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APPENDIX D.3

Review of Trigonometric Functions

D17

APPE N DIX

D.3

Review of Trigonometric Functions

Angles and Degree Measure · Radian Measure · The Trigonometric Functions · Evaluating Trigonometric Functions · Solving Trigonometric Equations · Graphs of Trigonometric Functions

Angles and Degree Measure

An angle has three parts: an initial ray, a terminal ray, and a vertex (the point of intersection of the two rays), as shown in Figure D.24. An angle is in standard position if its initial ray coincides with the positive x-axis and its vertex is at the origin. It is assumed that you are familiar with the degree measure of an angle.* It is common practice to use (the Greek lowercase theta) to represent both an angle and its measure. Angles between 0 and 90 are acute, and angles between 90 and 180 are obtuse. Positive angles are measured counterclockwise, and negative angles are measured clockwise. For instance, Figure D.25 shows an angle whose measure is 45 . You cannot assign a measure to an angle by simply knowing where its initial and terminal rays are located. To measure an angle, you must also know how the terminal ray was revolved. For example, Figure D.25 shows that the angle measuring 45 has the same terminal ray as the angle measuring 315 . Such angles are coterminal. In general, if is any angle, then n 360 , n is a nonzero integer

r Te

Vertex

m

in

al

ra

y

Initial ray

Standard position of an angle Figure D.24

is coterminal with . An angle that is larger than 360 is one whose terminal ray has been revolved more than one full revolution counterclockwise, as shown in Figure D.26. You can form an angle whose measure is less than 360 by revolving a terminal ray more than one full revolution clockwise.

315° - 45°

405°

45°

Coterminal angles Figure D.25

Coterminal angles Figure D.26

NOTE It is common to use the symbol to refer to both an angle and its measure. For instance, in Figure D.26, you can write the measure of the smaller angle as 45 .

*For a more complete review of trigonometry, see Precalculus, 6th edition, by Larson and Hostetler (Boston, Massachusetts: Houghton Mifflin, 2004).

D18

APPENDIX D

Precalculus Review

Radian Measure

To assign a radian measure to an angle , consider to be a central angle of a circle of radius 1, as shown in Figure D.27. The radian measure of is then defined to be the length of the arc of the sector. Because the circumference of a circle is 2 r, the circumference of a unit circle (of radius 1) is 2 . This implies that the radian 2 radians. measure of an angle measuring 360 is 2 . In other words, 360 Using radian measure for , the length s of a circular arc of radius r is s r , as shown in Figure D.28.

The arc length of the sector is the radian measure of . Arc length is s = r .

r=1

r

Unit circle Figure D.27

Circle of radius r Figure D.28

You should know the conversions of the common angles shown in Figure D.29. For other angles, use the fact that 180 is equal to radians.

30° = 6

45° = 4

60° = 3

90° = 2

180° = 360° = 2

Radian and degree measure for several common angles Figure D.29

EXAMPLE 1 a. 40 b. 540 c. 270

Conversions Between Degrees and Radians rad 2 radian 180 deg 9 rad 540 deg 3 radians 180 deg 270 deg

40 deg

rad 3 radians 180 deg 2 180 deg radians rad 90 d. 2 2 rad 180 deg 360 2 rad 114.59 e. 2 radians rad f. 9 radians 2 9 rad 2 180 deg rad 810

APPENDIX D.3

Review of Trigonometric Functions

D19

The Trigonometric Functions

Hy p n ote

Adjacent

Opposite

us

e

There are two common approaches to the study of trigonometry. In one, the trigonometric functions are defined as ratios of two sides of a right triangle. In the other, these functions are defined in terms of a point on the terminal side of an angle in standard position. The six trigonometric functions, sine, cosine, tangent, cotangent, secant, and cosecant (abbreviated as sin, cos, etc.), are defined below from both viewpoints.

Sides of a right triangle Figure D.30

y

Definition of the Six Trigonometric Functions Right triangle definitions, where 0 <

x 2 + y2

r= (x, y) y r x

sin csc

x

opposite hypotenuse hypotenuse opposite

cos sec

(see Figure D.30). 2 adjacent opposite tan hypotenuse adjacent hypotenuse adjacent cot adjacent opposite is any angle (see Figure D.31). x r r , x x 0 tan cot y , x x x , y y 0 0

<

Circular function definitions, where sin csc y r r , y y 0 cos sec

An angle in standard position Figure D.31

The following trigonometric identities are direct consequences of the definitions. is the Greek letter phi.

Trigonometric Identities

Pythagorean Identities:

[Note that sin2

is used to represent sin

Reduction Formulas:

2.]

sin2 cos2 1 2 2 1 tan sec 2 1 cot csc2

Sum or Difference of Two Angles:

sin cos tan

sin cos tan

sin cos tan

sin cos tan

HalfAngle Formulas:

DoubleAngle Formulas:

sin cos tan

± ± ±

sin cos cos cos

± cos

sin sin

sin2 cos2

1 1

sin

cos 2 2 cos 2 2

sin 2 cos 2

2 sin 2 cos2

cos 1

tan ± tan 1 tan tan

Reciprocal Identities:

1 2 sin2 cos2 sin2

Quotient Identities:

Law of Cosines:

a2

b

b2

c2

a c

2bc cos A

csc sec cot

A

1 sin 1 cos 1 tan

tan cot

sin cos cos sin

D20

APPENDIX D

Precalculus Review

Evaluating Trigonometric Functions

There are two ways to evaluate trigonometric functions: (1) decimal approximations with a calculator and (2) exact evaluations using trigonometric identities and formulas from geometry. When using a calculator toevaluate a trigonometric function, remember to set the calculator to the appropriate mode--degree mode or radian mode. EXAMPLE 2

(x, y) r=1 y

Exact Evaluation of Trigonometric Functions

Evaluate the sine, cosine, and tangent of . 3

Solution

x

3 in standard position, as shown in Begin by drawing the angle 3 radians, you can draw an equilateral triangle Figure D.32. Then, because 60 with sides of length 1 and as one of its angles. Because the altitude of this triangle 1 bisects its base, you know that x 2. Using the Pythagorean Theorem, you obtain y r2 x2 1 1 2

2

3 4

3 . 2

r=1

y

Now, knowing the values of x, y, and r, you can write the following. sin cos tan 3 3 3 y r x r y x 3 2 3 1 2 1 2 1 1 2 3 2 3 1 2

60° x=

1 2

The angle 3 in standard position Figure D.32

NOTE All angles in this text are measured in radians unless stated otherwise. For example, when sin 3 is written, the sine of 3 radians is meant, and when sin 3 is written, the sine of 3 degrees is meant.

45° 2 1

The degree and radian measures of several common angles are shown in the table below, along with the corresponding values of the sine, cosine, and tangent (see Figure D.33).

Common First Quadrant Angles

45° 1

Degrees Radians

30°

0 0 0 1 0

30 6 1 2 3 2 3 3

45 4 2 2 2 2 1

60 3 3 2 1 2 3

90 2 1 0 Undefined

3

2

sin cos tan

60° 1

Common angles Figure D.33

APPENDIX D.3

Review of Trigonometric Functions

D21

y

Quadrant II sin : + cos : - tan : -

Quadrant I sin : + cos : + tan : +

x

The quadrant signs for the sine, cosine, and tangent functions are shown in Figure D.34. To extend the use of the table on the preceding page to angles in quadrants other than the first quadrant, you can use the concept of a reference angle (see Figure D.35), with the appropriate quadrant sign. For instance, the reference angle for 3 4 is 4, and because the sine is positive in Quadrant II, you can write sin 3 4 sin 4 2 . 2

Quadrant III sin : - cos : - tan : +

Quadrant IV sin : - cos : + tan : -

Similarly, because the reference angle for 330 is 30 , and the tangent is negative in Quadrant IV, you can write tan 330 tan 30 3 . 3

Quadrant signs for trigonometric functions Figure D.34

Reference angle:

Reference angle:

Reference angle:

Quadrant II = - (radians) = 180° - (degrees)

Quadrant III = - (radians) = - 180°(degrees)

Quadrant IV = 2 - (radians) = 360° - (degrees)

Figure D.35

EXAMPLE 3

Trigonometric Identities and Calculators

Evaluate each trigonometric expression. a. sin

Solution

3

b. sec 60

c. cos 1.2

a. Using the reduction formula sin sin sin 3 . 2

sin , you can write

3

3

b. Using the reciprocal identity sec sec 60 1 cos 60 1 1 2 2.

1 cos , you can write

c. Using a calculator, you obtain cos 1.2 0.3624.

Remember that 1.2 is given in radian measure. Consequently, your calculator must be set in radian mode.

D22

APPENDIX D

Precalculus Review

Solving Trigonometric Equations

0? You know that 0 is one solution, How would you solve the equation sin but this is not the only solution. Any one of the following values of is also a solution. . . ., 3 , 2 , , 0, , 2 , 3 , . . .

You can write this infinite solution set as n : n is an integer . EXAMPLE 4 Solving a Trigonometric Equation

Solve the equation sin

Solution

y

3 . 2

To solve the equation, you should consider that the sine is negative in Quadrants III and IV and that sin 3 3 . 2

y = sin

1

So, you are seeking values of in the third and fourth quadrants that have a reference angle of 3. In the interval 0, 2 , the two angles fitting these criteria are

- 2 3 - 3

- 2

2

4 3

3 2

x 2

3

5 3

4 3

and

2

3

5 . 3

y=- 3 2

By adding integer multiples of 2 to each of these solutions, you obtain the following general solution. 4 3 2n or 5 3 2n , where n is an integer.

Solution points of sin u Figure D.36

3 2

See Figure D.36. EXAMPLE 5 Solve cos 2

Solution

Solving a Trigonometric Equation 2 3 sin , where 0 2 . 1 2 sin2 , you can rewrite the

Write original equation. Trigonometric identity

Using the double-angle identity cos 2 equation as follows. 1 cos 2 2 sin2 0 0 2 3 sin 2 3 sin 2 sin2 3 sin 2 sin 1 sin

1 1

Quadratic form Factor.

If 2 sin then sin 6 ,

1 0, then sin 1 2 and 1 and 2. So, for 0 5 , 6 or 2

6 or 5 6. If sin 1 2 , there are three solutions.

0,

APPENDIX D.3

Review of Trigonometric Functions

D23

Graphs of Trigonometric Functions

f x A function f is periodic if there exists a nonzero number p such that f x p for all x in the domain of f. The smallest such positive value of p (if it exists) is the period of f. The sine, cosine, secant, and cosecant functions each have a period of 2 , and the other two trigonometric functions, tangent and cotangent, have a period of , as shown in Figure D.37.

y 6 5 4 3 2 1 y

Domain: all reals Range: [-1, 1] Period: 2

y

6 5 4 3

Domain: all reals Range: [-1, 1] Period: 2

5 4 3 2 1

Domain: all x + n 2 Range: (-, ) y = tan x Period:

y = sin x

x

2

y = cos x

x -3

x

2

-1

2

-1

y Domain: all x n 4 3 2 1

Range: (-, -1] and [1, ) Period: 2

y Domain: all x 2 + n 4 3 2

y 5 4 3 2

Range: (-, -1] and [1, ) Period: 2

Domain: all x n Range: (- , ) Period:

-1

2

x -1 -2 -3

2

x

1

2

x

1 y = csc x = sin x

y = sec x =

1 cos x

1 y = cot x = tan x

The graphs of the six trigonometric functions Figure D.37

Note in Figure D.37 that the maximum value of sin x and cos x is 1 and the minimum value is 1. The graphs of the functions y a sin bx and y a cos bx oscillate between a and a, and so have an amplitude of a . Furthermore, because bx 0 when x 0 and bx 2 when x 2 b, it follows that the functions y a sin bx and y a cos bx each have a period of 2 b . The table below summarizes the amplitudes and periods for some types of trigonometric functions.

Function y y y a sin bx or a tan bx or a sec bx or y y y a cos bx a cot bx a csc bx Period 2 b b 2 b Amplitude a Not applicable Not applicable

D24

APPENDIX D

Precalculus Review

EXAMPLE 6

y

Sketching the Graph of a Trigonometric Function 3 cos 2x.

f (x) = 3 cos 2x (0, 3)

Sketch the graph of f x

Solution

Amplitude = 3

x

3

The graph of f x 3 cos 2x has an amplitude of 3 and a period of 2 2 . Using the basic shape of the graph of the cosine function, sketch one period of the function on the interval 0, , using the following pattern. Maximum: 0, 3 Minimum: 2 , ,3 3

-1 -2 -3

2

3 2

2

Period =

Maximum:

Figure D.38

By continuing this pattern, you can sketch several cycles of the graph, as shown in Figure D.38. Horizontal shifts, vertical shifts, and reflections can be applied to the graphs of trigonometric functions, as illustrated in Example 7. EXAMPLE 7 Shifts of Graphs of Trigonometric Functions

Sketch the graph of each function. a. f x

Solution

sin x

2

b. f x

2

sin x

c. f x

2

sin x

4

a. To sketch the graph of f x sin x 2 , shift the graph of y sin x to the left 2 units, as shown in Figure D.39(a). b. To sketch the graph of f x 2 sin x, shift the graph of y sin x upward two units, as shown in Figure D.39(b). c. To sketch the graph of f x 2 sin x 4 , shift the graph of y sin x upward two units and to the right 4 units, as shown in Figure D.39(c).

y 6 5 4 3 2 y y

f (x) = sin x + 2 2

(

(

x

6 5 4 3 2

f (x) = 2 + sin x 2

x

6 5 4 3 2

f (x) = 2 + sin x - 4 4

(

(

2

x

-2

y = sin x

-2

y = sin x

-2

y = sin x

(a) Horizontal shift to the left

(b) Vertical shift upward

(c) Horizontal and vertical shifts

Transformations of the graph of y Figure D.39

sin x

APPENDIX D.3

Review of Trigonometric Functions

D25

EXERCISES FOR APPENDIX D.3

In Exercises 1 and 2, determine two coterminal angles in degree measure (one positive and one negative) for each angle. 1. (a)

= 36°

10. Angular Speed A car is moving at the rate of 50 miles per hour, and the diameter of its wheels is 2.5 feet. (a) Find the number of revolutions per minute that the wheels are rotating.

(b)

= -120°

(b) Find the angular speed of the wheels in radians per minute. In Exercises 11 and 12, determine all six trigonometric functions for the angle . 11. (a)

y

2. (a)

= 300°

(b)

= - 420°

(b)

(3, 4)

y

x

x

In Exercises 3 and 4, determine two coterminal angles in radian measure (one positive and one negative) for each angle. 3. (a)

= 9

(-12, -5)

(b) = 4

3

12. (a)

y

(b)

x

y

x

4. (a)

= - 9 4

(b)

= 8 9

(8, -15)

(1, -1)

In Exercises 13 and 14, determine the quadrant in which 13. (a) sin (b) sec In Exercises 5 and 6, rewrite each angle in radian measure as a multiple of and as a decimal accurate to three decimal places. 5. (a) 30 6. (a) 20 (b) 150 (b) 240 (c) 315 (c) 270 (d) 120 (d) 144 14. (a) sin (b) csc < 0 and cos > 0 and cot > 0 and cos < 0 and tan < 0 < 0 < 0 > 0

lies.

In Exercises 1518, evaluate the trigonometric function. 15. sin

1 2

16. sin tan

1 3

In Exercises 7 and 8, rewrite each angle in degree measure. 3 7. (a) 2 7 8. (a) 3 7 (b) 6 (b) 11 30 (c) 7 12 11 (c) 6 (d) 2.367

cos

2

3 1

1

(d) 0.438

9. Let r represent the radius of a circle, the central angle (measured in radians), and s the length of the arc subtended by the angle. Use the relationship s r to complete the table. r s 8 ft 12 ft 1.6 3 4 15 in. 85 cm 96 in. 4 8642 mi 2 3

17. cos cot

4 5

18. sec csc

13 5

5 13

4

5

D26

APPENDIX D

Precalculus Review

In Exercises 1922, evaluate the sine, cosine, and tangent of each angle without using a calculator. 19. (a) 60 (b) 120 (c) (d) 4 5 4 225 5 3 11 6 20. (a) 30

39. Airplane Ascent An airplane leaves the runway climbing at an angle of 18 with a speed of 275 feet per second (see figure). Find the altitude a of the plane after 1 minute.

(b) 150 (c) (d) 6 2

18° a

21. (a) 225 (b) (c) (d)

22. (a) 750 (b) 510 (c) (d) 10 3 17 3 40. Height of a Mountain In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is 3.5 . After you drive 13 miles closer to the mountain, the angle of elevation is 9 . Approximate the height of the mountain.

In Exercises 2326, use a calculator to evaluate each trigonometric function. Round your answers to four decimal places. 23. (a) sin 10 (b) csc 10 25. (a) tan (b) tan 9 10 9 24. (a) sec 225 (b) sec 135 26. (a) cot 1.35 (b) tan 1.35

3.5° 13 mi

9°

Not drawn to scale

In Exercises 4144, determine the period and amplitude of each function. 41. (a) y

y

2 sin 2x

(b) y

y 1

1 2

sin x

In Exercises 2730, find two solutions of each equation. Give your answers in radians 0 < 2 . Do not use a calculator. 27. (a) cos (b) cos 29. (a) tan (b) cot 1 3 (b) sin In Exercises 3138, solve the equation for 31. 2 sin 33. tan2 35. sec csc 37. cos

2 2

3 2 1 x 1 -1 2

2 2 2 2

28. (a) sec (b) sec

2 2

4

-3

2

3 4

5 4

x

30. (a) sin

3 2 3 2 0 3 cos cos 1

42. (a) y

y 3

3 x cos 2 2

(b) y

y

2 sin

x 3

< 2

.

2 x -1 -2 -3 x -1 -2

1 tan 0 2 csc sin 1

32.

tan2

34. 2 cos2 36. sin 38. cos 2

2

3

2

cos

1 43. y 44. y 3 sin 4 x 2 x cos 3 10

APPENDIX D.3

Review of Trigonometric Functions

D27

In Exercises 4548, find the period of the function. 45. y 47. y 5 tan 2x sec 5x 46. y 48. y 7 tan 2 x csc 4x

67. Sales The monthly sales S (in thousands of units) of a seasonal product are modeled by S 58.3 32.5 cos t 6

Writing In Exercises 49 and 50, use a graphing utility to

graph each function f in the same viewing window for c 2, c 1, c 1, and c 2. Give a written description of the change in the graph caused by changing c. 49. (a) f x (b) f x (c) f x c sin x cos cx cos x c 50. (a) f x (b) f x (c) f x sin x c cos x c c

where t is the time (in months) with t 1 corresponding to January. Use a graphing utility to graph the model for S and determine the months when sales exceed 75,000 units. 68. Investigation Two trigonometric functions f and g have a period of 2, and their graphs intersect at x 5.35. (a) Give one smaller and one larger positive value of x where the functions have the same value. (b) Determine one negative value of x where the graphs intersect. (c) Is it true that f 13.35 answer. g 4.65 ? Give a reason for your

sin 2 x

In Exercises 5162, sketch the graph of the function. 51. y 53. y 55. y 57. y 59. y 61. y 62. y x sin 2 sin csc x 2 2 x 3 52. y 54. y 56. y 58. y 60. y 2 cos 2x

Pattern Recognition In Exercises 69 and 70, use a graphing

2 tan x tan 2x csc 2 x cos x 3 utility to compare the graph of f with the given graph. Try to improve the approximation by adding a term to f x . Use a graphing utility to verify that your new approximation is better than the original. Can you find other terms to add to make the approximation even better? What is the pattern? (In Exercise 69, sine terms can be used to improve the approximation and in Exercise 70, cosine terms can be used.) 69. f x 2

y

2 sec 2x sin x 1 1 cos x sin x

4

sin x

1 sin 3 x 3

2

2 1 x 3 -2

In Exercises 63 and 64, find a, b, and c such that the graph of the function matches the graph in the figure. 63. y

y 4 2 x 1

Graphical Reasoning

a cos bx

c

64. y

y

a sin bx

c

70. f x

y

1 2

4

2

cos x

1 cos 3 x 9

-4

3

-1 2 -1

2

3 4

x

2 1 x -1 1 2 3 4

65. Think About It Sketch the graphs of f x sin x, gx sin x , and h x sin x . In general, how are the graphs of f x and f x related to the graph of f ? 66. Think About It car is h 51 The model for the height h of a Ferris wheel

-2

50 sin 8 t

where t is measured in minutes. (The Ferris wheel has a radius of 50 feet.) This model yields a height of 51 feet when t 0. Alter the model so that the height of the car is 1 foot when t 0.

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