man is 300.195 mtr above the ground.

Example 2: A man is observing a pole of height 55 foot. According to his measurement, pole cast a 23 feet long shadow. Can you help him to know the angle of elevation of the sun from the tip of shadow?

Let x be the angle of elevation of the sun, then

tan x = 55/23 = 2.391

x = tan -1 (2.391)

or x = 67.30 degrees

Trigonometry Questions

Practise these questions given here to get a deep knowledge of Trigonometry. Use the formulas and table given in this article wherever necessary.

Q.1: In △ABC, right-angled at B, AB=22 cm and BC=17 cm. Find:

(a) sin A Cos B

(b) tan A tan B

Q.2: If 12cot θ= 15, then find sec θ.

Q.3: In Δ PQR, right-angled at Q, PR + QR = 30 cm and PQ = 10 cm. Determine the values of sin P, cos P and tan P.

Frequently Asked Questions on Trigonometry

What do you mean by trigonometry.

Trigonometry is one of the branches of mathematics which deals with the relationship between the sides of a triangle (right triangle) with its angles. There are 6 trigonometric functions for which the relation between sides and angles are defined. Learn more about trigonometry now by visiting BYJU’S.

What are the six basic Trigonometric Functions?

There are 6 trigonometric functions which are:

Sine function
Cosine function
Tan function
Sec function
Cot function
Cosec function

What is the formula for six trigonometry functions?

What is the primary function of trigonometry, who is the founder of trigonometry, what are the applications of trigonometry in real life.

One of the most important real-life applications of trigonometry is in the calculation of height and distance. Some of the sectors where the concepts of trigonometry are extensively used are aviation department, navigation, criminology, marine biology, etc. Learn more about the applications of trigonometry here.

Learn about Trigonometry in a simple manner with detailed information, along with step by step solutions to all questions, only at BYJU’S. Download the app to get personalised videos.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

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Trigonometry Worksheets

Free worksheets with answer keys.

Enjoy these free sheets. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

(This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle)

Law of Sines
Ambiguous Case of the Law of Sines
Law Of Cosines
Sine, Cosine, Tangent, to Find Side Length
Sine, Cosine, Tangent Chart
Inverse Trig Functions
Real World Applications of SOHCATOA
Mixed Review
Vector Worksheet
Unit Circle Worksheet
Graphing Sine and Cosine Worksheet

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

Popular pages @ mathwarehouse.com.

$Math = Love$

27 Hands-On Trigonometry Activities

By: Author Sarah Carter

Posted on Published: September 22, 2022 - Last updated: December 7, 2023

Categories Trigonometry

This blog post contains Amazon affiliate links. As an Amazon Associate, I earn a small commission from qualifying purchases.

Want to check out my favorite Amazon items for the classroom? Check out my Amazon favorites page !

Looking for hands-on trigonometry activities? Here’s a collection of 27 fun and engaging trig activities that I have created over the past decade while teaching trigonometry and pre-calculus to high school math students.

What is a Radian? Hands-On Activity

I created this hand-on exploration for my trigonometry students to explore the question “What is a Radian?”

Angle Spinner for Sketching Angles in Standard Position

This angle spinner for sketching angles in standard position is one of my favorite pages in our trigonometry interactive notebooks.

Coterminal Angles Card Sort Activity

I created this coterminal angles card sort activity for my trigonometry students to complete and glue in their interactive notebooks. Students have to sort the angle cards into groups that are coterminal with one another.

Trig Mini Poster Project

This Trig Mini Poster Project gives students lots of practice with graphing trigonometric functions.

Evaluating Trig Functions Square Puzzle Activity

I used this evaluating trig functions square puzzle recently with my trigonometry students. We glued our completed puzzles in our interactive notebooks. This puzzle was created and shared by NCTM Illuminations under the name “Trigonometry Square.”

Unit Circle Paper Plate Activity

Make the unit circle memorable with this unit circle paper plate activity.

Radians and Degrees War

Help students practice converting between radians and degrees with a fun game of radians and degrees war.

Clinometer Activity and Foldable

I love this clinometer activity for giving my trigonometry students extra practice finding missing angles and sides of right triangles. Plus, it answers the question of “When will we ever use this?”

Leap Frog Game for Evaluating Trig Functions

This Leap Frog style game gives students much-needed practice evaluating trig functions using the unit circle.

Unit Circle Bingo Game

I decided to create a Unit Circle Bingo Game to give my Pre-Calculus students some much-needed practice with evaluating trig functions using the unit circle. Technically, this is more of a game of connect 4 than bingo, but bingo just sounds more fun.

Unit Circle Project Examples

After years of seeing other teachers share pictures of the unit circle projects their students created, I decided to finally take the plunge. I assigned my trig students the task of creating a visual representation (2d or 3d) of the unit circle in lieu of a semester test in trigonometry. Check out the results!

Trig Identities Challenge Activity

This Trig Identities Challenge Activity will engage students and give them tons of practice rewriting trig expressions using trig identities.

Trig Identities Matching Activity

Introduce the concept of trig identities with this hands-on trig identities matching activity.

Trig Ratios Puzzle

Check out this trig ratios puzzle which I discovered on Open Middle and transformed into a printable puzzle for my students to tackle.

Double Angle Identities Joke Worksheet

Use double angle identities to rewrite each expression and answer the joke of “Why was the mathematician late for work?”

Unit Circle Magnets

I created these unit circle magnets to use with my dry erase magnetic unit circle from EAI Education. I plan to use the magnets both as a way to have a large unit circle on the wall of my classroom and for various activities throughout the year.

Radian Arts and Crafts Activity

This radian arts and crafts activity is designed to help students conceptually understand what a radian is. As a bonus, it creates a beautiful, mathematical art display for your classroom.

Trigonometry Puzzle

A couple of years ago, I created this trigonometry puzzle to keep my students busy when we unexpectedly finished our lesson with more time to spare than I had planned. This trigonometry puzzle is really the Twisted Wires Puzzle by Cliff Pickover in disguise.

Deriving the Unit Circle Foldable

Several years ago, I created this deriving the unit circle foldable for my trigonometry students to work through and glue in their interactive notebooks.

One or Negative One Trig Identities Worksheet

I really like the simplicity of this One or Negative One? Trig Identities Task that I found in the 8th Edition of Demana and Waits’ Pre-Calculus Textbook (Section 5.1). That’s why I decided to turn it into a worksheet!

Odd One Out Coterminal Angles Activity

coterminal angles activity with odd one out structure and paper plate angle spinners

I created this odd one out activity to give my pre-calculus students extra practice identifying coterminal angles. In this activity, students were placed in groups of 4. Each group was given a set of four angle cards. They must work together to determine the Odd One Out.

The Great Quadrant Guessing Game

great quadrant guessing game for trigonometry

I created The Great Quadrant Guessing Game several years ago to give my Pre-Calculus students some extra practice determining which quadrant of the coordinate plane was represented by two trigonometric facts.

Trigonometry Calculator Skills Pop Quiz

Several years ago, I created this trigonometry calculator skills pop quiz to give to my Pre-Calculus students before we learn to use our calculators for evaluating trig functions. Students tend to think that these calculator-based problems are super easy and that they don’t need to be taught how to type trig functions into their calculator.

Printable Radian Sectors

Several summers ago, I created a set of printable radian sectors to illustrate what different amounts of radians looked like when graphed in standard form.

Quadrants Unlocked Activity

quadrants unlocked trigonometry activity

I created this fun quadrants unlocked activity to give my pre-calculus students some much needed practice determining the signs of trig functions in various quadrants.

Parent Graphs of Trig Functions Clothespin Matching Activity

This parent graphs of trig functions clothespin matching activity has been in the making for multiple years. Students must match each statement with the appropriate parent functions.

Fill in the Blank Unit Circle Chart

I created this fill-in-the-blank unit circle chart for my pre-calculus classes to use as they practice constructing the unit circle from memory.

Word Ladder Puzzles for Introducing Trig Identities

Sarah Newton shares a fun hook for introducing trig identities – a word ladder puzzle. I have tried this activity with my students, and they had so much fun!

More Activities for Teaching Trigonometry

paper plate angle spinner for trigonometry

Free Printable trigonometry Worksheets for 9th Grade

Trigonometry-focused math worksheets for Grade 9 students, created to help educators discover effective teaching resources. Download and print these free materials to enhance your lessons.

Explore trigonometry Worksheets by Grades

Explore trigonometry worksheets for grade 9 by topic, explore other subject worksheets for grade 9.

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Explore printable trigonometry worksheets for 9th Grade

Trigonometry worksheets for Grade 9 are an essential resource for math teachers looking to engage their students in the world of angles, triangles, and trigonometric functions. These worksheets provide a comprehensive and structured approach to learning trigonometry, ensuring that students grasp the fundamental concepts and develop a strong foundation in this crucial area of mathematics. With a variety of topics covered, such as sine, cosine, tangent, and their applications in real-world problems, these worksheets cater to the diverse needs and learning styles of Grade 9 students. Teachers can easily incorporate these worksheets into their lesson plans, using them as in-class activities, homework assignments, or even as assessment tools. By utilizing trigonometry worksheets for Grade 9, math educators can effectively enhance their students' understanding and mastery of this vital subject.

Quizizz is an innovative platform that offers a wide range of resources for teachers, including trigonometry worksheets for Grade 9, math quizzes, and interactive games. This platform allows educators to create engaging and personalized learning experiences for their students, making it an ideal tool for teaching trigonometry and other mathematical concepts. With Quizizz, teachers can easily track their students' progress and identify areas where they may need additional support or practice. In addition to trigonometry worksheets for Grade 9, the platform also offers resources for other math topics, ensuring that teachers have access to a comprehensive suite of materials to support their instruction. By incorporating Quizizz into their teaching strategies, educators can foster a dynamic and effective learning environment that promotes the success of their Grade 9 math students.

Easy Peasy All-in-One High School

An extension of the easy peasy all-in-one homeschool, trigonometry and precalculus, found a problem check here ..

Credits: 1

Prerequisite: Geometry, Algebra 2

Recommended : 11th, 12th

Test Prep: CLEP exam

Course Description: This course offers a quarter (45 lessons) of trigonometry followed by a semester of precalculus. In trigonometry, students will not only learn the basic trigonometric functions and how to apply them to solve real-life problems, but will explore a number of topics from trigonometry including: triangle properties, radian, identities, solving complex equations, inverse functions, vectors, and the polar coordinate system. Students will study and write proofs and learn about the Law of Sines and Cosines and their applications. Functions are the focus of the pre-calculus including: linear, polynomial, rational, logarithmic, and exponential functions. Students will graph functions, combine functions, interpret roots and find the maxima and minima of functions. Students will apply their knowledge to model and to solve real-world applications.

Students will be using Khan Academy and Yay Math! (trigonometry) and OpenTextbook (pre-calculus).

You will need a calculator. You can use one online . On Lesson 7 you will receive some instruction on how to use a calculator to solve trigonometric functions. You can learn about other calculator options (handheld, mobile, etc.) on this page under “Tips/Suggestions” .

Print out some graph paper to have on hand. Regular Polar (for unit 6)

You may give yourself half credit on a question done correctly with one small “typo” type error.

There are quizzes and assignments, but the finals will have their questions all come from previous work in the course.

Work in a single spiral notebook, or keep all loose-leaf paper in a single binder for the course. Hold onto your written notes and exercises. Use them to study and as a record of what you’ve done.

Lesson 1(*)

If you didn’t get here through My EP. Please make an account to track your progress in the course. It’s free to create an account .
The first quarter of this course is on trigonometry, though it will come back in the second part of the course, which is pre-calculus.
When you do the reading, make sure you do the practice questions.
When you watch a video, grab your notebook and do the problems along with him.
There will be quizzes and tests.
(*)Print out your first quarter grading sheet , or use the excel version .
Learn the introduction to the trigonometric ratios.
Do the four activities . Continue through the whole thing.
Take the quiz . You’ll always need to pay attention to your score.
Record your score out of 5.
Learn about solving for a side in a right triangle using the trigonometric ratios : Right triangles & trigonometry.
Complete the lesson through the practice exercise. I don’t know what it looks like on a mobile device, but on a computer, there is a menu on the left that shows you all the parts to a lesson. Sometimes we’ll split up a long lesson, but mostly, we’ll complete a whole lesson.
Learn about solving for an angle in a right triangle using the trigonometric ratios.
Complete the lesson .
Choose a way to mark questions you got wrong or lesson you had trouble with. On a review day, go back and try again. You can make a list in the back of your notebook, keep a list on your computer, highlight problems with red in your notebook.
Learn about sine and cosine of complementary angles.
Learn about modeling with right triangles.
Take the quiz .
Learn about the reciprocal trigonometric ratios.
You can take some time to review. Your test should take around 20 minutes.
Take the unit test .
Record your score out of 9.

Lesson 9(*)

We’re going to switch it up for a few days.
(*)Print the worksheet and follow along with the lesson (or do a good job writing down what he’s doing.
Watch the video .

Lesson 10(*)

Take the quiz . You are NOT emailing to anyone. Record your score out of 10.
Take another quiz .
Record your score out of 12.

Lesson 11 (Materials: scientific calculator)

He’s going to talk about using a calculator.
Get out a notebook and calculator and work along with him.
Learn about unit circles .
Complete the lesson.
Learn about radians .
Learn about the Pythagorean theorem .
Learn about special trigonometric values in the first quadrant .
Make sure you are keeping track of lessons you need to practice and/or individual questions you got wrong that you should retry.
Learn about trigonometric values on the unit circle .
Record your score out of 7.
Learn about the graphs of sin(x), cos(x), and tan(x).
There is no exercise. Take notes. Draw graphs.
Learn about amplitude, midline, and period .
Learn about graphing sinusoidal functions .
Learn about transforming sinusoidal graphs .
Go through the example of transforming sinusoidal graphs .
Try the questions .
Learn about sinusoidal models .
Make sure you are keeping track of what lessons gave you trouble or individual problems you got wrong and could try again on a practice day.
Record your score out of 6.
Record your score out of 4. (room for extra credit)
Record your score out of 10. (room for extra credit)
Learn about the law of sines .
Learn about the law of cosines .
Learn about solving general triangles .
Take the unit test . (10-20 minutes)
Learn about inverse trigonometric functions .
Complete the lesson. This is a longer one. Keep moving.
Learn about sinusoidal equations .
Learn about angle addition identities .
Learn about using trigonometric identities .
Try the challenging trigonometric questions !
How many can you get?
Review. This unit starts with inverse trigonometric functions.
Take the unit test . (20 minutes)
Take the challenge . (30-45 minutes)
Record your score out of 30.

Lesson 34(**)

Let’s break it up with some Yay Math!
This is review. You should be pretty good at this. By the time we go through it again now, you’ll really have this.
(*)Print the worksheet or get out your notebook and copy down what he’s doing.
Watch the video on the trig functions .
Record your score out of 17.
(*)Print the worksheet or write down the problems as he does them.
Watch the video on trigonometric functions with right triangles .
Record your score out of 4 points.

Lesson 35(*)

Watch the lesson on trig function with general angles .
Record your score out of 29 points.

Lesson 36(**)

(*)Print the worksheet and work along with the video or write down the problems as he does them and work along.
Watch the lesson on the law of sines . You can stop when he finishes the problem around 20:20.
(*)Print the worksheet or write the problems down as he does them.
Watch the video on the law of cosines .

Lesson 37(**)

(*)Print the worksheet and work along with the video or write down the problems as he does them.
Watch the lesson on circular functions and the unit circle.
Record your score out of 23.
(*)Print out the worksheet or use your notebook.
Watch the lesson on graphing sine and cosine .

Lesson 38(**)

Watch the lesson on graphing CSC and SEC .
Record your score out of 21.
Watch the lesson on graphing TAN and COT .

Lesson 39(**)

Watch the lesson on the inverse trig function.
Record your score out of 10.
Watch the rest of the lesson .

Lesson 40(*)

Inverse Trig Functions — record your score out of 9.
Graphing SIN and COS — record your score out of 15.
Graphing SEC and CSC — record your score out of 20.
Watch the lesson on trig identities.

Lesson 41(*)

Watch the lesson on trig identities .
Take the quiz.
Record your score out of 8.

Lesson 42(*)

Watch the lesson on verifying trig identities .

Lesson 43(*)

Watch the lesson on the sum and differences of angles .

Lesson 44(**)

Watch the lesson on the double angle formula .
Watch the lesson on half-angle formula .

Lesson 45(*)

Watch the lesson on solving trigonometric equations .
If you want to improve your score on the course challenge from lesson 33, you can take it again and change your score if you improved. 🙂

Compute your grade for the first quarter. Hold onto your written work for studying and as a record of what you did for this course.

Lesson 46(*)

(*)Print out your second quarter grading sheet or use the Excel version.
Today we begin a new course, Pre-Calculus. I’m going to be linking to each chapter of the online textbook individually for using online, but if you’d like to download the whole text now for offline use, you can use the link found here. It also has a link to the answers for the whole book and links to buy the book.
This course will start with some review of what you’ve learned already in algebra.
We will also be doing some SAT practice and review. Make sure you are still keeping track of what you need to practice.
Read this introduction from Saylor.org. “ This subunit launches the course by introducing an important basic concept of mathematics – the function. Functions are dependable – if you know the output for a particular input, you will always get that same output for that particular input. It’s an easy idea but it can be difficult to explain. Suppose you have a company building a particular toy car that has three wheels. You could write a function for wheels based on how many cars you produced. If you make one car, you need three wheels. If you make two cars, you need six wheels. If you make 300 cars, you need 900 wheels. The output depends upon the input and any one input will ALWAYS yield the same output.”
Read pages one and two . As you go, answer the “Try It Now” questions. (Answers are on page 12.)
Watch the video on function notation .
Evaluate the functions .
Read pages two through four in the online textbook . Do the “Try It Now” question 3 at the top of page 5. (Answers are on page 12.)
Read page five in the textbook. Do the TIN question 4. Answers are on page 12.
Watch the video on evaluating a function . Here’s an example to watch as well.
Practice evaluating expressions with function notation .
Watch this short video on the vertical line test .
Read through pages 5-7 in the online textbook. As before, always work through examples on paper.
Do the TIN question 5 on page 7. Answers are on page 12.
Let’s mix it up.
We’ll call these review days.
Go back and redo any questions you got wrong this week. Try them from scratch. You can also try a question from each lesson that wasn’t assigned. Make sure you can still do them without having the examples right in front of you and fresh in your mind.
Practice for the SATs . You can do this a little at a time. After you have done your practice review and fixing mistakes work, you can do one or two of these.
Here are ACT practice questions . This is more limited in what’s offered.
There will be a final, not to mention calculus maybe in your future. Make sure you know the material. Go back and redo lessons/questions that gave you trouble. Hopefully, you listened to me and have a list. Are you still keeping a list?
Binge watch with notebook and pencil in hand of course!
Read this introductory material from Saylor.org. “ Think of it: If you know the diameter of a circle, you can find the circumference. It is a function because one particular diameter will ALWAYS yield one particular circumference – it won’t change. This subunit looks at relations and how to rewrite them to clarify the functional relation. For example: if x + y = 5, you can rearrange the algebra to write y = 5 – x. This would identify a function we might call g, and the algebraic description would be g(x) = 5 – x. “
Read through pages 7-10 in the online textbook. As before, always work through examples on paper. Complete the TIN question on page 10. Check your answer on page 12.
Read through pages 10-12 in the online textbook. As before, always work through examples on paper. Pay attention to the shapes of the functions.
Make a chart of these toolkit functions and keep it in a handy place to refer to during the course.
Complete the odd problems in pages 13 – 16 in the online textbook.
Check your answers , 1.1. Understand your mistakes.
Complete the odd problems on pages 17 – 19 in the online textbook.
Check your answers , 1.1. Understand your mistakes. Your final exam will be made up of questions from the exercises you will be completing during the course.
Record your score out of 24.
Watch the video on set builder notation .
Read through pages 21-24 in the online textbook.
Do the TIN questions as you come to them.
Check your answers on page 30.
Watch the video on finding domain and range of a function using a graph.
Read through pages 24-27 in the online textbook. As always, complete the TIN question when you come to it.
Check your answer on page 30.
Watch the video on piecewise functions .
Read through pages 28-30 in the online textbook. As always, complete the TIN question when you come to it.
Check your answer.
Complete the odd problems numbers 1-35 on pages 31 through 33 in the online textbook.
Check your answers , 1.2. Score up to 2 points each.
Record your score out of 30. (potential for extra credit)
Watch the video on rate of change . ( YouTube )
Read through pages 34-37 in the online textbook. As always, complete the TIN questions when you come to them.
Check your answers on page 44.
Catch up – Review day
Go back to this past week’s lessons and redo any you got wrong. Try them from scratch. You can also try a question from each section that wasn’t assigned. Make sure you can still do them without having the examples right in front of you.
Recommended if you are planning on taking the SAT: Practice for the SATs .
There will be a final. You need to know these things. Go back and redo lessons/questions that gave you trouble. Hopefully, you listened to me and have a list. Are you still keeping a list?
Read through pages 38-43 in the online textbook. As always, complete the TIN questions when you come to them.
Complete the odd problems from pages 46 to 48 in the online textbook.
Check your answers , 1.3. (up to 2 points each)
Record your score out of 40. (potential for extra credit)
Watch this short video on the composition of functions .
Read pages 49-51 of the chapter on the composition of functions . As always, answer any TIN questions as you come to them.
Check your answers on page 56.
Watch the video on the composition of functions .
Read from page 51 to 55 of the chapter on the composition of functions . As always, answer any TIN questions as you come to them.
Do the odd-numbered exercises for chapter 1.4, numbers 1-43.
Check your answers , 1.4. (Score up to 2 points for each problem.)
Read the section on shifts in the online textbook , pages 61-67. Stop at “Reflections.”
Remember to work through examples and do the TIN questions.
Check your answers on page 81.
Read the section in your online textbook on reflections , pages 67-70.
Check the answers to the TIN question on page 81.
Watch the video on odd/even functions.
Work through the page on graphs and functions transformations.
Read pages 71-76 through the TIN question on 76.
Watch this lesson on transformations in combination .
Work through pages 76-80 .
Check your answer on page 80.
Do every fourth problem – 1, 5, 9, 13, 17,… from problems 1-99 starting on page 82.
Do problem 99 for up to 6 points of extra credit!
Check your answers , 1.5. (up to 2 points each)
Record your score out of 48.
Watch the video on function inverses .
Work through pages 90-95 .
Check your answers on page 95.
Starting on page 96, complete the odd problems .
Check your answers , 1.6. (up to 2 points each)
Here is chapter two of our online textbook. Today read pages 99-101. As always, work through the examples on paper and complete any “Try It Now” (or now, “Flashback”) questions. Today, complete the TIN question at the top of 102.
Check your answers on page 106.
Today read pages 102-106 in your online textbook . As always work through the examples on paper and complete any “Try It Now” (or now, “Flashback”) questions.
Watch the video on average rate of change . ( youtube )
Today complete the odd-numbered questions, #1-49, beginning on page 107 in your online textbook .
Check your answers , 2.1. Score up to 2 points per problem.
Record your score out of 50.
Today work through pages 111-117 in your online textbook. You can stop at Parallel and Perpendicular Lines.
Check your answers on page 121.
On My EP, click on Did More? on the Settings page near the top, and you’ll be able to have the next lesson pop up for you once you mark this one complete.
Today work through pages 117-121 in your online textbook.
There were six sets of exercises in chapter 1. Do two odd problems from the first set of exercises. You may use the online textbook to review how to do problems if you need to, but you may not use the solutions! (Take note of your score. This will be recorded on Lesson 93.)
Today complete the odd-numbered questions, #1-51, beginning on page 122 in your online textbook .
Check your answers , 2.2. Score up to 2 points per problem.
Record your score out of 50. (potential for extra credit)
Today work through pages 126-133 in your online textbook.
There were six sets of exercises in chapter 1. Do two odd problems from the second set of exercises. You may use the online textbook to review how to do problems if you need to, but you may not use the solutions! (Hold onto your cumulative scores for recording when we get to Lesson 93.)
Today complete the odd-numbered questions, #1-21, beginning on page 134 in your online textbook .
Check your answers , 2.3. Score up to 2 points per problem.
Record your score out of 20. (potential for extra credit)
Practice for the SATs .
Go through this introduction on trend lines and answer the questions. Scroll down for answers.
Work through pages 138-139 . Remember to answer Flashback questions as you would TIN questions.
Answers are on page 143.
Complete this matching activity .
There were six sets of exercises in chapter 1. Do two odd problems from the third set of exercises. You may use the online textbook to review how to do problems if you need to, but you may not use the solutions! (Hold onto your score from these problems, along with those from the first two sets. This will be recorded on Lesson 93.)
Play with correlation . After you guess, the top number is the actual correlation coefficient. What observations can you make about the correlation coefficient?
Work through pages 139-143 .
There were six sets of exercises in chapter 1. Do two odd problems from the fourth set of exercises. You may use the online textbook to review how to do problems if you need to, but you may not use the solutions! (Hold onto your score.)
Today complete the odd questions, #1-13, beginning on page 144 in your online textbook .
Check your answers , 2.4. Score up to 4 points per problem.
Record your score out of 28.
Work through pages 146-149 .
Answers are on page 152.
There were six sets of exercises in chapter 1. Do two odd problems from the fifth set of exercises. You may use the online textbook to review how to do problems if you need to, but you may not use the solutions! Hold onto your score. We’ll record them all together on Lesson 93.

Compute your grade for the second quarter. You have one last quarter to raise your grade or to keep it up. Finish strong. Hold onto your written work for studying and as a record of what you did for this course.

Lesson 91(*)

(*) Print out your third quarter grading sheet or use the Excel version .
Watch the video on the connection between distance and absolute value .
Work through pages 149-152 .
There were six sets of exercises in chapter 1. Do two odd problems from the sixth set of exercises. You may use the online textbook to review how to do problems if you need to, but you may not use the solutions! (Add up all your scores from the twelve problems and hold onto it for Lesson 93.)
Today complete the odd questions, #1-25, beginning on page 153 in your online textbook .
Check your answers , 2.5. Score up to 2 points per problem. (ANSWER CORRECTION: #3 should be: f(x)= -3|x-3|+3 )
Record your score out of 25. (potential for one point extra credit)
I had to change the order of some exercises, which is why the score you are recording today got spread out to exercises over the last couple of weeks. You have saved your scores right? Good news! Each correct answer is worth up to 4 points. You can record it now. I hope that served as a good review of the first chapter.
Do two SAT questions.
Read the introduction from Saylor.org. Unit 2 planted seeds of ideas that are extended in Unit 3 to more complex functions. A linear function is a simple, special case of a polynomial function. In a polynomial function, the unknowns will have whole number exponents. We began with linear functions because they offered a simple model with which to explore the basic concepts of function and representations of functions. Linear functions, however, can often be solved with simple arithmetic. Unit 3 extends the function ideas to the more complex functions called polynomial and rational that are more easily solved with an algebraic approach. A rational function involves an algebraic fraction made with a polynomial numerator and a polynomial denominator. Their graphs are more complex than polynomial functions, and the solution process is more intricate than the functions in the previous chapter. Here you will explore concepts of functions and models as the course presents symbolic and graphical ways to investigate a number of situations. The unit begins with power and polynomial functions with special attention to quadratic functions. The middle of the unit presents graphs of polynomial functions. After that you will address rational functions and their graphs. The unit closes with the inverses of polynomial functions, which are called radical functions. This is a power-packed unit with complex concepts, so take a deep breath and be ready to think powerfully and work hard. Let’s get going!
Work through pages 155 to 157 in chapter 3 . Stop at Polynomials. As usual, answer the “Try It Now” questions when you come to them.
Work through pages 157 to 161 in chapter 3 . As usual, answer the “Try It Now” questions when you come to them.
Watch the video on the “ End Behavior or Long Run Behavior of Functions .”
Do the odd questions on page 162, #1-33.
Check your answers , 3.1. Score up to 2 points per problem.
Watch the video on graphing quadratic functions .
Work through pages 163 through to the end of 165 . As usual, answer the “Try It Now” questions when you come to them.
Check your answers on page 171.
Work through pages 166 to 168 . As usual, answer the “Try It Now” questions when you come to them. Stop at “Short Run Behavior.”
Watch this video on the “ Vertex and Root of Parabola .”
Answer the questions to Factoring Quadratic Polynomials .
Work through pages 168 to 171 . As usual, answer the “Try It Now” questions when you come to them.
Do the odd questions #1-39 starting on page 172. (Skip #9)
Check your answers , 3.2. Score up to 2 points for each question. Give yourself the 2 points for #9!
Record your score out of 40.
Find the roots of the polynomial functions . Here is a second example .
Work through pages 176 and 177 , stopping after the TIN question.
Answers are on page 184.
If you found today’s lesson short, take your extra time to review your trig!
A polynomial function has only one vertical intercept, because otherwise it would not be a function. It can, however, have multiple horizontal intercepts. At a horizontal intercept, the function may change sign or loop back into the same vertical half of the coordinate grid. Mathematicians, scientists, businesspeople, and others develop functions to model real-world behavior, such as the trajectory of a meteor, the growth of income in a company, the intensity of earthquakes, and climate change. A function model makes it possible to study phenomena and to better predict future outcomes.
Watch this video on “ Graphs of Polynomial – End Behavior, Zeros, Multiplicity .”
Work through pages 177 to 179 . Stop at Solving Polynomial Equations.
Watch the video on Solving a Polynomial Inequality .
Work through pages 179 to the end of the section on 181 .
Work through pages 181 to 184 .
Do the odd problems from pages 185-187, #1-49.
Do number 51 for extra credit.
Check your answers , 3.3. Score up to 2 points for each problem.
Rational Functions look like fractions, and their graphs are complex. Brushing up on your earlier fraction skills will help enormously in this subunit because algebraic fractions work in a parallel way – like when you need a common denominator. This subunit begins by defining a rational function and identifying its characteristics. It continues by exploring graphs of rational functions and helps you determine asymptotes and intercepts.
Watch the video on the Asymptotes of Rational Functions .
Work through pages 188-191 until the next section begins. As always, do the TIN questions as you come to them.
Check your answers on page 199.
Work through pages 191-195 .
Answers are on page 200.
Watch the video on Determining Vertical and Horizontal Asymptotes .
Work through pages 195-199 .
Do the odd-numbered problems #1-43 starting on page 201. SKIP number 31.
Do number 40 for up to 4 points extra credit.
Check your answers , 3.4. Score up to 2 points per problem.
Record your score out of 42.
Work through pages 206-211 .
Answers are on page 211.
Do the odd-numbered problems #1-23 starting on page 212.
Check your answers , 3.5. Score up to 2 points per problem.
This is a good habit I am teaching you. When you are in college, you will only have classes a few times a week. It’s a great practice to do your homework on the day that it is assigned. Try to complete it. Then when you have an extra day, you can go back to anything you got wrong and make sure you have it figured out.
There were five sets of exercises in chapter 2. Do four odd problems from each set of exercises. It’s better for you to choose each one from a different section to get more review. You may use the textbook review, but not the solutions!
Check your answers . (up to 2 points each)
Review exponent rules . (If you need to review this, you can use these links.)
Work through pages 215-219 (until the start of the next section.) As usual, complete the TIN questions as you come to them.
Answers are on page 227.
Watch the video on Exponential Growth Functions . ( youtube )
Work through pages 219-221 . As usual, complete the TIN questions as you come to them.
Watch the video introduction to compound interest . ( youtube )
Work through pages 222-224 (to the start of the next section). As usual, complete the TIN questions as you come to them.
There will be a final. You need to know these things. Go back and redo lessons/questions that gave you trouble.
Finish the unit working through pages 224-227 . As usual, complete the TIN questions as you come to them.
Do problem 37 on page 231.
Check your answers .
Record your score out of 4. (potential for one point extra credit)
Do the odd-numbered questions 1-35 , starting on page 228.
Check your answers , 4.1. Score up to 2 points per question.
Record your score out of 35. (potential for one point extra credit)
Watch the video on graphing exponential functions . ( youtube )
Work through pages 232-235 (to the start of the next section). As usual, complete the TIN questions as you come to them.
Answers are on page 239.
Watch the video on transformations of graphs . Take notes. Pause and draw graphs.
Work through pages 235-239 (to the start of the next section). As usual, complete the TIN questions as you come to them.
Do the odd-numbered questions 1-35 , starting on page 240.
Check your answers , 4.2. Score up to 2 points per question.
Work through pages 242-244 . As usual, complete the TIN questions as you come to them.
Answers are on page 250.
Check your understanding of logarithms with these problems. (This is a Shockwave Flash file. If you can’t use it, you could do the first four example problems on this page.) You could also use that second link if you want to hear someone else explain logarithms to you. They do have a point! They make some complicated equations easier to deal with.
Work through pages 244-245 . As usual, complete the TIN questions as you come to them.
Watch the video on calculators for natural logarithms. ( YouTube )
Work through pages 245-247 . As usual, complete the TIN questions as you come to them.
Watch the video on change of base rules.
Work through pages 247-250 . As usual, complete the TIN questions as you come to them.
Watch the video on solving exponential equations.
Do the odd-numbered questions 1-35 , starting on page 251.
Check your answers , 4.3. Score up to 2 points per question.
Do the odd-numbered questions 37-71 , starting on page 251.
Work through pages 253-254 . As usual, complete the TIN questions as you come to them and work through examples.
Answers are on page 259.
Read the page on understanding logarithms . Watch the video at the bottom of the page on logarithm properties. ( youtube )
Practice logs .
Work through pages 255-256 . As usual, complete the TIN questions as you come to them and work through examples.
Watch the video on solving logarithmic equations .
Watch another video on solving logarithmic equations .
Work through pages 256-259 .
Do the odd-numbered questions 1-25 , on page 260.
Check your answers , 4.4. Score up to 2 points per question.
Do the odd-numbered questions 27-47 , on page 261.
Record your score out of 22.
Watch the video on graphing natural logarithms . ( YouTube )
Watch the video on graphing logarithmic functions . ( YouTube )
Work through pages 262 and 263 in the online textbook .
Answers are on page 267.
Watch the video on transformations of logarithmic functions .
Do the odd-numbered questions 1-23 , on page 268.
Check your answers , 4.5. Score up to 2 points per question.
Work through pages 270 – 274 in the online textbook .
Answers are on Lesson 282.
Your final for Pre-Calculus is on Lesson 180. All problems will be from the test/exercises you’ve been doing.
Work through pages 274 – 276 in the online textbook .
Work through pages 276 – 282 in the online textbook .
Do the odd-numbered questions 1-39 , on page 283.
Answer Note: I’m getting different answers for number 11, close, but not the same. 24,000 and 15,084 (or 15,119). You can count them as correct answers.
Look at, and understand, the four graphs , linear and logarithmic.
Work through pages 289 – 292 in the online textbook .
Answers are on page 294.
Work through pages 292 – 294 in the online textbook .
Do the odd-numbered questions 1-15 , on page 295.
Check your answers , 4.7. Score up to 2 points per question.
Record your score out of 16.
Your final exams are on Lesson 159 and 160, trigonometry and precalculus.
Review your work from the year.
Review for the final. You’ve been doing review throughout the year. You should be good at this. Are there things you need to go over?
A good idea is to take one question from each unit and try it to see if you can do it without the examples.
Review for the finals, trig on Lesson 159 and precalculus on Lesson 160.
Take the Trig final .
Take your final .
Check your answers: chapter 1 , chapter 2 , chapter 3 , chapter 4 .
Record your score out of 100.
Congratulations on finishing Pre-Calculus with Trigonometry!
Take the polls .

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Geometry (all content)

Course: geometry (all content) > unit 13.

Hypotenuse, opposite, and adjacent

Intro to the trigonometric ratios

Trigonometric ratios in right triangles

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Video transcript

IMAGES

Right Triangle Trigonometry Worksheet
10th class trigonometry basics in easy explanation/exercise 8.1 #NCERT
Assignments For Class 10 Trigonometry
Right Triangle Trigonometry Worksheet
Gr 10 Applied Math: Trigonometry Problems
Trigonometry Homework Assistance

VIDEO

010A
Mastering Your Homework
Grade 10
011A
012B
010B

COMMENTS

Trigonometry
Trigonometry 4 units · 36 skills. Unit 1 Right triangles & trigonometry. Unit 2 Trigonometric functions. Unit 3 Non-right triangles & trigonometry. Unit 4 Trigonometric equations and identities. Course challenge. Test your knowledge of the skills in this course. Start Course challenge. Math.
Trigonometry
Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more! Right-Angled Triangle. The triangle of most interest is the right-angled triangle.The right angle is shown by the little box in the corner:
Trigonometry lessons
Find legs of right triangles, and don't get pulled over! Sec, Csc, and Cot. Secant and friends (three more trig functions) 3. Special angles. 45 degrees. Calculate the sine, cosine, and tangent of 45 degrees. 30/60 degrees. Calculate the sine, cosine, and tangent of 30 and 60 degrees.
Practice Trigonometry
This course will teach you all of the fundamentals of trigonometry, starting from square one: the basic idea of similar right triangles. In the first sequences in this course, you'll learn the definitions of the most common trigonometric functions from both a geometric and algebraic perspective. In this course, you'll master trigonometry by solving challenging problems and interacting with ...
Trigonometry
Unit test. Level up on all the skills in this unit and collect up to 1,700 Mastery points! Let's extend trigonometric ratios sine, cosine, and tangent into functions that are defined for all real numbers. You might be surprised at how we can use the behavior of those functions to model real-world situations involving carnival rides and ...
Trigonometry
Unit test. Level up on all the skills in this unit and collect up to 1,300 Mastery points! Knowing trig identities is one thing, but being able to prove them takes us to another level. In this unit, we'll prove various trigonometric identities and define inverse trigonometric functions, which allow us to solve trigonometric equations.
Trigonometry: Lessons and Worksheets
Trigonometry is the study of the relationships between the sides and angles of triangles. These lessons and worksheets will help you to learn concepts like sine, cosine, and tangent. Trigonometry has wide applications in fields like physics, engineering, and computer graphics.
1.E: Right Triangle Trigonometry Angles (Exercises)
1.3 Exercises. 1.3.1 From a position $150 $ ft above the ground, an observer in a building measures angles of depression of $12^\circ $ and $34^\circ $ to the top and bottom, respectively, of a smaller building, as in the picture on the right. Use this to find the height $h $ of the smaller building. 1.3.2 Generalize Example 1.12: A person standing $a $ ft from the base of a mountain ...
Trigonometry (Functions, Table, Formulas & Examples)
The trigonometry angles which are commonly used in trigonometry problems are 0°, 30°, 45°, 60° and 90°. The trigonometric ratios such as sine, cosine and tangent of these angles are easy to memorize. We will also show the table where all the ratios and their respective angle's values are mentioned.
Trigonometry Worksheets
Plus each one comes with an answer key. Law of Sines and Cosines Worksheet. (This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle) Law of Sines. Ambiguous Case of the Law of Sines. Law Of Cosines.
Trigonometry For Beginners!
This math video tutorial provides a basic introduction into trigonometry. It covers trigonometric ratios such as sine, cosine, and tangent. It explains how...
27 Hands-On Trigonometry Activities
14 Trig Ratios Puzzle. 15 Double Angle Identities Joke Worksheet. 16 Unit Circle Magnets. 17 Radian Arts and Crafts Activity. 18 Trigonometry Puzzle. 19 Deriving the Unit Circle Foldable. 20 One or Negative One Trig Identities Worksheet. 21 Odd One Out Coterminal Angles Activity. 21.1 The Great Quadrant Guessing Game.
Right triangles & trigonometry
Trigonometry 4 units · 36 skills. Unit 1 Right triangles & trigonometry. Unit 2 Trigonometric functions. Unit 3 Non-right triangles & trigonometry. Unit 4 Trigonometric equations and identities. Course challenge. Test your knowledge of the skills in this course. Start Course challenge. Math.
Free Printable trigonometry Worksheets for 9th Grade
Explore trigonometry Worksheets for grade 9 by Topic. converse of pythagoras theorem. pythagorean theorem. trigonometric identities. law of sines. trigonometric ratios sin cos tan csc sec and cot. inverse trigonometric functions. radians and degrees. radians and arc length.
PDF Trigonoinetry
The radian measure of a central angle is the ratio of the length sof the arc subtending that angle to the radius rof the circle: s r. For a central angle of measure 1 radian, s 1. rr. That is, an angle of 1 radian is the angle subtended by an arc with length equal to the radius of the circle. Figure 1.1. 19.
Trigonometry Practice Questions
Answers - Version 1. Answers - Version 2. Practice Questions. The Corbettmaths Practice Questions on Trigonometry.
Trigonometry full course for Beginners
Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of #triangles. Throughout history, #trigonometry has been ...
PDF Name: Period: Date: Trigonometry Assignment
Name: _____ Period: _____ Date: _____ Author: Rafae Created Date: 3/21/2017 3:40:08 PM
Trigonometric equations and identities
In this unit, you'll explore the power and beauty of trigonometric equations and identities, which allow you to express and relate different aspects of triangles, circles, and waves. You'll learn how to use trigonometric functions, their inverses, and various identities to solve and check equations and inequalities, and to model and analyze problems involving periodic motion, sound, light, and ...
Trigonometry and Precalculus
Prerequisite: Geometry, Algebra 2. Recommended: 11th, 12th. Test Prep: CLEP exam. Course Description: This course offers a quarter (45 lessons) of trigonometry followed by a semester of precalculus. In trigonometry, students will not only learn the basic trigonometric functions and how to apply them to solve real-life problems, but will explore ...
Right Triangle Trigonometry Assignment Flashcards
A. 0.32 rad. Describe three ways to determine the measure of segment YZ. •Solve the equation where the sine of 30 degrees is equal to YZ/50. •Solve 180- (30+90)=60 to find the measure of angle Z. Then use the cosine of Z to write and solve an equation. •Use the 30°-60°-90° triangle theorem to find that YZ=50/2. Study with Quizlet and ...
Intro to the trigonometric ratios (video)
Intro to the trigonometric ratios. Sin, cos, and tan are trigonometric ratios that relate the angles and sides of right triangles. Sin is the ratio of the opposite side to the hypotenuse, cos is the ratio of the adjacent side to the hypotenuse, and tan is the ratio of the opposite side to the adjacent side. They are often written as sin (x ...

Introduction to Trigonometry

Right-Angled Triangle

Why a Right-Angled Triangle?

Sine, Cosine and Tangent

Example: What is the sine of 35°?

Example: How Tall is The Tree?

Try Sin Cos and Tan

Unit Circle

Degrees and Radians

Repeating Pattern

Example: what is the cosine of 370°?

Example: what is the sine of −3 radians?

Solving Triangles

Example: Find the Missing Angle "C"

Other Functions (Cotangent, Secant, Cosecant)

Trigonometric and Triangle Identities

Trigonometry

Trigonometry

End of Unit 1

End of Unit 2

End of Unit 3

End of Unit 4

End of Unit 5

Course description

Topics covered

Prerequisites and next steps

Prerequisites

Trigonometry

Trigonometry (Mathematics Curriculum)

Where Does the Word Trigonometry Come From?

Pythagoras' Theorem

Sine Function

Cosine Function

Tangent Function

Area of a Triangle Using Trigonometry

Trigonometry Curriculum

1.E: Right Triangle Trigonometry Angles (Exercises)

1.1 Exercises

1.2 Exercises

1.3 Exercises

1.4 Exercises

1.5 Exercises

Trigonometry

What is Trigonometry?

Trigonometry Ratios-Sine, Cosine, Tangent

Six Important Trigonometric Functions

Even and Odd Trigonometric Functions

Trigonometry Angles

Trigonometry Table

Unit Circle

List of Trigonometry Formulas

Trigonometry Identities

Euler’s Formula for trigonometry

Trigonometry Basics

Trigonometry Examples

Applications of Trigonometry

Video Lesson on Applications of Trigonometry

Trigonometry Problems and Solutions

Trigonometry Questions

Frequently Asked Questions on Trigonometry

What are the six basic Trigonometric Functions?

What is the formula for six trigonometry functions?

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Trigonometry Worksheets

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

27 Hands-On Trigonometry Activities

What is a Radian? Hands-On Activity

Angle Spinner for Sketching Angles in Standard Position

Coterminal Angles Card Sort Activity

Trig Mini Poster Project

Evaluating Trig Functions Square Puzzle Activity

Unit Circle Paper Plate Activity

Radians and Degrees War

Clinometer Activity and Foldable

Leap Frog Game for Evaluating Trig Functions

Unit Circle Bingo Game

Unit Circle Project Examples

Trig Identities Challenge Activity

Trig Identities Matching Activity