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APPENDIX A: TrigonometryBasics

Trigonometry Basics A2Degree Measure A2Right-Triangle Trigonometry A4Unit-Circle Trigonometry A8Radian Measure A10Angle Measure Conversions A14Sine and Cosine Functions A15

A.1 Concept Inventory A19A.1 Activities A19

Answers to Appendix A A25

Copyright © Houghton Mifflin Company. All rights reserved.

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A2 APPENDIX A

Trigonometry Basics

This appendix gives supplementary material on degree measure and right-triangletrigonometry. It can be used in connection with Section 8.1.

Degree Measure

One complete revolution of a circle is divided into 360 equal parts called degrees.Talking about “small parts of a rotation” is awkward, so we call partial rotationsangles. The measure of an angle is described by the amount of rotation in the turn.For example, a 90-degree angle is one-fourth of a complete counterclockwise revolu-tion (see Figure A.1). A small circle as a superscript on the angle measure is used as asymbol to indicate degrees (90°).

FIGURE A.1 A 90° angle is

of a complete revolution.

The selection of 360 as the number of divisions is rooted in history and mayreflect the fact that the Earth completes one revolution about the Sun in approximately365 days. Because 365 does not have a large number of divisors (only 5 and 73), theinventors of the system probably picked 360 with its multitude of divisors (2, 3, 4, 5,6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180) as being close tothe number of days of the year and easy to use in computing. The fortunate choice of360° gives us a large number of even-degree angles that are simple fractions of a fullrevolution.*

14

90°


Degree Measurement

360 degrees � 360° � 1 complete revolution

1 degree � 1° � of a complete revolution1

360

*It is possible to divide one revolution into 400 equal parts. Incidentally, some do. These parts are calledgradians or grads, and this angle measure is available on many calculators. Of course, you might prefer tobe especially patriotic and divide one revolution into 1776 equal parts. In this case, you would be invent-ing a new angle measure.

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Trigonometry Basics A3

EXAMPLE 1 Measuring Angles in Degrees

a. Express of a complete rotation in degrees.

b. Express of a complete rotation in degrees.

Solution Keeping in mind that one complete rotation is 360°, we have

a. (360°) � 45° (See Figure A.2a.)

b. (360°) � 120° (See Figure A.2b.)

(a) of a revolution (b) of a revolutionis a 45° angle. is a 120° angle.

FIGURE A.2 ●

In the study of trigonometry, we consider revolutions around a particular circle.Specifically, we consider the unit circle—that is, the circle with radius 1, centered atthe origin. We say that an angle is in standard position when the vertex of the angleis at the origin and one of its sides is drawn along the positive x-axis. The side that isdrawn on the positive x-axis is called the initial side. The other side of the angle iscalled the terminal side. Refer to Figure A.3.

Figure A.4 shows a 90° angle, a 160° angle, and a 250° angle.

FIGURE A.3 FIGURE A.4

Because a 90° angle is one-quarter of a rotation (one quarter of the way around thefull circle), a 180° angle would be one-half of a rotation, a 270° angle would be

x

y

90°160°

250°

x

y

θ Radius

= 1

Initial side

Terminal side

13

18

45°120°

13

18

13

18


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A4 APPENDIX A

three-quarters of a rotation, and a 360° angle is one full rotation. Note that in FigureA.4, the 160° angle is drawn between a quarter rotation and a half rotation, and the250° angle is drawn between a half rotation and a three-quarter rotation. You shouldalso note that all of the rotations are drawn in the counterclockwise direction. Weconsider counterclockwise rotation to define a positive angle and clockwise rotationto define a negative angle. For example, Figure A.5 illustrates a �90° angle and a �200°angle.

When angle measures are larger than 360°, they describe more than one rotationaround the circle. For example, an 810° angle describes two full rotations plus one-quarter of a rotation in the counterclockwise direction (2 � 360 � 90 � 810). A�505° angle describes one full rotation plus one-quarter of a rotation plus an extra�55° angle in the clockwise direction (�360 � 90 � 55 � �505). These two angles areshown in Figure A.6.

FIGURE A.6

Before defining trigonometric functions for angles on a unit circle, we take a brieflook at the trigonometric functions defined in terms of a right triangle.

Right-Triangle Trigonometry

From the early use of trigonometry (meaning “triangle measurement”) throughpresent-day applications, right triangles have provided a method for solving prob-lems that involve indirect measurements. A right triangle is a triangle with one 90°angle. In applications, we generally label one of the remaining angles of the right tri-angle as angle �. (See Figure A.7.)

We define the sine of the angle � as the ratio of the length of the leg opposite theangle to the length of the hypotenuse (the side opposite the right angle). The cosineof the angle � is the ratio of the length of the adjacent leg to the length of thehypotenuse. The tangent of the angle � is the ratio of the length of the leg oppositethe angle � to the length of the adjacent leg. We abbreviate cosine � by cos �, sine � bysin �, and tangent � by tan �.

x

y

810°

x

y

-505°


x

y

-90°-200°

FIGURE A.5

Leg adjacent to θ

Leg

opp

osite

θ

Hypotenuse

θ

FIGURE A.7

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These three functions are illustrated in Figure A.8.

FIGURE A.8

Three other functions that are commonly studied in trigonometry are the secant,cosecant, and cotangent functions. They are defined as follows:

secant �: sec � � �

cosecant �: csc � � �

cotangent �: cot � � �

We study primarily sin �, cos �, and tan � because the other three trigonometricfunctions are defined in terms of these first three.

If we are given any right triangle for which we know the angle � and know thelength of one side of the triangle, we can use the trigonometric functions sin �, cos �,and tan � to obtain the lengths of the other two sides of the triangle. For example, ifwe have a right triangle with a 25° angle and hypotenuse of length 2 inches, we canuse sin 25° and cos 25° to find the lengths of the two legs. (See Figure A.9.)

length of the adjacent leglength of the opposite leg

1tan �

length of the hypotenuselength of the opposite leg

1sin �

length of the hypotenuselength of the adjacent leg

1cos �

Opp

osite

leg

Hypotenuse

θ

sin θ = opposite leghypotenuse

Adjacent leg

Hypotenuse

θ

cos θ = adjacent leghypotenuse

Adjacent leg

Opp

osite

leg

θ

tan θ = opposite legadjacent leg


Right-Triangle Trigonometric Definitions

For a right triangle with one of the non-90° angles having measure �, thesine, cosine, and tangent of � are defined in terms of the sides of the triangle.

sin � �

cos � �

tan � �length of the leg opposite the angle �

length of the leg adjacent to the angle �

length of the leg adjacent to the angle �length of the hypotenuse

length of the leg opposite the angle �length of the hypotenuse

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A6 APPENDIX A

FIGURE A.9

To find the length of the leg adjacent to the 25° angle, we use the fact that cos 25° isthe ratio of the length of the adjacent leg to the length of the hypotenuse. We will usea to represent the length of the adjacent leg, so cos 25° � . Solving for a, we have

a � 2 cos 25° � 2(0.90631) � 1.81262 inches

That is, the leg adjacent to the 25° angle is approximately 1.8 inches long. Similarly,the length of side b can be found as

b � 2 sin 25° � 2(0.42262) � 0.84524 inch

Trigonometry is often used in applications where indirect measurement of thelength of one or more legs of a triangle is necessary. Land surveying often relies onthe use of an instrument called a sextant to measure angles and on trigonometry todetermine distances using right triangles.

EXAMPLE 2 Using Trig to Determine Measurement

Castle Measurement A soldier in ancient times making use of triangle ratios isillustrated in Figure A.10. He is using a sextant and sighting it in line with the top ofthe castle wall. The sextant gives an angle of 14.6°. The soldier then counts the bricksin the wall and estimates the distance d without having to enter the battle zonebetween his safe spot and the castle wall.

FIGURE A.10

a. Estimate the height of the wall if each brick is 11 inches tall.

b. Find the distance to the castle.

c. Find the distance from the soldier to the top of the castle wall.

h

θ = 14.6°d

a2

a

b2 inches

25°


A.1

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Solution

a. There are 22 layers of bricks in the wall, so the wall is (22 bricks)(11 inches perbrick) � 242 inches � 20.2 feet tall.

b. Using the definition of the tangent of an angle, we know that

tan 14.6° �

so we have

d � � � 77.4 feet

The soldier is approximately 77.4 feet away from the castle.

c. The Pythagorean Theorem yields

h � � � 80 feet

The soldier is approximately 80 feet away from the top of the castle wall. ●

Two special right triangles that occur in applications using trigonometry are the30°�60° right triangle and the 45°�45° right triangle. The values of sine, cosine,and tangent for one of these triangles are explored in the next example. The other tri-angle is left for an activity.

EXAMPLE 3 Determining Ratios for a Special Triangle

The 30°–60° right triangle is so named because in a right triangle with a 30° angle,the remaining angle must be 60°. A reflection of the triangle across the leg betweenthe 30° angle and the right angle forms an equilateral triangle (one whose three sidesare of equal length). The line of reflection is the perpendicular bisector of the base.(See Figure A.11.)

a. Use the Pythagorean Theorem to find the length h in terms of s.

b. Use the sides h and s to find the sine, cosine, and tangent for the 30° angle in a 30°–60° right triangle.

c. Use the sides h and s to find the sine, cosine, and tangent for the 60° angle.

Solution

a. The side opposite the 30° angle has length s that is equal to half of the length 2s ofthe hypotenuse. Applying the Pythagorean Theorem yields

h2 � s2 � (2s)2

Solving for h, we find the remaining leg length to be

h � � � � s�3�3s 2�4s2 � s 2�(2s)2 � s 2

�6400.7�20.22 � 77.42

20.2 feet0.26048

20.2 feettan 14.6°

20.2 feetd


30°

2s

h

s

s

60°

60°

FIGURE A.11

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A8 APPENDIX A

b. Referring to Figure A.12, we find the values of sine, cosine, and tangent for the30° angle in the right triangle.

sin 30° � �

cos 30° � �

tan 30° � �

c. Similarly, the values of sine, cosine, and tangent for the 60° angle follow:

sin 60° � �

cos 60° � �

tan 60° � � ●

Unit-Circle Trigonometry

Right-triangle trigonometry is useful in many applications. However, as we previ-ously noted, the trigonometric functions are not restricted to use with angles that areless than 90°.

We can now define the trigonometric functions sine, cosine, and tangent for allangles by referring to our knowledge of right-triangle trigonometry. When the angle� is between 0° and 90°, we draw a right triangle in the unit circle such that thehypotenuse of the triangle is along the terminal side of the angle � from the origin tothe unit circle, and the other two legs of the right triangle are drawn along the x-axisand perpendicular to the x-axis, as shown in Figure A.13.

If we let (x, y) represent the point where the terminal edge of angle � intersectsthe unit circle, then the leg drawn along the x-axis has length x and the leg drawn per-pendicular to the x-axis has length y. The hypotenuse has length 1 (the radius of theunit circle). We define the trigonometric functions sin �, cos �, and tan � as we didfor the right triangle, so that

sin � � � y

cos � � � x

tan � �

We define the trigonometric functions similarly for angles that are not between0° and 90°. First, draw the angle on the unit circle. Then draw a right triangle suchthat the hypotenuse of the triangle is along the terminal side of the angle � from theorigin to the unit circle, and the other two legs of the right triangle are drawn along

yx

x1

y1

�3�3s

s

12

s2s

�32

�3s2s

1

�3

s

�3s

�32

�3s2s

12

s2s


x

y

x

y1

θ

FIGURE A.13 Right triangleassociated with angle �

30°

2s s60°

√3s

FIGURE A.12

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the x-axis (possibly in the negative x direction) and perpendicular to the x-axis (pos-sibly in the negative y direction) as shown in Figures A.14a and b.

FIGURE A.14

Once again, call the point where the terminal edge of angle � intersects the unitcircle (x, y), and let a represent the length of the leg drawn along the x-axis and let brepresent the length of the leg drawn perpendicular to the x-axis. The hypotenuse haslength 1 (the radius of the unit circle). However, we must indicate whether the legs ofthe triangle are drawn in the negative x and/or the negative y direction. We indicatethis by writing the negative of the length in the appropriate direction. For example,the triangle in Figure A.14a has one leg along the negative portion of the x-axis. Thisleg has length a in the negative x direction, so x � �a. The other leg is drawn up fromthe x-axis; thus it has positive direction and has length y � h.

We define the trigonometric functions sin �, cos �, and tan � as we have done pre-

viously, so that sin � � � y, cos � � � x, and tan � � . Thus, for the angle

drawn in Figure A.14a, sin � � b, cos � � �a, and tan � � � . Similarly, for the

angle drawn in Figure A.14b, sin � � �d, cos � � �c, and tan � � � .

In general, for any angle �, we define sine, cosine, and tangent as follows:

Because the unit circle is given by the equation x2 � y2 � 1, it is true that for anyangle �, cos2 � � sin2 � � 1.

dc

�d�c

�ba

b�a

yx

x1

y1

x

y

-a

b1

(x, y) = (-a, b)

θx

y

-d

-c

1

θ

(x, y) = (-c, -d)

(a) (b)


Unit-Circle Trigonometric Definitions

Let (x, y) be the point on the unit circle where the terminal side of the angle �intersects the circle. Then the sine, cosine, and tangent of the angle � are

sin � � y

cos � � x

tan � �yx

A.2

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A10 APPENDIX A

EXAMPLE 4 Determining Ratios for a Specific Angle

A �420° angle is shown in Figure A.15.

a. Draw the right triangle associated with a �420° angle. Label the angle in the trian-gle between the terminal side and the x-axis.

b. Find the sine, cosine, and tangent of a �420° angle.

Solution

a.

FIGURE A.16 Right triangle associated with the �420° angle

b. This right triangle is one of the special triangles mentioned in Example 3. It is a

30°-60° right triangle. We know from Example 3 that sin 60° � and

cos 60° � . Thus the leg on the x-axis is positive and has length cos 60° � , so

x � . The other leg is drawn down from the x-axis, so it’s measure is negative and

has length sin 60° � , so y � . Place these values on Figure A.16. (See FigureA.17.) Now we can read the trigonometric values from the figure:

sin(�420°) � y �

cos(�420°) � x �

tan(�420°) � � � ●

We have now defined trigonometric functions as they apply to right trianglesand, in the much broader sense, as they apply to unit circles. However, our discussionof unit-circle trigonometry would not be complete if we did not consider anotherangle measure called radian measure. In fact, we cannot use the trig functions in cal-culus without considering radian measure of angles.

Radian Measure

Another unit of angle measurement is called a radian. Picture a circular pizza. We candescribe a slice of pizza in terms of the angle of the wedge formed by the slice. For

�3yx

12

��32

��32

�32

12

12

12

�32

x

y

60°


x

y

-420°

FIGURE A.15

x

y

12

1-√3

2

FIGURE A.17

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instance, in angular terms we would describe one-eighth of a pizza as a 45° wedge. Aone-sixth slice would be described as a 60° wedge (see Figure A.18).

Another way to think of a slice of pizza is in terms of the amount of crust aroundthe edge. If there were something special in the crust, like a cheese filling, then wemight want to focus on the length of the crust around the circular edge of the slice. Indoing so, we would be talking about an arc of a circle—that is, the slice’s portion ofthe circumference. (A portion of the circle is called an arc, and the complete distancearound the circle is its circumference.) If the radius r of the pizza in Figure A.18 were1 foot, then the circumference of the pizza would be 2�r � 2�(1 foot) � 2� feet �6.3 feet. Basic geometry tells us that the arc length is the same fractional part of thecircumference of the circle as the angle is of one complete rotation. Thus the 45°

wedge would form an arc of (2� feet) � 0.8 foot of the special cheese crust. The

60° wedge would form an arc of (2� feet) � 1 foot.We define the radian measure of an angle to be the ratio of the length of the arc s

cut out of the circumference of a circle by the angle � to the radius r of the circle. (SeeFigure A.19.) Because the total arc length (that is, the circumference) of a circle is 2�

times its radius, a full revolution has radian measure of � 2�. (See Figure A.20.)Note that 2� is a real number approximately equal to 6.283. When we are using aunit circle, the radius is 1, so the radian measure is simply the corresponding arc

length: � � � � s.

It is important to recognize that the radian measure of an angle is a real numberwith no units attached. In the definition of radian measure, both s and r measurelength and must have the same units, which causes � to be a unitless quantity. How-ever, we sometimes report this measure as � radians.

Figure A.21 shows some common radian measures on a circle.

s1

sr

2�rr

60°360°

45°360°


FIGURE A.18 ofa pizza is a 60° wedge.

16

�

r

s

FIGURE A.19 Theradian measure of anangle is the ratio ofthe arc length to theradius.

� �sr

Radian Measurement

2� radians � 1 complete counterclockwise rotation

1 radian � counterclockwise rotation (a little less than revolution)

FIGURE A.20

= 2π radians = 1 radian

(a) (b)

16

12�

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A12 APPENDIX A

EXAMPLE 5 Understanding Radian Measure of Angles

a. Draw angles with radian measure and 5 on a unit circle.

b. Estimate the radian measure of the two angles shown in Figures A.22a and b.

FIGURE A.22

Solution

a. An angle of is half of

a clockwise rotation plus another of a clockwise rotation

(see Figure A.23a). An angle of 5 is between of a counter-

clockwise rotation and a complete rotation

(2� � 6.3) (see Figure A.23b).

�3�2 � 4.7�

34

18

-5�4 � ��

�4 � � �1

2(2�) �18(2�)�

(a) (b)

-5�4

7π4

π4

5π4

π

π2

0, 2π

3π4

3π2


FIGURE A.21 The angle � corre-

sponds to of a complete rotation,

or (2�) � units around the

circle. Thus its radian measure is .3�4

3�4

38

38

FIGURE A.23

−5π 4

=

= 5

(a) (b)

4362_HMC_App A_A001-A026 12/26/03 2:13 PM


b. The angle in Figure A.22a appears to be approximately of a rotation, or

radian. The angle in Figure A.22b is �3�: one complete clockwise rotation (�2�)plus a half of a clockwise rotation (��). ●

EXAMPLE 6 Measuring Angles in Radians

a. Express of a complete rotation in radians.

b. Express of a complete rotation in radians.

c. Express 4� radians in terms of complete rotations.

Solution Keeping in mind that 1 complete rotation is 2� radians, we have

a. (2�) � radians

b. (2�) � radians

c. Because 1 radian � of a complete revolution, 4� radians is 4� � 2

complete rotations. ●

The odometer on an automobile converts turns of the drive shaft into distancesso that you can observe the number of miles you have traveled. Although the nextexample explores this idea, it is not unique to automobiles. Whenever wheels rotateto cause movement, the same type of relationship between rotations of the wheelsand distance traveled holds true.

EXAMPLE 7 Using Radius to Determine Arc Length

Rotating Wheels Consider a front-wheel-drive automobile with wheels that are20 inches in diameter.

a. How far will 1 rotation of the wheels cause the automobile to travel?

b. How many times will the wheels revolve when the automobile travels 1 mile?

Solution

a. As an automobile travels, its drive shaft and front tires turn at the same rate; thatis, 1 revolution in one causes 1 revolution in the other. If the wheels are 20 inchesin diameter (radius � 10 inches), then each rotation causes the automobile tomove forward 2�r � (2�)(10 inches) � 62.8 inches � 5.2 feet. Each rotation ofthe wheels causes the automobile to travel about 62.8 inches or 5.2 feet.

b. There are 5280 12 � 63,360 inches in a mile. So

63,360 � 1008.4

The wheels must revolve approximately 1008.4 times in each mile traveled. ●

revolutionsmile�1 revolution

2��10 inches��inchesmile�

�inchesfoot��feet

mile�

� 12��1

2�

�2

14

�4

18

14

18

2�12 � 0.5

112


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A14 APPENDIX A

Angle Measure Conversions

Whenever you have two measurement scales that can be applied to the same object, itis important to have a conversion technique. In this case, 1 revolution is both 360°

and 2� radians. Thus degrees are converted to radians by multiplying by � ,

and radians are converted to degrees by multiplying by . Our conversion for-

mula is

Up to this point, we have used the word radian to specify our choice of anglemeasure, not the units of the angle. You may have noted that it is cumbersome towrite the word radians each time angle measure is used. Therefore, we adopt the fol-lowing convention.

That is, an angle with measure 60 is quite different than one with measure 60°. Anangle of 60 radians is more than 9 full revolutions, whereas an angle of 60° is onlyone-sixth of a revolution. If you mean an angle of 60 degrees, be certain that you usethe degree symbol with the angle.

EXAMPLE 8 Converting Angles

a. Express in degrees.

b. Express 135° in radians.

Solution

a. � � 210°

b. 135° � (135°) � ●3�

4��

180°��180°

��7�

6�7�

6

7�6

Angle Measure Convention

All angles are understood to be measured in radians unless the degree symbolis used to specify degree measure of the angle.

Angle Conversion Formula

� radians � 180°

180�

�180

2�360


A.3a, b

4362_HMC_App A_A001-A026 12/26/03 2:13 PM


Sine and Cosine Functions

We now define sine and cosine functions of angles in standard position on a unit cir-cle. The box also contains a very important trig identity that follows from the factthat the equation of the unit circle at the origin of the x- and y-axes is x2 � y2 � 1.

Figure A.25 shows the four points where the unit circle intersects the axes, thecorresponding angles in radians, and the values of the sine and cosine functions.

FIGURE A.25

Any point on the unit circle has multiple corresponding angles with the same initial

and terminal sides. For example, an angle of is 1 complete revolution plus a quar-

ter of a revolution 2� � , resulting in the same point on the unit circle as the

angle . Hence cos � cos � 0.�2

5�2

�2

��2�

5�2

� = , ,

(1, 0)(-1, 0)

(0, 1)

(0, -1)

cos 0 = 1, sin 0 = 0� = 0,sin π = 0� = π, cos π = -1,

π2_ π

2_cos = 0 π

2_sin = 1

� = , 3π2_3π

2_ cos = 0 3π

2_sin = -1,


Trigonometric Values for a General Angle

Consider an angle � in standardposition. The terminal side of theangle intercepts the unit circle ata point (x, y). The sine andcosine functions are defined asfollows:

f(�) � sin � is the functionwhose output is the y-coordinate.

g(�) � cos � is the functionwhose output is the x-coordinate.

Identity: cos2 � � sin2 � � 1 for any angle �

FIGURE A.24

�

(x, y) = (cos �, sin �)

(1, 0)x

y

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A16 APPENDIX A

The values of the sine and cosine functions are not obvious for points other thanthose shown in Figure A.25. You can use a calculator or computer to find other sineand cosine values, as illustrated in Example 2. Be sure your technology is set in radianmode.

EXAMPLE 9 Calculating Sine and Cosine Values

a. Find sin and cos .

b. Interpret your answer to part a in terms of the coordinates of a point on the unitcircle.

c. Find two other angles whose sine and cosine are the same as those in part a.

Solution

a. Using a calculator or computer you should find that sin � �0.38268 and

cos � �0.92388.

FIGURE A.26

b. The cosine and sine values are the x- and y-coordinates of the point where the ter-

minal side of the angle intersects the unit circle. This angle and the correspond-

ing point are shown in Figure A.26. Note that because the x- and y-coordinates ofthe point are negative, both the cosine and the sine of this angle are negative.

c. We seek two other angles corresponding to the point where the terminal side of the angle shown in Figure A.26 intersects the unit circle. One such angle is

obtained by a clockwise rotation from the positive x-axis. This angle is 2� � �

and is expressed as a negative number to denote the direction. Thus the

angle is , and sin � sin and cos � cos .

It is also possible to reach the point under consideration by going around the

circle 1 full rotation counterclockwise and then an additional radians. This

angle is 2� � � . Thus sin � sin and cos � cos . There are 9�8

25�8

9�8

25�8

25�8

9�8

9�8

9�8��7�

8�9�8��7�

8��7�8

7�8

9�8

9�8

(-0.92388, -0.38268)

x

y

9π8_

9�8

9�8

9�8

9�8


Instructions for using technologywith this example are given inSection 8.1.1 of the technologysupplement.

4362_HMC_App A_A001-A026 12/26/03 2:13 PM


infinitely more positive and negative angles corresponding to the point indicatedin Figure A.26, all with a sine value of approximately �0.38268 and a cosine valueof approximately �0.92388. ●

So far we have referred to f(�) � sin � and g(�) � cos � as functions, but we havenot verified that this is the case. To verify that f(�) � sin � is a function, we must ask,“Can sin � have more than one output value for a particular angle � (the input)?”Because the terminal side of the angle � intersects the unit circle at a single point,there will be only one output f(�) � sin � corresponding to each angle �. Thus f(�) � sin � is indeed a function. Similar reasoning confirms that g(�) � cos � is alsoa function.

Be careful not to confuse angle inputs of these functions. Even though angles

such as and have the same initial and terminal sides and therefore the same

trigonometric function outputs, the two angles are not the same: is of a

revolution, whereas represents 1 complete revolution plus additional

revolution. For cyclic functions, infinitely many distinct inputs correspond to thesame output.

In a graph of the sine function, the horizontal axis represents the angle measuresand the vertical axis represents the y-coordinates on the unit circle. For the cosinefunction, the horizontal axis represents the angle measure and the vertical axis repre-sents the x-coordinate on the unit circle.

Refer to Figure A.25. If we plot the angles shown in that figure and the corre-sponding y-coordinates, we obtain the graph shown in Figure A.27. We also add thepoint (2�, 0) corresponding to 1 complete revolution.

We add more points to this graph by calculating some intermediate values forthe sine function. Table A.1 shows selected angles between 0 and 2� (correspondingto sixteenths of a revolution) and the associated y-coordinates (to five decimalplaces) on a unit circle. Figure A.28 shows the points in Table A.1 added to the graphin Figure A.27.

TABLE A.1

112

13�6

112

�6

13�6

�6


FIGURE A.27 Points on the graphof f(�) � sin �

�

f (�)

π2_ 3π

2_

1

-1

π 2π

� sin � � sin �

0.38268 �0.38268

0.70711 �0.70711

0.92388 �0.92388

0.92388 �0.92388

0.70711 �0.70711

0.38268 �0.38268

� 0 2� 0

15�

87�

8

7�

43�

4

13�

85�

8

11�

83�

8

5�

4�

4

9�

8�

8

4362_HMC_App A_A001-A026 12/26/03 2:13 PM

A18 APPENDIX A

FIGURE A.28 FIGURE A.29

Because the y-coordinates increase, decrease, and increase again and take on allreal number values between �1 and 1 as we move around the circle, we expect the sinefunction graph to increase and decrease in a smooth, continuous manner, taking onall values between �1 and 1. We therefore connect these points with a smooth curve toobtain the graph of the sine function shown in Figure A.29.

Once the angle exceeds 2�, we begin retracing the unit circle, and the sine func-tion begins repeating itself. Figure A.30 shows a sine graph with more of the repeti-tions. This graph extends infinitely far in both directions.

FIGURE A.30

The sine function is periodic because it repeats itself every 2� input units. Theperiod of the sine function is 2�. The sine function is also cyclic, because it variescontinuously, alternating between �1 and 1. The portion of the sine function overone period—that is, the part that keeps repeating itself—is called a cycle of thefunction.

Although we have viewed the sine and cosine functions as having angle measureas input, we are not restricted to this interpretation. In fact, we can consider the sineand cosine functions as having any real number as the input. Because most applica-tions to which we will apply a sine model have input that is not an angle measure,we will no longer use � to denote the input. Instead we use the notation y � sin xand y � cos x. Do not confuse x and y here with points on the unit circle. The vari-able x is simply the input, which can be interpreted as an angle measured in radians,and the variable y is the output, which can be interpreted as the x- or y-coordinate onthe unit circle, depending on whether we are considering a cosine function or a sinefunction.

�

f (�)

π2_π

2_ 3π

2_3π

2_

1

-1

π-π 2π 5π2_ 3π 7π

2_ 4π----2π

f (�) = sin �

�

f (�)

π2_ 3π

2_

1

-1

π 2π

f (�) = sin �

�

f (�)

π2_ 3π

2_

1

-1

π 2π


4362_HMC_App A_A001-A026 12/26/03 2:13 PM



A.1 Concept Inventory

Degree measure of angles

Initial and terminal sides and standard position of angles

Right-triangle trigonometry: sine, cosine, and tangent

Unit-circle trigonometry: sine, cosine, and tangent

Radian measure of angles

Angle measure conversion

Sine and cosine functions of a general angle

A.1 Activities

1. Because the number 360 has so many integer divi-sors, many fractional revolutions have nice expres-sions when expressed as angles measured in degrees.Complete the table of fractional rotations or fullrotations and their associated degree measures.

2. There are many easily expressed fractions of a fullturn, and the measure of these expressions in radi-ans is easy to determine. Complete the table of frac-tional rotations or full rotations and their associ-ated radian measures.

Rotation (in turns)

Angle measure (in degrees)

15° 45° 60° 120°

Rotation (in turns)


135° 180° 225° 240° 270° 300°

Rotation (in turns) 1 3 60


720° 3780°

78

56

23

12

38

13

14

112

124

Table for Activity 1

Rotation (in turns)

Angle measure (in radians)

Rotation (in turns)


�

Rotation (in turns) 1 3 60


4� 10.5�

78

5�

33�

25�

4

56

23

12

38

�

3�

4�

12

13

14

112

124

Table for Activity 2

4362_HMC_App A_A001-A026 12/26/03 2:13 PM

A20 APPENDIX A

3. Convert the following angles in degrees into anglesin radians, and sketch the angles on a unit circle.

a. 110° b. 700° c. 90° d. 0.01°

4. Convert the following angles in radians to angles indegrees, and sketch the angles on a unit circle.

a. 2� b. c. d. 30

5. Cycling Superbike II, the $15,000 baby of the U.S.Cycling Federation (USCF), made its debut in the1996 Olympics. Superbike II’s front wheel is 23.62inches in diameter, and its rear wheel is 27.56inches in diameter. Over a distance of 1 mile, howmany turns does each wheel make?

6. Auto Wheels Use degree measure to write theangle that the tenths wheel on the odometer turnswhen the drive shaft of the automobile in Example6 completes one turn.

7. Ferris Wheel Consider a Ferris wheel on whichare 30 equally spaced seats.

a. Through what angle does the wheel movebetween stops to release passengers in two con-secutive seats?

b. What is the radian measurement of this angle?

c. If the Ferris wheel is 100 feet in diameter, what isthe distance traveled by each of the seats as theoperator stops between consecutive seats toexchange passengers?

8. Auto Wheels Suppose the owner of the automo-bile in Example 6 replaced the tires with oversizedtires that were 22 inches in diameter.

a. What would be the error created on the odome-ter readings?

b. If the tires were replaced with tires that have an18-inch diameter, what would be the error causedon the odometer?

c. If the regular tires (20-inch diameter) lost 0.001inch of their diameter through wear, how wouldthe odometer readings be affected?

9. For finer measurements, the degree is furtherdivided into 60 parts, and each such part is called aminute. The symbol used to indicate minutes is thesingle prime; for example, 15 minutes is expressedas 0°15. An angle consisting of 200 minutes couldalso be expressed as 3°20. For even finer measure-ments, the minute is further divided into 60 equal

7�16

3�8


parts called seconds. The symbol for a second is thedouble prime, so an angle of 30 degrees 14 minutes7 seconds would be written as 30°147�. You mighttry to estimate how small a second is to realize howmuch precision is used when some jobs stateacceptable tolerances in terms of a seconds of arc.Convert into decimal degree equivalents the follow-ing angles given in degrees, minutes, and seconds.(Note: The Saturn missile used in the moon shotshad guidance computers placed on a stable gim-baled platform that had to stay within 4 seconds ofarc for the first 15 minutes of the launch.)

a. 5°3030� b. 35°1217�

10. Refer to the definitions in Activity 9, and convertinto measures using degrees, minutes, and secondsthe following angles given in decimal degrees.

a. 5.1234° b. 125.365°

11. Suppose you were to divide a rotation into 1776pieces. We will call each piece a “patriotic unit” andwill use the abbreviation PU.

a. How many rotations does each of the followingpatriotic unit measurements represent?

i. 1776 PU ii. 1332 PU

iii. 888 PU iv. 444 PU

v. 222 PU vi. 111 PU

b. Convert each of the patriotic unit measurementsin part a to radians.

c. Find the value of each of the following:

i. cos(1776 PU) ii. sin(1776 PU)

iii. cos(666 PU) iv. sin(666 PU)

12. Clock Creating different angle measurement sys-tems such as degrees and grads for partial turns andcalculating their conversion factors is like making a25-hour clock. Mr. Morton Rachofsky has built a 25-hour clock for the Circadian Clock Company.This clock divides the 86,400 seconds in the standardday into 25 equal-length periods called “hours.”Noon on the clock is the same as noon on our regu-lar time scale, but the other hour marks are different.Scientists conducting experiments in the 1930sobserved people in caves where they could not seethe sun. These people developed activity cycles thatlasted 25 hours. Because we cannot change the solarday, Mr. Rachofsky said,“Why not change the clock?”(Source: New York Times. October 27, 1996, p. 47.)

4362_HMC_App A_A001-A026 12/26/03 2:13 PM


a. How long are Mr. Rachofsky “hours” in our reg-ular minutes?

b. What time on Mr. Rachofsky’s clock is the regu-lar clock time 3:00 P.M.?

c. What is the regular clock time when it is 6:00P.M. on Mr. Rachofsky’s clock?

For Activities 13 through 18, identify the trigonometricfunction you would use to find the length of the indi-cated side. Assume that the triangle given is a right tri-angle with one of the other two angles identified asangle �.

13. Given the length of the hypotenuse, find the lengthof the leg adjacent to angle �.

14. Given the length of the leg opposite angle �, findthe length of the leg adjacent to angle �.

15. Find the length of the leg opposite angle �, giventhe length of the leg adjacent to angle �.

16. Find the length of the hypotenuse, given the lengthof the leg opposite angle �.

17. Given the length of the leg adjacent to angle �, findthe length of the hypotenuse.

18. Given the length of the hypotenuse, find the lengthof the leg opposite angle �.

For Activities 19 through 24, solve for the length of theindicated side. Assume that the triangle given is a righttriangle.

19. One angle of the triangle is 20°. The leg oppositethis angle is 5 inches long.

a. Find the length of the leg adjacent to the givenangle.

b. Find the length of the hypotenuse.

20. One angle of the triangle is 78°. The leg adjacent tothis angle is 1 meter long.

a. Find the length of the hypotenuse.

b. Find the length of the leg opposite the givenangle.

21. One angle of the triangle is 15.2°. The hypotenuseis 12 centimeters long.




22. One angle of the triangle is 30.75°. The leg oppositethis angle is 5 inches long.


b. Find the length of the hypotenuse.

23. One angle of the triangle is 85.4°. The leg adjacentto this angle is 1.5 miles long.

a. Find the length of the hypotenuse.


24. One angle of the triangle is 10.2°. The hypotenuseis 3 centimeters long.



25. Roof A gable is the triangular segment of a wallcreated by the roof. The pitch of a gable is its heightdivided by its width. See figure. (That is, pitch isequal to half the slope of the roof.) Consider a gable40 feet wide feet with an angle of 130° at its peak.

a. How tall is the attic at its center?

b. What is the pitch of the gable?

c. How much board would be needed to cover theroof (not including the overhanging portion ofthe roof) if the house is 40 feet long?

26. Nuts Find the diameter of the smallest iron rodfrom which a hexagonal nut with side 4mm can becut. (Hint: The angle between two adjacent sides ofthe nut is 120°.)

27. Stairway A stairway is to be constructed on a hillwith a 34° incline.

a. If each step is to have a 7-inch rise, what must beits tread (horizontal depth)?

Width

Height

4362_HMC_App A_A001-A026 12/26/03 2:13 PM

A22 APPENDIX A

b. How many 7-inch steps will be needed if thelength of the hill (measured up the slope) is 4feet 2 inches?

28. Navigation A boat that is sailing S 47°E is sailingon a trajectory that is along an angle 47° east of truesouth. A ship sails 12.8 nautical miles S 47°E fromits starting position.

a. How far south has the ship sailed?

b. How far east has it sailed?

29. Navigation Air Force pilots mark their bearing asthe clockwise angle measured from the north. Forexample, a bearing of 90° is east and a bearing of180° is south. There is a landing strip 5 miles awayfrom a plane at bearing 332°.

a. What bearing is west?

b. In what direction must the plane fly to reach thelanding strip?

c. If the plane were to fly directly north or southand then east or west to reach the landing strip,how far in each direction would the plane haveto fly?

30. Because the sum of the three angles of any triangleis 180°, a right triangle with one 45° angle mustcontain another 45° angle. As shown, this type oftriangle is formed by two sides and a diagonal of asquare. Thus the two legs of the triangle are ofequal length. By the Pythagorean Theorem, thehypotenuse must have length

� �

Using the lengths s and , determine the exactvalues of the following trigonometric ratios.

a. sin 45° b. cos 45° c. tan 45°

31. Use the values for sin 45° and cos 45° to determinethe sine, cosine, and tangent values of each of thefollowing angles.

a. 315° b. �135° c. �225°

�2s

s

ss√2

45°

45°

�2s�2s2�s2 � s2

For Activities 32 through 39, sketch the given angle onthe unit circle, and draw the appropriate right trianglecorresponding to this angle. Calculate the sine andcosine of the angle, and indicate these values appropri-ately on the sketch of the triangle.

32. 160° 33. 415° 34. 920° 35. 310°

36. �489° 37. �945° 38. �280° 39. �37°

40. Calculate sin 180° and cos 180° and explain youranswers in terms of the unit circle. Do the samething for sin 90° and cos 90°.

41. a. Use the graph to estimate sin 4, cos 4, sin ,

and cos .

b. Mark the estimated locations of the angles

� � �1 and � � on the figure, and use the

grid to estimate the values of sin(�1), cos(�1), sin

, and cos .

42. a. Use the graph to estimate sin 2.5, cos 2.5,

sin , and cos .

b. Mark the estimated locations of the angles

� � and � � on the figure, and use the

grid to estimate the values of sin , cos ,

sin , and cos .��5�12��5�

12��1

2��12�

�5�12

�12

��15�16��15�

16�

144�16

144�16

144�16

1

−1

−1

1 x

y

� = 4–�8

� =

��8�

��8�


4362_HMC_App A_A001-A026 12/26/03 2:13 PM



43. a. Find the values of sin and cos .

b. Interpret each answer to part a in terms of thecoordinates of a point on the unit circle.

c. Give two other angles whose sine and cosine val-

ues are the same as those for � � .

44. a. Find the values of sin 15 and cos 15.

b. Interpret each answer to part a in terms of thecoordinates of a point on the unit circle.

c. Give two other angles whose sine and cosine val-ues are the same as those for � � 15.


b. Interpret the answers to part a in terms of theunit circle.

c. Give three other angles whose sine and cosine

values are the same as those for � � .


b. Interpret the answers to part a in terms of theunit circle.

c. Give three other angles whose sine and cosine

values are the same as those for � � .

47. Is it possible for the trigonometric functions of anangle � to be cos � � 0.5 and sin � � �0.5? Explain.

48. Is it possible for the trigonometric functions of anangle � to be cos � � 0.35 and sin � � 0.82?Explain.

4�9

4�9

4�9

3�2

3�2

3�2

2�3

2�3

2�3

1

−1

−1 1 x

y

� =

� = 2.5

–15�16

49. a. Without using technology, indicate whether eachof the following values is positive, negative, or zeroby using its interpretation as the y-coordinate ofa point on the unit circle.

b. Use your calculator or computer to complete thefollowing table. Compare the signs of the valuesto your results from part a.

c. Plot the values you obtained in part b as a func-tion of the angle. How is this plot related to thegraph of the function f(x) � sin x?

50. a. Without using technology, indicate whethereach of the following values is positive, negative,or zero by using its interpretation as the x-coordinate of a point on the unit circle.

Trig value sin 0 sin sin sin sin �

Sign

Trig value sin sin sin sin 2 �

Sign

7�4

3�2

5�4

3�4

�2

�4

Trig value sin 0 sin sin sin sin �

Decimal value

Trig value sin sin sin sin 2�

Decimal value

7�4

3�2

5�4

3�4

�2

�4

b. Use technology to complete the following table.Compare the signs of the values to your resultsfrom part a.

Trig value cos 0 cos cos cos cos �

Sign

Trig value cos cos cos cos 2 �

Sign

7�4

3�2

5�4

3�4

�2

�4

Trig value cos 0 cos cos cos cos �

Decimal value

Trig value cos cos cos cos 2 �

Decimal value

7�4

3�2

5�4

3�4

�2

�4

4362_HMC_App A_A001-A026 12/26/03 2:13 PM

4362_HMC_App A_A001-A026 12/26/03 2:13 PM

Answers to Appendix A A25

Answers to Appendix A

TRIGONOMETRY BASICS

1. Rotation (in turns): , , , , 2,

Angle measure (in degrees): 30, 90, 315, 360, 1080, 21,600

3. a. radians

b. radians

c. radians

d. 0.000056 radian

5. The front wheel makes approximately 853.86 turns in 1

mile; the rear wheel makes approximately 731.79 turns in

1 mile.

7. a. 12° b. radian c. Approximately 10.47 feet

9. a. Approximately 5.508°

b. Approximately 35.2047°

11. a. i. 1 rotation ii. rotation iii. rotation

iv. rotation v. rotation vi. rotation

b. i. 2� radians ii. 1.5� radians iii. � radians

iv. radians v. radians vi. radians

c. i. 1 ii. 0 iii. �0.707 iv. 0.707

13. cos �

15. tan �

17. cos �

19. a. 13.737 inches

b. 14.619 inches

21. a. 11.580 centimeters

b. 3.146 centimeters

23. a. 18.703 miles

�8

�4

�2

116

18

14

12

34

�15

0.000056

π2_�

2

35π9

___35�

9

11π18___11�

18

212

34

58

16

18

b. 18.643 miles

25. a. 9.326 feet

b. 0.233

c. 1765.4 square feet

27. a. 10.38 inches

b. Four steps

29. a. Bearing 270°

b. Northwest

c. 4.415 miles north and 2.347 miles west

31.

33.

35.

37.

39.

33.

35.

x

y

310°

-0.766

0.643

x

y

415°

0.574

0.819


� sin � cos � tan �

45° 1

a. 315° �1

b. �135° 1

c. �225° �1��2

2�22

��22

��22

�22

��22

�22

�22

� sin � cos �

415° 0.819 0.574

310° �0.766 0.643

�945° 0.707 �0.707

�37° �0.602 0.799

4362_HMC_App A_A001-A026 12/26/03 2:13 PM

A26 APPENDIX A


37.

39.

41. a. sin 4 � �0.8, cos 4 � �0.7, sin � �0.4,

cos � 0.9 (Answers may vary.)

b. sin (�1) � �0.8, cos(�1) � 0.5, sin � 0,

cos � �1 (Answers may vary.)144�

16

144�16

��8�

��8�

x

y

-0.602

0.799

-37°

x

y

-945°0.707

-0.707

43. a. sin � 0.866, cos � �0.500

b. The point on the unit circle corresponding to an angle

of is approximately (�0.5, 0.866).

c. Two possible answers are and .

45. a. sin � �1, cos � 0

b. The point on the unit circle corresponding to an angle

of is (0, �1).

c. Possible answers include , , and .

47. It is not possible to have an angle � such that cos � � 0.5

and sin � � �0.5 because cos � and sin � are x- and y-

coordinates on the unit circle and must satisfy the equa-

tion cos2 � � sin2 � � 1.

49. a. Zero, positive, positive, positive, zero

Negative, negative, negative, zero

b. 0, 0.707, 1, 0.707, 0

�0.707, �1, �0.707, 0

11�2

7�2

��2

3�2

3�2

3�2

�4�3

8�3

2�3

2�3

2�3

1

–1

–1 1 x

y

144�16

� =

� = –1

4362_HMC_App A_A001-A026 12/26/03 2:13 PM

appendix a: trigonometry basicscollege.cengage.com/mathematics/latorre/calculus...tion (see figure...

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