4.1 Angles and Rotation

Introduction

So far we have studied angles as parts of triangles, but we can also use angles to describe rotation. For example, think of the minute hand on a clock. Every hour, the minute hand moves through one complete rotation, or $360 degrees$ . In two hours, the minute hand rotates through $720 degrees$ .

The volume control on an amplifier is a dial with ten settings, as shown at right. Through how many degrees would you rotate the dial to increase the volume level from 0 to 7?

$0.7$ of $360 degrees$ , or $252 degrees$

Angles in Standard Position

The degree measure of an angle depends only on the fraction of a whole rotation between its sides, and not on the location or position of the angle. To compare and analyze angles, we place them in standard position, so that the vertex of the angle is located at the origin and its initial side lies on the positive $x$ -axis. If the terminal side rotates counter-clockwise, the degree measure of the angle is positive; it is negative if the terminal side rotates clockwise. The figure below shows four angles and how each appears in standard position.

One-half a complete revolution is $180 degrees$ , and three-quarters of one revolution is $270 degrees$ . Thus, for angles between $180 degrees$ and $270 degrees$ in standard position, the terminal side lies in the third quadrant, and for angles between $270 degrees$ and $360 degrees$ , the terminal side lies in the fourth quadrant.

Find the degree measure of each angle below, and sketch the angle in standard position.

angle difference between 250 degrees and straight angle

a. $120 degrees$

b. $70 degrees$

Trigonometric Ratios for All Angles

In Chapter 3 we defined the sine, cosine, and tangent for obtuse angles by placing the angle in a Cartesian coordinate system. We can do the same for angles that represent rotations.

First, we place the angle $θ$ in standard position, with its vertex at the origin. We picture the terminal side sweeping counter-clockwise around a circle to form the angle.
Next, we choose a point $P$ with coordinates $(x, y)$ on the terminal side, as shown at right. The distance from the origin to $P$ is then $r = \sqrt{x^{2} + y^{2}}$ . The trigonometric ratios of $θ$ are defined as follows.

We can choose any point on the terminal side of the angle, and the trig ratios defined by its coordinates will be the same. (Can you explain why? Think about similar triangles.) And, as we noted when we first defined the trig ratios, they are functions of the input angle, so there is only one sine, cosine, or tangent for a given angle.

Because it is the distance from the origin to $P$ , $r$ is always positive. However, $x$ and $y$ can be positive or negative (or zero), depending on the angle $θ$ . For example, in the second quadrant, $x$ is negative but $y$ is positive, so the cosine and the tangent of angles between $90 degrees$ and $180 degrees$ are negative, but their sines are positive.

Example 3

Give the sign of each of the three trigonometric ratios of the angles.

$200 degrees$
$300 degrees$

a. In standard position, the terminal side of an angle of $200 degrees$ lies in the third quadrant. (See figure (a) below.) In the third quadrant, $x less than 0$ and $y less than 0$ , but $r greater than 0$ . Thus, $sin ⁡ open parenthesis 200 degrees close parenthesis$ is negative, $cos ⁡ open parenthesis 200 degrees close parenthesis$ is negative, and $tan ⁡ open parenthesis 200 degrees close parenthesis$ is positive.

b. The terminal side of $300 degrees$ lies in the fourth quadrant, so $x greater than 0$ and $y less than 0$ , and $r greater than 0$ . Thus, $sin ⁡ open parenthesis 300 degrees close parenthesis$ is negative, $cos ⁡ open parenthesis 300 degrees close parenthesis$ is positive, and $tan ⁡ open parenthesis 300 degrees close parenthesis$ is negative.

For angles in each of the four quadrants shown below, explain why the indicated trig ratios are positive. Then complete the table.

Quadrant	Degrees	Sine	Cosine	Tangent
First	$0 degrees less than θ less than 90 degrees$	positive	positive	positive
Second	$90 degrees less than θ less than 180 degrees$
Third	$180 degrees less than θ less than 270 degrees$
Fourth	$270 degrees less than θ less than 360 degrees$

Quadrant	Degrees	Sine	Cosine	Tangent
First	$0 degrees less than θ less than 90 degrees$	positive	positive	positive
Second	$90 degrees less than θ less than 180 degrees$	positive	negative	negative
Third	$180 degrees less than θ less than 270 degrees$	negative	negative	positive
Fourth	$270 degrees less than θ less than 360 degrees$	negative	positive	negative

Example 4

Find the sine, cosine, and tangent of the angle shown at right.

The $y$ -coordinate of the point $P$ is $minus 5$ , and $r equals 6$ , so

$\sin (θ) = \frac{y}{r} = \frac{- 5}{6}$

To find the $x$ -coordinate of $P$ , we use the equation of a circle of radius $6$ , $x^{2} + y^{2} = 36$ .

$\begin{aligned} x^{2} + (- 5)^{2} & = 36 \\ x^{2} & = 36 - 25 = 11 \\ x & = \pm \sqrt{11} \end{aligned}$

Because $P$ is in the third quadrant, $x = - \sqrt{11}$ . Thus,

$\cos (θ) = \frac{x}{r} = \frac{- \sqrt{11}}{6} and \tan (θ) = \frac{y}{x} = \frac{- 5}{- \sqrt{11}} = \frac{5}{\sqrt{11}}$

Find the sine, cosine, and tangent of the angle shown at right. The circle has radius 4.

We know that $r equals 4$ and $x = - 3$ , so

$y = \sqrt{4^{2} - (- 3)^{2}} = \sqrt{7}$

Thus,

$\sin (θ) = \frac{\sqrt{7}}{4}, \cos (θ) = \frac{- 3}{4}, \tan (θ) = \frac{- \sqrt{7}}{3}$

Reference Angles

In Section 3.1 we learned that the trig ratios for angles in the second quadrant are the same as the trig ratios for their supplements, except for sign. For example, you can use your calculator to verify that

$sin ⁡ open parenthesis 130 degrees close parenthesis equals 0.7660$	$sin ⁡ open parenthesis 50 degrees close parenthesis equals 0.7660$
$\cos (130^{\circ}) = - 0.6428$	$cos ⁡ open parenthesis 50 degrees close parenthesis equals 0.6428$
$\tan (130^{\circ}) = - 1.1918$	$tan ⁡ open parenthesis 50 degrees close parenthesis equals 1.1918$

The trig ratios for $130 degrees$ and $50 degrees$ have the same absolute value because the two triangles formed by the angles are congruent, as shown above.

$△ O P^{'} Q^{'}$ is called a reference triangle for $130 degrees$ , and $50 degrees$ is called the reference angle.

The trig ratios for angles between $180 degrees$ and $360 degrees$ , whose terminal sides lie in the third and fourth quadrants, are also related to the trig ratios of familiar angles in the first quadrant. We "refer" the angle to a first quadrant angle with a congruent reference triangle.

We can construct reference triangles for angles in any of the four quadrants, and the trig ratios of the angle are the same as the trig ratios of its reference angle, up to sign. Here is how to construct a reference triangle for an angle:

The figure below shows angles $θ$ between $0 degrees$ and $360 degrees$ , and the reference angle, $\tilde{θ}$ , for each. Study the figures, and make sure you understand the formula for finding the reference angle in each quadrant.

Activity

Reference Angles

The circle below has radius 1 and is centered at the origin.

1. Use a protractor to draw an angle of $56 degrees$ in standard position. Draw its reference triangle.
2. Use your calculator to find the sine and cosine of $56 degrees$ , rounded to two decimal places. Label the sides of the reference triangle with their lengths.
3. What are the coordinates of the point $P$ where your angle intersects the circle?
4. Draw the reflection of your reference triangle across the $y$ -axis, so that you have a congruent triangle in the second quadrant.
1. You now have the reference triangle for a second-quadrant angle in standard position. What is that angle?
2. Use your calculator to find the sine and cosine of your new angle. Label the coordinates of the point $Q$ where the angle intersects the circle.
3. Draw the reflection of your triangle from part (1d) across the $x$ -axis, so that you have a congruent triangle in the third quadrant.
1. You now have the reference triangle for a third-quadrant angle in standard position. What is that angle?
2. Use your calculator to find the sine and cosine of your new angle. Label the coordinates of the point $R$ where the angle intersects the circle.
3. Draw the reflection of your triangle from part (2c) across the $y$ -axis, so that you have a congruent triangle in the fourth quadrant.
1. You now have the reference triangle for a fourth-quadrant angle in standard position. What is that angle?
2. Use your calculator to find the sine and cosine of your new angle. Label the coordinates of the point where the angle intersects the circle.

Generalize: All four of your angles have the same reference angle, $56 degrees$ . For each quadrant, write a formula for the angle whose reference angle is $θ$ .

Quadrant I:
Quadrant II:
Quadrant III:
Quadrant IV:

Example 5

Find the reference angle for $200 degrees$ .
Sketch $200 degrees$ and its reference angle in standard position, along with their reference triangles. Verify that both angles have the same trigonometric ratios, up to sign.

In standard position, an angle of $200 degrees$ lies in the third quadrant. Its reference angle is
$200^{\circ} - 180^{\circ} = 20^{\circ}$
Both angles are shown at right. Note that the reference triangle for $200 degrees$ is congruent to the reference triangle for $20 degrees$ . You can use your calculator to verify the following values.

$\begin{aligned} \sin (20^{\circ}) & = 0.3420 & \sin (200^{\circ}) = - 0.3420 \\ \cos (20^{\circ}) & = 0.9397 & \cos (200^{\circ}) = - 0.9397 \\ \tan (20^{\circ}) & = 0.3640 & \tan (200^{\circ}) = 0.3640 \end{aligned}$

Find the reference angle for $285 degrees$ .
Sketch $285 degrees$ and its reference angle in standard position, along with their reference triangles. Verify that both angles have the same trigonometric ratios, up to sign.

The terminal side of an angle of $285 degrees$ lies in the fourth quadrant, so its reference angle is $360 - 285 = 75^{\circ}$
$\sin (285^{\circ}) = - \sin (75^{\circ}) = - 0.9659$ $\cos (285^{\circ}) = \cos (75^{\circ}) = 0.2588$ $\tan (285^{\circ}) = - \tan (75^{\circ}) = - 3.7321$

Using Reference Angles

Here is a summary of our discussion about reference angles.

Any acute angle $θ$ is the reference angle for four angles between $0 degrees$ and $360 degrees$ , one in each quadrant. The figure below shows the four angles in standard position whose reference angle is $35 degrees$ . Note that each angle is found by measuring $35 degrees$ from the $x$ -axis in the appropriate quadrant, and that the four angles together make a “bow-tie” shape.

From the figure, you can see that the angles in each quadrant with a given reference angle are computed as follows.

Note

To find an angle $θ$ with a given reference angle $\tilde{θ}$ :

Quadrant I: $θ = \tilde{θ}$		Quadrant II: $θ = 180^{\circ} - \tilde{θ}$
Quadrant III: $θ = 180^{\circ} + \tilde{θ}$		Quadrant IV: $θ = 360^{\circ} - \tilde{θ}$

Example 6

Find two angles between $0 degrees$ and $360 degrees$ whose cosine is $\frac{5}{8}$ . Round your answers to the nearest tenth of a degree.
Find two angles between $0 degrees$ and $360 degrees$ whose cosine is $\frac{- 5}{8}$ . Sketch the reference triangle for each angle.

One of the angles we want is $\cos^{- 1} (\frac{5}{8})$ . Use your calculator to find the first quadrant angle, $51.3 degrees$ . The cosine is also positive in the fourth quadrant, so we look for the angle in the fourth quadrant with reference angle $51.3 degrees$ . That angle is $360^{\circ} - {51.3}^{\circ} = {308.7}^{\circ}$ , as shown at right.
Because the cosine is negative in the second and third quadrants, we would like angles $θ$ in those quadrants whose reference angle is $θ ~ equals 51.3 degrees$ . In the second quadrant, we find
$\begin{aligned} θ & = 180^{\circ} - \tilde{θ} \\ = 180^{\circ} - {51.3}^{\circ} = {128.7}^{\circ} \end{aligned}$
And in the third quadrant,

$\begin{aligned} θ & = 180^{\circ} + \tilde{θ} \\ = 180^{\circ} + {51.3}^{\circ} = {231.3}^{\circ} \end{aligned}$
Both angles are shown above. You can check that the cosines of $128.7 degrees$ and $231.3 degrees$ are both approximately $minus 0.625$ .

Find an angle in the third quadrant whose tangent is $3.66$ . Round your answer to the nearest tenth of a degree.
Use reference angles to find two angles whose tangent is $minus 3.66$ .

$\tan^{- 1} (3.66) = {74.7}^{\circ}$ . The angle in the third quadrant with reference angle $74.7 degrees$ is $180^{\circ} + {74.7}^{\circ} = {254.7}^{\circ}$
The tangent is negative in the second and fourth quadrants, and the angles in those quadrants with reference angle $74.7 degrees$ are $105.3 degrees$ and $285.3 degrees$

The Special Angles

Recall that the angles $30 degrees, 45 degrees$ and $60 degrees$ are called the special angles because we can express the exact values of their trigonometric ratios in terms of radicals. There are special angles in all four quadrants; namely, those whose reference angles are $30 degrees, 45 degrees$ and $60 degrees$ .

Example 7

Find exact values for the sine, cosine, and tangent of $210 degrees$ .

An angle of $210 degrees$ lies in the third quadrant, and its reference angle is

$\tilde{θ} = 210^{\circ} - 180^{\circ} = 30^{\circ}$

In the third quadrant, the sine and cosine are negative, so we have

$\sin (θ) = \frac{- 1}{2}, \cos (θ) = \frac{- \sqrt{3}}{2}, \tan (θ) = \frac{1}{\sqrt{3}}$

Alternatively, consider the reference triangle in the figure at right, where $r equals 2$ . Because the sides of a 30-60-90 triangle are in the ratio $1: the square root of 3: 2$ , the point $P$ has coordinates $(- \sqrt{3}, - 1)$ . Using the definitions of the trig ratios, we can calculate the values above.

Sketch an angle of $300 degrees$ in standard position, and its reference triangle. Find the reference angle for $300 degrees$ .
Find exact values for the sine, cosine, and tangent of $300 degrees$ .

Reference angle: $60 degrees$
$\sin (300^{\circ}) = \frac{- \sqrt{3}}{2}$ , $\cos (300^{\circ}) = \frac{1}{2}$ , $\tan (300^{\circ}) = - \sqrt{3}$

All of the special angles are shown at right. In the Homework Problems you will calculate the three trigonometric ratios for all the special angles, and it will be useful to be familiar with these values, and be able to calculate them quickly. You may want to review the two "special triangles" in Section 2.3.

Coterminal Angles

Because $360 degrees$ represents one complete revolution, we can add or subtract a multiple of $360 degrees$ to any angle, and the terminal side will arrive at the same position. For example, the angles $70 degrees$ and $430 degrees$ have the same terminal side because $430^{\circ} = 70^{\circ} + 360^{\circ}$ . Such angles are called coterminal.

The angle $790 degrees$ is also coterminal with $70 degrees$ , because if we add two revolutions to $70 degrees$ , we get $790^{\circ} = 70^{\circ} + 2 (360^{\circ})$ , as shown below.

Because coterminal angles have the same standard position, their trigonometric ratios are equal. For example, you can verify that, to four decimal places,

$\cos (790^{\circ}) = \cos (70^{\circ}) = 0.3420$

Example 8

Find an angle between $0 degrees$ and $360 degrees$ that is coterminal with $520 degrees$ .
Use a calculator to verify that the trig ratios of $520 degrees$ are the same as the ratios of the coterminal angle.

We subtract multiples of $360 degrees$ from $520 degrees$ until the remainder is less than $360 degrees$ .
$520^{\circ} - 360^{\circ} = 160^{\circ}$
Because $160 degrees$ is between $0 degrees$ and $360 degrees$ , this is the angle we want.
You can use your calculator to see that
$\begin{aligned} \sin (160^{\circ}) & = \sin (520^{\circ}) = 0.3420 \\ \cos (160^{\circ}) & = \cos (520^{\circ}) = - 0.9397 \\ \tan (160^{\circ}) & = \tan (520^{\circ}) = - 0.3640 \end{aligned}$

If the direction of rotation is important, we let positive angles represent rotation in the counter-clockwise direction, and negative angles represent rotation in the clockwise direction. For example, the angle $minus 60 degrees$ shown at right lies in the fourth quadrant. It is coterminal with $- 60^{\circ} + 360^{\circ} = 300^{\circ}$ .

Find two angles coterminal with $102 degrees$ , one positive and one negative.

Add $360 degrees$ to get $462 degrees$ , and subtract $360 degrees$ to get $minus 258 degrees$

Review the following skills you will need for this section.

Algebra Refresher

Evaluate the function.

f ( x ) = x 2 − 2 x
1. $f open parenthesis minus 3 close parenthesis$
2. $f open parenthesis a minus 3 close parenthesis$
3. $f open parenthesis a close parenthesis minus 5$
4. $f open parenthesis a close parenthesis minus f open parenthesis 5 close parenthesis$
g ( x ) = x + 4
1. $g open parenthesis 9 close parenthesis$
2. $g open parenthesis 4 h close parenthesis$
3. $g open parenthesis 0 close parenthesis plus g open parenthesis 1 close parenthesis$
4. $g (c^{2})$
F ( x ) = 2 x
1. $F (\frac{- 1}{2})$
2. $F (\frac{w}{2})$
3. $F open parenthesis w plus 2 close parenthesis$
4. $F open parenthesis w close parenthesis plus F open parenthesis 2 close parenthesis$
G ( x ) = 2 x
1. $G open parenthesis minus 3 close parenthesis$
2. $G open parenthesis a plus 3 close parenthesis$
3. $G open parenthesis a close parenthesis plus G open parenthesis 3 close parenthesis$
4. $G (\frac{3}{2})$

$\underset{―}{}$

Algebra Refresher Answers

1. $15$
2. $a^{2} - 8 a + 15$
3. $a^{2} - 2 a - 5$
4. $a^{2} - 2 a - 15$
1. $the square root of 13$
2. $2 \sqrt{h + 1}$
3. $2 + \sqrt{5}$
4. $\sqrt{c^{2} + 4}$
1. $minus 4$
2. $\frac{4}{w}$
3. $\frac{2}{w + 2}$
4. $\frac{2}{w} + 1$
1. $\frac{1}{8}$
2. $8 open parenthesis 2 to the power a close parenthesis$
3. $2^{a} + 8$
4. $2 the square root of 2$

Section 4.1 Summary

Vocabulary

Standard position
Reference angle
Reference triangle
Coterminal angle

Concepts

We can use angles to describe rotation. Positive angles indicate rotation in the counter-clockwise direction; negative angles describe clockwise rotation.
We define the trigonometric ratios of any angle by placing the angle in standard position and choosing a point on the terminal side, with $r = \sqrt{x^{2} + y^{2}}$ .
Note
The Trigonometric Ratios
If $θ$ is an angle in standard position, and $(x, y)$ is a point on its terminal side, with $r = \sqrt{x^{2} + y^{2}}$ , then
$\sin (θ) = \frac{y}{r} \cos (θ) = \frac{x}{r} \tan (θ) = \frac{y}{x}$
To construct a reference triangle for an angle:
1. Choose a point $P$ on the terminal side.
2. Draw a line from point $P$ perpendicular to the $x$ -axis.
The reference angle for $θ$ is the positive acute angle formed between the terminal side of $θ$ and the $x$ -axis.
The trigonometric ratios of any angle are equal to the ratios of its reference angle, except for sign. The sign of the ratio is determined by the quadrant.

Note

To find an angle $θ$ with a given reference angle $\tilde{θ}$ :

Quadrant I: $θ = \tilde{θ}$		Quadrant II: $θ = 180^{\circ} - \tilde{θ}$
Quadrant III: $θ = 180^{\circ} + \tilde{θ}$		Quadrant IV: $θ = 360^{\circ} - \tilde{θ}$

There are always two angles between $0 degrees$ and $360 degrees$ (except for the quadrantal angles) with a given trigonometric ratio.
Coterminal angles have equal trigonometric ratios.

Study Questions

Explain why $cos ⁡ open parenthesis θ close parenthesis less than or equal to 1$ for any angle $θ$ .
Is it true that $tan ⁡ open parenthesis θ close parenthesis less than or equal to 1$ for any angle $θ$ ? Explain.
Sketch a figure showing how to compute the reference angle for angles in each of the four quadrants.
True or false: If $β > α$ , then $\sin (β) > \sin α$ .
How many angles have a cosine equal to 0.4? How many angles between $0 degrees$ and $360 degrees$ have a cosine equal to 0.4?

Skills

Use angles to represent rotations #1-6
Sketch angles in standard position #7-12
Find coterminal angles #13-24
Find and use reference angles #25-44, 55-64
Find trigonometric ratios for the special angles #45-54

Homework 4.1

How many degrees are in each angle?

$\frac{3}{5}$ of one rotation
$\frac{3}{10}$ of one rotation
$\frac{4}{3}$ of one rotation
$\frac{8}{3}$ of one rotation

$216 degrees$
$108 degrees$
$480 degrees$
$960 degrees$

How many degrees are in each angle?

$\frac{5}{6}$ of one rotation
$\frac{3}{8}$ of one rotation
$\frac{7}{4}$ of one rotation
$\frac{7}{12}$ of one rotation

What fraction of a complete rotation is represented by each angle?

$45 degrees$
$300 degrees$
$540 degrees$
$420 degrees$

$\frac{1}{8}$
$\frac{5}{6}$
$\frac{3}{2}$
$\frac{7}{6}$

What fraction of a complete rotation is represented by each angle?

$60 degrees$
$240 degrees$
$450 degrees$
$150 degrees$

Through what angle does the hour hand of a clock rotate between 2 pm and 10 pm?
Through what angle does the hour hand of a clock rotate between 2 am and 10 pm?

$\frac{2}{3}$
$\frac{5}{3}$

Through what angle does the minute hand of a clock rotate between 3:25 am and 3:50 am?
Through what angle does the minute hand of a clock rotate between 4:10 pm and 6:25 pm?

For Problems 7–12, calculate the degree measure of the unknown angle, and sketch the angle in standard position.

$60 degrees$

$60 degrees$

$14 degrees$

For Problems 13–18, find two angles, one positive and one negative, that are coterminal with the given angle.

$40 degrees$

$400 degrees$ and $minus 320 degrees$ (Answers vary.)

$160 degrees$

$215 degrees$

$575 degrees$ and $minus 145 degrees$ (Answers vary.)

$250 degrees$

$305 degrees$

$665 degrees$ and $minus 55 degrees$ (Answers vary.)

$340 degrees$

For Problems 19–24, find a positive angle between $0 degrees$ and $360 degrees$ that is coterminal with the given angle.

$minus 65 degrees$

$295 degrees$

$minus 140 degrees$

$minus 290 degrees$

$70 degrees$

$minus 325 degrees$

$minus 405 degrees$

$315 degrees$

$minus 750 degrees$

For Problems 25–26, use the grid provided below.

Draw two different angles $α$ and $β$ in standard position whose sine is $0.6$ . Note that the radius of the circle is 1,

Use a protractor to measure $α$ and $β$ .
Find the reference angles for both $α$ and $β$ . Draw in the reference triangles.

$36.9 degrees, 143.1 degrees$

Draw two different angles $θ$ and $ϕ$ in standard position whose sine is $minus 0.8$ .

Use a protractor to measure $θ$ and $ϕ$ .
Find the reference angles for both $θ$ and $ϕ$ . Draw in the reference triangles.

For Problems 27–28, use the grid provided below.

Draw two different angles $α$ and $β$ in standard position whose cosine is $0.3$ .

Use a protractor to measure $α$ and $β$ .
Find the reference angles for both $α$ and $β$ . Draw in the reference triangles.

$72.5 degrees, 287.5 degrees$

Draw two different angles $θ$ and $ϕ$ in standard position whose cosine is $minus 0.4$ .

Use a protractor to measure $θ$ and $ϕ$ .
Find the reference angles for both $θ$ and $ϕ$ . Draw in the reference triangles.

For Problems 29–34, find the reference angle. Make a sketch showing the angle, the reference angle, and the reference triangle.

$100 degrees$

$80 degrees$

$125 degrees$

$216 degrees$

$36 degrees$

$242 degrees$

$297 degrees$

$63 degrees$

$336 degrees$

For Problems 35–40, find three angles between $90 degrees$ and $360 degrees$ with the given reference angle, and sketch all four angles on the same grid.

$15 degrees$

$165 degrees$ , $95 degrees$ , $345 degrees$

$26 degrees$

$40 degrees$

$140 degrees$ , $220 degrees$ , $320 degrees$

$50 degrees$

$68 degrees$

$112 degrees$ , $248 degrees$ , $292 degrees$

$75 degrees$

For Problems 41–48, use the values given below to find the trigonometric ratio. Do not use a calculator!

$\cos (23^{\circ}) = 0.9205 \sin (46^{\circ}) = 0.7193 \tan (78^{\circ}) = 4.7046$

$cos ⁡ open parenthesis 157 degrees close parenthesis$

$minus 0.9205$

$sin ⁡ open parenthesis 226 degrees close parenthesis$

$sin ⁡ open parenthesis 314 degrees close parenthesis$

$minus 0.7193$

$cos ⁡ open parenthesis 203 degrees close parenthesis$

$tan ⁡ open parenthesis 258 degrees close parenthesis$

$4.705$

$tan ⁡ open parenthesis 282 degrees close parenthesis$

$sin ⁡ open parenthesis minus 134 degrees close parenthesis$

$minus 0.7193$

$cos ⁡ open parenthesis minus 383 degrees close parenthesis$

On the circle in the figure, all angles are shown in standard position. Find the measure in degrees of the angles labeled (a)-(i).

$120 degrees$
$135 degrees$
$150 degrees$
$210 degrees$
$225 degrees$
$240 degrees$
$300 degrees$
$315 degrees$
$330 degrees$

Find the reference angle for each of your answers in Problem 45.

Draw three angles, one in each quadrant except the first, whose reference angle is $60 degrees$ .
Find exact values for the sine, cosine, and tangent of each of the angles in part (a).

$\sin (120^{\circ}) = \frac{\sqrt{3}}{2}, \cos (120^{\circ}) = \frac{- 1}{2}, \tan (120^{\circ}) = - \sqrt{3},$
$\sin (240^{\circ}) = \frac{- \sqrt{3}}{2}, \cos (240^{\circ}) = \frac{- 1}{2}, \tan (240^{\circ}) = \sqrt{3},$
$\sin (300^{\circ}) = \frac{- \sqrt{3}}{2}, \cos (300^{\circ}) = \frac{1}{2}, \tan (300^{\circ}) = - \sqrt{3}$

Draw three angles, one in each quadrant except the first, whose reference angle is $30 degrees$ .
Find exact values for the sine, cosine, and tangent of each of the angles in part (a).

Draw three angles, one in each quadrant except the first, whose reference angle is $45 degrees$ .
Find exact values for the sine, cosine, and tangent of each of the angles in part (a).

$\sin (135^{\circ}) = \frac{1}{\sqrt{2}}, \cos (135^{\circ}) = \frac{- 1}{\sqrt{2}}, (\tan 135^{\circ}) = - 1,$
$\sin (225^{\circ}) = \frac{- 1}{\sqrt{2}}, \cos (225^{\circ}) = \frac{- 1}{\sqrt{2}}, \tan (225^{\circ}) = 1,$
$\sin (315^{\circ}) = \frac{- 1}{\sqrt{2}}, \cos (315^{\circ}) = \frac{1}{\sqrt{2}}, \tan (315^{\circ}) = - 1$

Complete the table with exact values.

$θ$	$30 degrees$	$45 degrees$	$60 degrees$	$120 degrees$	$135 degrees$	$150 degrees$	$210 degrees$	$225 degrees$	$240 degrees$	$300 degrees$	$315 degrees$	$330 degrees$
$\cos (θ)$
$\sin (θ)$
$\tan (θ)$

In which two quadrants is the statement true?

The sine is negative.
The cosine is negative.
The tangent is positive.

III and IV
II and III
I and III

Find all angles between $0 degrees$ and $360 degrees$ for which the statement is true.

$\cos (θ) = - 1$
$\sin (θ) = - 1$
$\tan (θ) = - 1$

Find two angles, $0 less than or equal to θ less than 360 degrees$ , with $sin ⁡ open parenthesis θ close parenthesis equals 0$ .
Find two angles, $0 less than or equal to θ less than 360 degrees$ , with $cos ⁡ open parenthesis θ close parenthesis equals 0$ .

$0 degrees and 180 degrees$
$90 degrees and 270 degrees$

Find two angles, $0 less than or equal to θ less than 360 degrees$ , with $\sin (θ) = \cos (θ)$ .
Find two angles, $0 less than or equal to θ less than 360 degrees$ , with $\sin (θ) = - \cos (θ)$ .

For Problems 59–64, find a second angle between $0 degrees$ and $360 degrees$ with the given trigonometric ratio.

$sin ⁡ open parenthesis 75 degrees close parenthesis$

$105 degrees$

$cos ⁡ open parenthesis 32 degrees close parenthesis$

$tan ⁡ open parenthesis 84 degrees close parenthesis$

$264 degrees$

$sin ⁡ open parenthesis 16 degrees close parenthesis$

$cos ⁡ open parenthesis 47 degrees close parenthesis$

$313 degrees$

$tan ⁡ open parenthesis 56 degrees close parenthesis$

Explain why the definitions of the trigonometric ratios for a third-quadrant angle (between $180 degrees$ and $270 degrees$ ) are independent of the point $P$ chosen on the terminal side. Illustrate with a figure.

Sides of similar triangles are proportional.

Explain why the definitions of the trigonometric ratios for a fourth-quadrant angle (between $270 degrees$ and $360 degrees$ ) are independent of the point $P$ chosen on the terminal side. Illustrate with a figure.

Trigonometry by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later.