Precalculus concepts through functions a right triangle approach to trigonometry 3rd edition sulliva

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Ch. 6 Analytic Trigonometry 6.1 The Inverse Sine, Cosine, and Tangent Functions 1 Find the Exact Value of the Inverse Sine, Cosine, or Tangent Function MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the exact value of the expression.

3 1) sin-1 2 A)

2) cos-1 A)

π 3

π 4

3π 4

5) sin-1 (0.5) π A) 6

A)

C)

π 4

D)

3π 4

B)

7π 4

C)

π 6

D)

11π 6

B)

π 4

C)

-π 4

D)

-3π 4

B)

π 4

C)

5π 4

D)

7π 4

B)

7π 6

C)

7π 3

D)

π 3

B)

π 3

C)

4π 3

D)

π 4

2 2

4) tan-1 (-1) π A) 4

6) tan-1

2π 3

2 2

3) cos-1 A)

B)

3 3 π 6

2 Find an Approximate Value of an Inverse Sine, Cosine, or Tangent Function MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Use a calculator to find the value of the expression rounded to two decimal places.

1) sin-1 (0.7) A) 0.78

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B) 44.43

C) 0.80

D) 45.57

2) cos-1 (-0.6) A) 2.21

B) 126.87

C) -0.64

D) -36.87

3) tan-1 (2.8) A) 1.23

B) 70.35

C) 0.34

D) 19.65

1 7

4) sin-1

A) 0.14

B) 8.21

C) 1.43

D) 81.79

B) 112.02

C) -0.38

D) -22.02

B) 35.26

C) 0.96

D) 54.74

B) 121.95

C) -0.56

D) -31.95

3 5) cos-1 8 A) 1.96 3 3

6) sin-1

A) 0.62 7) cos-1 -

7 5

A) 2.13

3 Use Properties of Inverse Functions to Find Exact Values of Certain Composite Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the exact value of the expression. Do not use a calculator.

1) cos [cos-1 (-0.9372)] A) -0.9372

B) 0.9372

C) -0.4686

D) 0.4686

2) sin [sin-1 (0.8)] A) 0.8

B) 1.394

C) 1.25

D) 0.2

3) tan [tan-1 (0.8)] A) 0.8

B) 0.9712

C) 1.25

D) 0.2

B)

2π 5

C) -

3π 5

D) -

B)

3π 5

C) -

3π 5

D)

2π 5

B)

3π 5

C)

D)

5 2π

4) cos-1 cos A)

3π 5

3π 5

2π 5

3π 5) tan-1 tan 5 A) -

2π 5

6) sin-1 sin A)

3π 5

2π 5

7) sin [sin-1 (-0.3)] A) -0.3

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B) 0.3

5 3π

C) 2.7

D) -2.7

π 8) cos-1 cos 10 A)

π 10

9) sin-1 sin A)

B) 10π

C) -

π 10

D) -10π

π 5

π 5

B) -

π 5

C) 5π

D) -5π

C) 8π

D) -8π

π 10) tan-1 tan 8 A) -

π 8

B)

π 8

4 Find the Inverse Function of a Trigonometric Function MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the inverse function f-1 of the function f.

1) f(x) = 7 sin x - 6 x+6 A) f-1 (x) = sin-1 7 C) f-1 (x) = sin-1

x+7 6

B) f-1 (x) = cos

x+6 7

D) f-1 (x) = 7 sin-1 x - 6

2) f(x) = 8 cos x + 7 x-7 A) f-1 (x) = cos-1 8 C) f-1 (x) = cos-1

x+7 8

3) f(x) = 3 tan(7x) x 1 A) f-1 (x) = tan-1 3 7 C) f-1 (x) =

1 3 tan(7x)

4) f(x) = -9 cos(4x) 1 x A) f-1 (x) = cos-1 4 9 C) f-1 (x) = -

1 x cos-1 4 9

5) f(x) = - sin(x + 7) - 8 A) f-1 (x) = -sin-1 (x + 8) - 7 C) f-1 (x) = -sin-1 (x + 7) - 8

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B) f-1 (x) = sin

x-7 8

D) f-1 (x) = 8 cos-1 x + 7

B) f-1 (x) =

1 x tan-1 3 7

D) f-1 (x) = 3 tan-1 (7x)

B) f-1 (x) =

1 x cos-1 9 4

D) f-1 (x) = -9 cos-1 (4x)

B) f-1 (x) = sin-1 (x + 8) - 7 D) f-1 (x) = -sin-1 (x - 8) + 7

6) f(x) = cos(x - 2) - 4 A) f-1 (x) = cos-1 (x + 4) + 2 C) f-1 (x) = cos-1 (x - 2) - 4 7) f(x) = 8 tan(10x - 2) 1 x tan-1 A) f-1 (x) = +2 10 8 C) f-1 (x) = 8 tan-1 (10x - 2) 8) f(x) = -9 cos(10x + 6) 1 x cos-1 A) f-1 (x) = -6 10 9 C) f-1 (x) = -9 cos-1 (10x + 6) Find the domain of the function f and of its inverse function f-1 .

9) f(x) = 5 sin x - 7 A) Domain of f: (-∞, ∞) Domain of f-1 : [-12, -2] C) Domain of f: (-∞, ∞) Domain of f-1 : (-∞, ∞)

B) f-1 (x) = cos-1 (x - 4) - 2 D) f-1 (x) = cos-1 (x + 2) + 4

B) f-1 (x) =

1 x tan-1 +2 8 10

D) f-1 (x) =

1 x tan-1 -2 10 8

B) f-1 (x) =

1 x cos-1 -6 9 10

D) f-1 (x) = -

1 x cos-1 +6 10 9

B) Domain of f: (-∞, ∞) Domain of f-1 : [2, 12] D) Domain of f: [2, 12] Domain of f-1 : [-12, -2]

10) f(x) = 5 tan x + 7 A) Domain of f: x ≠

(2k + 1)π ; k an integer 2

Domain of f-1 : (-∞, ∞) C) Domain of f: (-∞, ∞) (2k + 1)π Domain of f-1 : x ≠ ; k an integer 2

B) Domain of f: (-∞, ∞) Domain of f-1 : [2, 12] D) Domain of f: x ≠

(2k + 1)π ; k an integer 2

Domain of f-1 : [2, 12]

11) f(x) = 2 sin(4x) A) Domain of f: (-∞, ∞) Domain of f-1 : [-2, 2]

B) Domain of f: (-∞, ∞) Domain of f-1 : [2, 6]

C) Domain of f: (-∞, ∞) Domain of f-1 : [-4, 4]

D) Domain of f: -

1 1 , 4 4

Domain of f-1 : (-∞, ∞)

12) f(x) = -5 cos(9x)

Page 4

A) Domain of f: (-∞, ∞) Domain of f-1 : [-5, 5]

B) Domain of f: (-∞, ∞) Domain of f-1 : [4, 14]

C) Domain of f: (-∞, ∞) Domain of f-1 : [-9, 9]

D) Domain of f: -

1 1 , 9 9

Domain of f-1 : (-∞, ∞)

13) f(x) = cos(x - 7) + 5 A) Domain of f: (-∞, ∞) Domain of f-1 : [4, 6]

B) Domain of f: (-∞, ∞) Domain of f-1 : (-∞, ∞)

C) Domain of f: (-∞, ∞) Domain of f-1 : [-6, -4]

D) Domain of f: [-7, 7] Domain of f-1 : (-∞, ∞)

14) f(x) = tan(x - 8) + 6 A) Domain of f: x ≠

(2k + 1)π + 6; k an integer 2

B) Domain of f: x ≠

(2k + 1)π ; k an integer 2

Domain of f-1 : (-∞, ∞)

Domain of f-1 : (-∞, ∞)

C) Domain of f: (-∞, ∞) Domain of f-1 : [-7, -5]

D) Domain of f: [-8, 8] Domain of f-1 : (-∞, ∞)

15) f(x) = -6 cos(10x + 8) A) Domain of f: (-∞, ∞) Domain of f-1 : [-6, 6]

B) Domain of f: (-∞, ∞) Domain of f-1 : [-10, 10]

C) Domain of f: [-6, 6] Domain of f-1 : (-∞, ∞)

D) Domain of f: (-∞, ∞) Domain of f-1 : (-∞, ∞)

16) f(x) = 7 sin(5x - 1) A) Domain of f: (-∞, ∞) Domain of f-1 : [-7, 7]

B) Domain of f: (-∞, ∞) Domain of f-1 : [-5, 5]

C) Domain of f: [-7, 7] Domain of f-1 : (-∞, ∞)

D) Domain of f: -

1 1 , 5 5

Domain of f-1 : (-∞, ∞)

5 Solve Equations Involving Inverse Trigonometric Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the exact solution of the equation.

1) cos-1 x = 0 A) x = 1 2) sin-1 x =

C) x = 0

D) x = -1

B) x = 0

C) x = π

D) x = -1

B) x = 1

C) x = 0

D) x = -

π 2

A) x = 1

Page 5

B) x = π

3) sin-1 x =

π 6

A) x =

1 2

1 2

4) 2 cos-1 x = π A) x = 0

B) x = 1

5) -sin-1 (4x) = A) x = -

C) x =

π 2

D) x =

C) x =

2 8

D) x = 0

C) x =

π 4

D) x = 0

3π 2

π 4

2 8

B) x =-

2 2

6) -4 tan-1 x = π A) x = -1

B) x = 1

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Solve the problem.

7) The formula D = 24 1 -

cos-1 (tan i tan θ) π

can be used to approximate the number of hours of daylight when the declination of the sun is i ° at a location θ° north latitude for any date between the vernal equinox and autumnal equinox. To use this formula, cos-1 (tan i tan θ) must be expressed in radians. Approximate the number of hours of daylight in Fargo, North Dakota, (46°52ʹnorth latitude) for vernal equinox (i = 0°).

8) The formula D = 24 1 -

cos-1 (tan i tan θ) π

can be used to approximate the number of hours of daylight when the declination of the sun is i ° at a location θ° north latitude for any date between the vernal equinox and autumnal equinox. To use this formula, cos-1 (tan i tan θ) must be expressed in radians. Approximate the number of hours of daylight in Flagstaff, Arizona, (35°13ʹ north latitude) for summer solstice (i = 23.5°).

Page 6

9) When light travels from one medium to anotherfrom air to water, for instanceit changes direction. (This is why a pencil, partially submerged in water, looks as though it is bent.) The angle of incidence θi is the angle in the first medium; the angle of refraction θr is the second medium. (See illustration.) Each medium has an index of refractionn i and n r, respectivelywhich can be found in tables. Snellʹs law relates these quantities in the formula n i sinθi = n r sin θr Solving for θr, we obtain ni sin θi θr = sin-1 nr Find θr for crown glass (n i = 1.52), water(n r = 1.33), and θi = 38°.

10) When light travels from one medium to anotherfrom air to water, for instanceit changes direction. (This is why a pencil, partially submerged in water, looks as though it is bent.) The angle of incidence θr is the angle in the first medium; the angle of refraction θr is the second medium. (See illustration.) Each medium has an index of refractionn i and n r, respectivelywhich can be found in tables. Snellʹs law relates these quantities in the formula n i sinθi = n r sin θr Solving for θr, we obtain ni sin θi θr = sin-1 nr Find θr for air (n i = 1.0003), methylene iodide (n r = 1.74), and θi = 14.7°.

Page 7

11) When light travels from one medium to anotherfrom air to water, for instanceit changes direction. (This is why a pencil, partially submerged in water, looks as though it is bent.) The angle of incidence θi is the angle in the first medium; the angle of refraction θr is the second medium. (See illustration.) Each medium has an index of refractionn i and n r, respectivelywhich can be found in tables. Snellʹs law relates these quantities in the formula n i sinθi = n r sin θr Solving for θr, we obtain ni sin θi θr = sin-1 nr Find θr for fused quartz (n i = 1.46), ethyl alcohol (n r = 1.36), and θi = 8.5°.

6.2 The Inverse Trigonometric Functions (Continued) 1 Find the Exact Value of Expressions Involving the Inverse Sine, Cosine, and Tangent Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the exact value of the expression.

1) sin (tan-1 2) A)

2 5 5

2) cos sin-1

7 8

4) cos sin-1 A)

Page 8

4 5

5 2 2

D) 5 2

15 2

C)

4 15 15

D)

2 15 15

3 8

D)

1 8

B)

3) cos 2 sin-1 A)

C)

1 4

15 4

A)

B) 2 5

1 4 B)

5 8

C)

B)

1 5

C) -

3 5 3 5

D) -

4 5

7π 5) cos-1 cos 6 A)

5π 6

6) cos-1 sin A)

B)

π 3

C)

π 6

D)

4π 5

B)

π 3

C)

π 6

D)

4π 5

B)

2 2 3

C) 2 3

B)

3 3

C)

7π 6

2π 3

1 7) tan cos-1 3 A) 2 2

8) cos tan-1

3 3

3 2

A)

9) cos-1 [cos (-0.9372)] A) 0.9372 10) sin-1 sin A)

A) -

A) 1

Page 9

B)

3π 5

C)

B)

3π 5

C) -

B) 1

D) -0.4686

5 3π

D)

5 2π

3π 5

D)

2π 5

C) 0

D)

2 2

1 2

3 3

A) 2

14) cot sin-1

C) 0.4686

π 3

3 2

A) 2

13) csc tan-1

D)

3π 5

2π 5

12) sec sin-1 -

B) -0.9372

1 2

3π 5

2π 5

11) tan-1 tan

D) -2 3

B)

2 3 3

C)

3

D)

B)

2

C)

2 2

D) 2

2 2

2 Define the Inverse Secant, Cosecant, and Cotangent Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the exact value of the expression.

1) cot-1 -1 3π A) 4 2) csc-1 -2 π A) 6

B)

π 4

C)

π 3

D)

2π 3

B)

π 6

C)

π 3

D)

2π 3

B) 0

C)

π 6

D) -

π 6

C)

4π 3

D) -

2π 3

3) sec -1 -1 A) π 4) sec -1 (-2) 2π A) 3

B) -

π 3

3 Use a Calculator to Evaluate sec^-1 x, csc^-1 x, and cot^-1 x MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Use a calculator to find the value of the expression in radian measure rounded to two decimal places.

6 1) csc-1 5 A) -0.99

B) -56.44

C) 2.56

D) 146.44

2) sec -1 - 4 A) 1.82

B) 104.48

C) -0.25

D) -14.48

B) 70.97

C) 0.33

D) 19.03

B) 0.50

C) 1.13

D) -2.01

B) 2 5

C)

5 2 2

D) 5 2

15 2

C)

4 15 15

D)

10 3) cot-1 29 A) 1.24 4) sec -1 -

7 3

A) 2.01 Find the exact value of the composition.

5) sin(tan-1 (2)) A)

2 5 5

6) cos sin-1 A)

Page 10

15 4

1 4 B)

2 15 15

7) sin (tan-1 2) 2 5 A) 5 8) cos (sin-1 A)

5 2 2

D) 5 2

3 5

D) -

B) 2 5

C)

B)

1 5

C) -

B)

π 3

C)

3 ) 5

4 5

4 5

7π 9) cos-1 (sin ) 6 A)

2π 3

π 6

D)

4π 5

4 Write a Trigonometric Expression as an Algebraic Expression MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Write the trigonometric expression as an algebraic expression in u.

1) sin (tan-1 u) u u2 + 1

A)

2) cos (tan-1 u) 1 A) u2 + 1

B)

u u2 - 1

C) u u2 + 1

D)

1 u2 + 1

B)

1 u2 - 1

C) u u2 + 1

D)

u u2 + 1

B)

u2 + 1

C)

u2 - 1

D)

u2 + 1 u

B)

1 u2 + 1

C)

u2 - 1

D)

u2 + 1 u

B)

u2 - 1 u

C)

u2 - 1

D)

1 u2 + 1

B)

u2 - 1 u

C) u

D)

u2 + 1 u

B)

1 - u2 u

C)

D)

u u2 + 1

3) cos (sin-1 u) 1 - u2

A)

4) cos (cot-1 u) u u2 + 1

A)

5) tan (csc-1 u) 1 u2 - 1

A)

6) sin (csc-1 u) A)

1 u

7) tan (sin-1 u) A)

Page 11

u 1 - u2

1 - u2

8) csc (tan-1 u) u2 + 1 A) u

B)

1 u2 - 1

C)

1 u2 + 1

D)

u u2 + 1

B)

u2 - 1 u

C)

1 - u2

D)

1 u2 - 1

B)

1 - u2 u

C)

1 - u2

D)

u u2 + 1

9) sec (sin-1 u) 1

A)

1 - u2

10) cot (cos-1 u) u

A)

1 - u2

6.3 Trigonometric Equations 1 Solve Equations Involving a Single Trigonometric Function MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the equation on the interval 0 ≤ θ < 2π.

3 1) sin (4θ) = 2 A)

π π 2π 7π 7π 13π 5π 19π , , , , , , , 12 6 3 12 6 12 3 12

C) 0,

π ,π 4

π ,π 4

3) csc (3θ) = 0 4π 2π , π, A) 0, 3 3 4)

2 cos (2θ) = 1 π 7π 9π 15π , , A) , 8 8 8 8

π 5π , 4 4

D) 0

2) 2 3 sin (4θ) = 3 π π 2π 7π 7π 13π 5π 19π , , , , , , , A) 12 3 12 12 6 3 12 6 C) 0,

B)

B)

π 5π , 4 4

D) 0

π 3π 5π 7π , , , 4 4 4 4

C)

π 9π , 8 8

D) No solution

2π 4π , π, 3 3

C)

π 3π 5π 7π , , , 4 4 4 4

D) No solution

B) π

C)

π 2

D)

3π 2

B) π

C) 2π

D)

3π 2

B)

B) 0,

5) cos θ - 1 = 0 A) 0 6) 6 csc θ - 2 = 4 π A) 2

Page 12

7) 2 cos (2θ) = 3 π 11π 13π 23π A) , , , 12 12 12 12 C)

π 2

8) 2 cos θ + 1 = 0 2π 4π , A) 3 3 π 2

9) cot 2θ -

B)

π 5π , 3 3

B)

π 11π , 6 6

D)

3π 2

C)

π 3π , 2 2

D)

3π 2

=1

A)

15π 3π 7π 11π , , , and 8 8 8 8

B)

3π 7π , 8 8

C)

π 5π 9π 13π , , , and 4 4 4 4

D)

3π 8

Solve the equation. Give a general formula for all the solutions.

10) cos θ = 1 A) θ = 0 + 2kπ

B) θ = π + 2kπ

C) θ =

π + 2kπ 2

D) θ =

3π + 2kπ 2

B) θ = π + 2kπ

C) θ = 0 + 2kπ

D) θ =

3π + 2kπ 2

B) θ = 0 + 2kπ

C) θ =

π + 2kπ 2

D) θ =

π + kπ 2

B) θ = 0 + 2kπ

C) θ =

π + 2kπ 2

D) θ = 0 + kπ

11) sin θ = 1 A) θ =

π + 2kπ 2

12) sin θ = 0 A) θ = 0 + kπ 13) cos θ = 0 A) θ =

14) sin θ =

π + kπ 2

3 2

A) θ =

π 2π + 2kπ, θ = + 2kπ 3 3

B) θ =

π 2π + kπ, θ = + kπ 3 3

C) θ =

5π π + 2kπ, θ = + 2kπ 6 6

D) θ =

π 5π + kπ, θ = + kπ 6 6

π + kπ 4

C) θ =

3π + 2kπ 4

D) θ =

π + 2kπ 4

B) θ = π + 2kπ

C) θ =

π + 2kπ 2

D) θ =

3π + 2kπ 2

15) tan θ = -1 3π A) θ = + kπ 4

B) θ =

16) cos θ - 1 = 0 A) θ = 2kπ

Page 13

17) 2 cos θ + 1 = 0 2π 4π A) θ = + 2kπ, θ = + 2kπ 3 3 C) θ =

3π π + 2kπ, θ = + 2kπ 2 2

18) cos (2θ) =

B) θ =

2π 4π + kπ, θ = + kπ 3 3

D) θ =

3π + kπ 2

2 2

A) θ =

π 7π + kπ, θ = + kπ 8 8

B) θ =

π 7π + 2kπ, θ = + 2kπ 8 8

C) θ =

3π π + kπ, θ = + kπ 4 4

D) θ =

2π 4π + kπ, θ = + kπ 3 3

C) θ =

π + 2kπ 9

19) csc

θ 2 3 = 3 3

A) θ = π + 6kπ

B) θ =

π + 6kπ 2

D) θ =

π + 2kπ 18

Solve the equation on the interval [0, 2 π).

20) Suppose f(x) = cos θ - 1. Solve f(x) = 0.

A) 0

B) π

21) Suppose f(x) = 5 csc θ - 1. Solve f(x) = 4. π B) π A) 2 22) Suppose f(x) = 2 cos θ + 1. Solve f(x) = 0. π 5π 2π 4π A) , B) , 3 3 3 3

Solve the problem using Snellʹs Law:

sin θ1 sin θ2

π 2

D)

3π 2

C) 2π

D)

3π 2

D)

3π 2

C)

=

C)

π 3π , 2 2

v1 . v2

23) A light beam in air travels at 2.99 × 108 meters per second. If its angle of incidence to a second medium is 38° and its angle of refraction in the second medium is 28°, what is its speed in the second medium (to two decimal places)?

C) 1.84 × 108 mps D) 1.40 × 108 mps A) 2.28 × 108 mps B) 3.92 × 108 mps 24) A ray of light near the horizon with an angle of incidence of 88° enters a pool of water and strikes a fishʹs eye. If the index of refraction is 1.33, what is the angle of refraction (to two decimal places)?

A) 48.71° B) 41.29° C) 47.35° D) 42.65° 25) The index of refraction of light passing from air into a second medium is 1.54. If the angle of incidence is 80°, what is the angle of refraction (to two decimal places)?

A) 39.75° B) 50.25° C) 37.60° D) 52.40° 26) A light beam traveling through air makes an angle of incidence of 41° upon a second medium. The refracted beam makes an angle of refraction of 31°. What is the index of refraction of the material of the second medium?

Give the answer to two decimal places.

A) 1.27 B) 0.79 C) 0.66 D) 0.52

Page 14

27) A light beam in air travels at 2.99 × 108 meters per second. If its angle of incidence to a second medium is 82° and its angle of refraction in the second medium is 72°, what is its speed in the second medium (to two decimal places)?

C) 2.96 × 108 mps D) 2.84 × 108 mps A) 2.87 × 108 mps B) 3.11 × 108 mps Solve the problem.

28) The function I(t) = 40 sin 60πt -

π 2

represents the amperes of current produced by an electric generator as a function of time t, where t is measured in seconds. Find the smallest value of t for which the current is 20 amperes. Round your answer to three decimal places, if necessary.

A) 0.011 B) 0.033 C) 0.017 D) 0.008 29) A weight suspended from a spring is vibrating vertically with up being the positive direction. The function 3πt π f(t) = 10 sin 4 4 represents the distance in centimeters of the weight from its rest position as a function of time t, where t is measured in seconds. Find the smallest positive value of t for which the displacement of the weight above its rest position is 5 cm. Round answer to three decimal places, if necessary.

A) 0.556 B) 0.222 C) 2.293 D) 1.586 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

30) You are flying a kite and want to know its angle of elevation. The string on the kite is 43 meters long and the kite is level with the top of a building that you know is 28 meters high. Use an inverse trigonometric function to find the angle of elevation of the kite. Round to two decimal places.

31) Before exercising, an athlete measures her air flow and obtains 2π t a = 0.65 sin 5 where a is measured in liters per second and t is the time in seconds. If a > 0, the athlete is inhaling; if a < 0, the athlete is exhaling. The time to complete one complete inhalation/exhalation sequence is a respiratory cycle.

Find the values of t for which the athleteʹs air flow is zero. Find all values of t for t < 20 seconds.

32) A mass hangs from a spring which oscillates up and down. The position P (in feet) of the mass at time t (in seconds) is given by P = 4 cos (4t) For what values of t, 0 ≤ t < π, will the position be 2 2 feet? Find the exact values. Do not use a calculator.

33) The path of a projectile fired at an inclination θ (in degrees) to the horizontal with an initial velocity v0 is a parabola. The range R of the projectile, that is, the horizontal distance that the projectile travels, is found by using the formula

R=

2 v0 g

sin (2θ)

where g is the acceleration due to gravity. Suppose the projectile is fired with an initial velocity of 400 feet per seconds and g = 32 feet per second2 . What angle θ, 0° ≤ θ < 90°, would you select for the range to be 2500 feet?

(There should be two values of θ.)

Page 15

34) Wildlife management personnel use predator-prey equations to model the populations of certain predators and their prey in the wild. Suppose the population M of a predator after t months is given by π M = 750 + 125 sin t 6 while the population N of its primary prey is given by π N = 12,250 + 3050 cos t 6 Find the values of t, 0 ≤ t < 12, for which the predator population is 875. Find the values of t, 0 ≤ t < 12, for which the prey population is 10,725.

35) A consumer notes the sinusoidal nature of her monthly power bills. In winter when she uses electricity to heat her home and in summer when she cools her home, the bills are high. In spring and fall, significantly less electricity is used and the bills are much smaller. The following function models this behavior. π π tC = 60 + 40 cos 3 3 Here C is the cost of power in dollars for the month t, 1 ≤ t ≤ 12, with t = 1 corresponding to January. For what values of t, 1 ≤ t ≤ 12, is the cost exactly $80?

36) The average daily temperature T of a city in the United States is approximated by 2π (t - 30) T = 55 - 23 cos 365 where t is in days, 1 ≤ t ≤ 365, and t = 1 corresponds to January 1. For what range of values of t is the average daily temperature above 70°F? Use a calculator and round answers to the nearest whole number.

2 Solve Trigonometric Equations Using a Calculator MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Use a calculator to solve the equation on the interval 0 ≤ θ < 2π. Round the answer to two decimal places.

1) sin θ = 0.35 A) 0.36, 2.78 B) 0.36, 5.92 C) 0.36, 3.50 D) 0.36, 1.93

Page 16

2) cos θ = 0.53 A) 1.01, 5.27

B) 1.01, 2.13

C) 1.01, 4.15

D) 1.01, 2.58

3) tan θ = 4.2 A) 1.34, 4.48

B) 1.34, 4.94

C) 1.34, 1.80

D) 1.34, 2.91

4) sin θ = -0.42 A) 3.58, 5.85

B) 0.43, 5.85

C) 0.43, 3.57

D) 0.43, 2.00

5) cos θ = -0.69 A) 2.33, 3.95

B) 2.33, 5.47

C) 0.81, 3.95

D) 0.81, 2.33

6) 2 csc θ = 5 A) 0.41, 2.73

B) 0.20

C) 0.41

D) 0.20, 2.94

3 Solve Trigonometric Equations Quadratic in Form MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the equation on the interval 0 ≤ θ < 2π.

1) cos2 θ + 2 cos θ + 1 = 0 A) π

2) 2 sin2 θ = sin θ π 5π A) 0, π, , 6 6 3) csc5 θ - 4 csc θ = 0 π 3π 5π 7π , , A) , 4 4 4 4 4) sin2 θ + sin θ = 0 3π A) 0, π, 2

B) 2π

C)

π 3π , 2 2

D)

π 7π , 4 4

B)

π 3π π 2π , , , 2 2 3 3

C)

π 5π , 6 6

D)

π 2π , 3 3

B)

π 3π π 5π , , , 4 4 6 6

C)

π 5π π 5π , , , 4 4 3 3

D)

π 3π π 5π , , , 4 4 3 6

B) 0, π,

4π 5π , 3 3

C) 0, π,

π 5π , 3 3

D) 0, π,

π 2π , 3 3

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Solve the problem.

5) A water wheel rotates through the angle θ, the water level L behind the wheel changes according to the equation L = 1 - sin θ - 2 cos2 θ where L is measured in inches. Determine the values of θ for which the water level is zero. Find the exact values. Do not use a calculator.

4 Solve Trigonometric Equations Using Fundamental Identities MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the equation on the interval 0 ≤ θ < 2π.

1) tan (2θ) - tan θ = 0 A) 0, π C)

B)

π π 2π 7π 7π 13π 5π , , , , , , 12 6 3 12 6 12 3

π 5π , 4 4

D) 0

2) tan θ + sec θ = 1 A) 0

3) sec

B)

π 4

5π 4

B)

π 5π , 4 4

θ θ = cos 2 2

A) 0 C)

Page 17

C)

π π 2π 7π 7π 13π 5π , , , , , , 12 6 3 12 6 12 3

D) 0,

π 5π , π, 4 3

D) No solution

4) csc (3θ) = 0 2π 4π , π, A) 0, 3 3 5) sin2 (2θ) = 1 π 3π 5π 7π A) , , , 4 4 4 4 6) sec 2 θ - 2 = tan 2 θ π A) 3 7) sin2 θ - cos2 θ = 0 π 3π 5π 7π A) , , , 4 4 4 4 8) cos (2θ) = 2 - cos (2θ) π 7π 9π 15π , , A) , 8 8 8 8 9) sin (2θ) + sin θ = 0 4π 2π , π, A) 0, 3 3 10) 3 cot2 θ - 4 csc θ = 1 π 5π A) , 6 6

B)

π 3π 5π 7π , , , 4 4 4 4

C)

π 9π , 8 8

D) No solution

2π 4π , π, 3 3

C)

π 9π , 8 8

D) No solution

B) 0,

B)

π 6

C)

π 4

D) No solution

B)

π π , 4 6

C)

π π , 4 3

D)

C)

π 3π 5π 7π , , , 4 4 4 4

D) No solution

B) 0,

2π 4π , π, 3 3

π 4

B)

π 9π , 8 8

C)

π 3π 5π 7π , , , 4 4 4 4

D) No solution

B)

7π 11π , 6 6

C)

π 6

D)

7π 6

Solve the problem.

11) The altitude of a projectile in feet (neglecting air resistance) is given by 16 y = (tan θ)x x2 , v2 cos2 θ where x is the horizontal distance covered in feet and v is the initial velocity of the projectile at an angle θ from the horizontal. Find the firing angle (in degrees) of a projectile fired at an initial velocity of 100 feet per second so that it strikes the ground 312.5 feet from the firing point.

A) 45° B) 30° C) 22.5° D) 50° 5 Solve Trigonometric Equations Using a Graphing Utility SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Use a calculator to solve the equation on the interval 0 ≤ x < 2π. Round the answer to one decimal place if necessary.

1) x + 3 sin x = 1 2) 2x - 3 cos x = 0 3) ex = cos x 4) 2x2 - 3x sin x = 2

Page 18

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Use a graphing utility to solve the equation on the interval 0 ° ≤ x < 360°. Express the solution(s) rounded to one decimal place.

5) 2 + 13 sin x = 14 cos2 x A) 34.9°, 214.9°

B) 34.9°, 145.2°

C) 55.2°, 124.9°

D) 214.9°, 325.2°

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

6) -11 + 24 sin x = 16 cos 2 x MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

7) sin2 x - 8 sin x + 16 = 0 A) 28.2°, 151.8°, 208.2°, 331.8° C) 28.2°, 151.8°

B) 208.2°, 331.8° D) No solution

8) sin2 x + 8 sin x + 16 = 0 A) 28.2°, 151.8°, 208.2°, 331.8° C) 28.2°, 151.8°

B) 208.2°, 331.8° D) No solution

9) sin2 x - 8 sin x - 4 = 0 A) 208.2°, 331.8° C) 28.2°, 151.8°

B) 28.2°, 151.8°, 208.2°, 331.8° D) No solution

10) sin2 x + 8 sin x - 4 = 0 A) 28.2°, 151.8° C) 208.2°, 331.8°

B) 28.2°, 151.8°, 208.2°, 331.8° D) No solution

11) tan2 x + 5 tan x + 3 = 0 A) 103.1°, 145.1°, 283.1°, 325.1° C) 49.8°, 130.2°, 229.8°, 310.2°

B) 70.5°, 109.5°, 180.0° D) 51.8°, 128.2°

12) 3 cos2 x + 2 cos x = 1 A) 70.5°, 180.0°, 289.5° C) 49.8°, 130.2°, 229.8°, 310.2°

B) 103.2°, 145.2°, 283.2°, 325.2° D) 51.8°, 128.2°

13) 7 cot2 x - 5 = 0 A) 49.8°, 130.2°, 229.8°, 310.2° C) 70.5°, 109.5°, 180.0°

B) 103.2°, 145.2°, 283.2°, 325.2° D) 51.8°, 128.2°

14) cos2 x + cos x - 1 = 0 A) 51.8°, 308.2° C) 70.5°, 109.5°, 180.0°

B) 103.2°, 145.2°, 283.2°, 325.2° D) 49.8°, 130.2°, 229.8°, 310.2°

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Solve the problem.

15) A weight is suspended on a system of spring and oscillates up and down according to P = 0.1[3 cos (8t) - sin (8t)] where P is the position in meters above or below the point of equilibrium (P = 0) and t is time in seconds. Find the time when the weight is at equilibrium. Find all values of t, 0 ≤ t ≤ 1, rounded to the nearest 0.01 second.

Page 19

16) The ground movement of an earthquake near a fault line is modeled by the equation π 2M 1d = D tan 2 S where M is the horizontal movement (in meters) at a distance d (in kilometers) from the earthquake, D is the depth (also in kilometers) below the surface of the center of the earthquake, and S is the total horizontal displacement (also in meters) at the fault line. What is the horizontal movement 5 kilometers from an earthquake centered 3 kilometers below the surface with a total horizontal displacement of 4 meters? Round the answer to the nearest 0.01 meter.

17) The seasonal variation in the length of daylight can be represented by a sine function. For example, the daily 41 5 2πx number of hours of daylight in a certain city in the U.S. can be given by h = , where x is the + sin 4 3 365 number of days after March 21 ( disregarding leap year). On what day(s) will there be about 10 hours of daylight?

6.4 Trigonometric Identities 1 Use Algebra to Simplify Trigonometric Expressions SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Simplify the trigonometric expression by following the indicated direction.

1) Rewrite in terms of sine and cosine: cot x · tan x 2) Multiply

cos θ 1 + sin θ by 1 - sin θ 1 + sin θ

3) Rewrite over a common denominator:

4) Multiply and simplify:

5) Factor and simplify:

1 1 + 1 - cos θ 1 + cos θ

(cot θ + 1)(cot θ + 1) - csc2 θ cot θ

4 sin2 θ + 5 sin θ + 1 sin2 θ - 1

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Simplify the expression. cos θ 6) + tan θ 1 + sin θ A) sec θ 7) (1 + cot θ)(1 - cot θ) - csc2 θ A) -2 cot2 θ

B) cos θ + sin θ

C) 1

D) sin2 θ

B) 0

C) 2

D) 2 cot2 θ

2 Establish Identities SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Establish the identity.

1) (sin x)(tan x cos x - cot x cos x) = 1 - 2 cos 2 x

Page 20

2) cot2 x = (csc x - 1)(csc x + 1) 3) cos x csc x tan x = 1 4) 1 + sec2 x sin2 x = sec2 x 5)

1 + csc x = cos x + cot x sec x

6) tan2 x = sec2 x - sin2 x - cos2 x 7)

sin x sin x + = 2 csc x 1 - cos x 1 + cos x

8) cos2 x - sin2 x = 1 - 2 sin2 x

9)

cot2 x 1 + sin x = csc x - 1 sin x

10) cot 2 x + csc 2 x = 2 csc 2 x - 1 11)

cot x csc x - 1 = 1 + csc x cot x

12)

cot 2 x 1 - sin x = csc x + 1 sin x

13) -tan 2 x + sec 2 x = sec 2 x cos 2 x 14) sec 4 x - tan 4 x = sec 2 x + tan 2 x 15)

1 - sin t cos t = cos t 1 + sin t

16)

cos t 1 + sin t + = 2 sec t 1 + sin t cos t

17)

sin x + cos x 1 + 2 sin x cos x = sin x - cos x 2 sin 2 x - 1

18)

csc x - 1 cot 2 x = csc x + 1 csc 2 x + 2 csc x + 1

19) csc 4 x - cot 4 x = csc 2 x + cot 2 x

20)

Page 21

1 + cot t tan2 t + 2 - csc2 t tan t + = tan t tan t - 1 1 - cot t

21) sin3 x cos2 x = sin x (cos2 x - cos4 x) 22) csc3 x tan2 x = csc x (1 + tan2 x) 23) cot x sec 4 x = cot x + 2 tan x + tan3 x 24)

sin x sin x + = 2 tan 2 x csc x - 1 csc x + 1

25)

cos x 2 cos x cos x = sec x - 1 sec x + 1 tan2 x

26)

1 - 2 sec x - 3 sec 2 x 1 - 3 sec x = 1 - sec x -tan 2 x

27)

5 csc2 x + 4 csc x - 1 5 csc x - 1 = csc x - 1 cot 2 x

28) cot3 x = cot x (csc2 x - 1) 29) tan5 x = tan x (sec 4 x - 2 sec 2 x + 1) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Complete the identity.

1 30) sec θ =? sec θ A) sin θ tan θ

B) 1 + cot θ

C) -2 tan2 θ

D) sec θ csc θ

B) sin θ tan θ

C) -2 tan2 θ

D) sec θ csc θ

B) 1 + cot θ

C) sin θ tan θ

D) sec θ csc θ

B) -2 tan2 θ

C) 1 + cot θ

D) sin θ tan θ

C) sin θ tan θ

D) sec θ csc θ

C) -sec 2 θ

D) 1 - sin θ

31) csc θ(sin θ + cos θ) = ?

A) 1 + cot θ 32)

sin θ sin θ =?

1 + sin θ 1 - sin θ A) -2 tan2 θ

33)

sin θ cos θ + =? cos θ sin θ A) sec θ csc θ

34) tan2 θ - 3 sin θ tan θ sec θ = ?

A) -2 tan2 θ B) 1 + cot θ 35)

(sin θ + cos θ)2 =?

1 + 2 sin θ cos θ A) 1

Page 22

B) 0

36)

1 1 =? cos2 θ cot2 θ B) 0

C) -sec 2 θ

D) 1 - sin θ

B) 1

C) 0

D) 1 - sin θ

B) -sec 2 θ

C) 1

D) 0

B) 1 - sin θ

C) -sec 2 θ

D) 1

B) sin θ + 1

C) sin θ

D) cos2 θ

B) sec 2 θ (1 + tan 2 θ)

C) sec 2 θ + tan 2 θ

D) 2

B) sin θ + cos θ

C) csc θ tan θ

D) csc2 θ + tan 2 θ

B) sin2 θ + 1

C) cot2 θ + 1

D) cot2 θ - 1

44) sec 4 θ + sec2 θ tan 2 θ - 2 tan 4 θ = ?

A) 3 sec2 θ - 2 B) 4 sec2 θ

C) sec 2 θ + 2

D) tan2 θ - 1

45) tan4 θ - sec4 θ = ?

A) -2 tan2 θ - 1

B) tan 2 θ - sec2 θ

C) sec 2 θ + tan 2 θ

D) sec 2 θ

B) 1

C) csc2 θ

D) sec 2 θ

B) tan2 θ

C) 1

D) csc2 θ

48) cos θ - cos θ sin2 θ = ?

A) cos3 θ

B) tan2 θ

C) sec 2 θ

D) sin θ

49) sin2 θ + tan 2 θ + cos2 θ = ?

A) sec 2 θ

B) tan2 θ

C) cos3 θ

D) sin θ

A) 1 37) 2 tan θ - (1 + tan θ)2 = ?

A) -sec 2 θ 38) tan θ(cot θ - cos θ) = ?

A) 1 - sin θ 39)

sec θ sin θ -1=? tan θ A) 0

40)

sin2 θ - 1 =? sin θ + 1 A) sin θ - 1

41) sec 4 θ - 2 sec 2 θ tan 2 θ + tan 4 θ = ?

A) 1 42)

csc θ (sin2 θ + cos2 θ tan θ) =? sin θ + cos θ A) 1

43) sin2 θ + sin2 θ cot2 θ = ?

A) 1

46)

csc θ cot θ =? sec θ A) cot2 θ

47)

1 + sec θ cos θ = ? cot2 θ A) sec 2 θ

Page 23

50)

7 + 7 sin θ =?

-5 cos θ A)

7 cos θ -5(1 - sin θ)

B)

7 sin θ -5(1 + cos θ)

C)

7 sin θ -5(1 - cos θ)

D)

7 cos θ -5(1 + sin θ)

51) ln cot x = ?

A) ln cos x - ln sin x C)

B) ln sin x - ln cos x

ln cos x ln sinx

52) ln csc x = ?

A) - ln sin x 53) sin (tan-1 v) = ? v v2 + 1 A) v2 + 1 54) cos (tan-1 v) = ? v2 + 1 A) v2 + 1

D)

ln sin x ln cos x

B) - ln cos x

C) ln sin x

D) ln cos x

B)

v v2 - 1 v2 - 1

C) v v2 + 1

D)

v2 + 1 v2 + 1

B)

v2 - 1 v2 - 1

C) v v2 + 1

D)

v v2 + 1 v2 + 1

B)

v2 + 1

C)

v2 - 1

D)

v2 + 1 v

55) cos (sin-1 v) = ?

A)

1 - v2

6.5 Sum and Difference Formulas 1 Use Sum and Difference Formulas to Find Exact Values MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the exact value of the expression.

11π 1) sin 12 A)

2) cos

2( 3 - 1) 4

B)

64

2

C)

2+ 4

6

D) -

6+ 4

2

B) -

2( 3 - 1) 4

C)

2( 3 - 1)

D) - 2( 3 - 1)

B) 2 +

3

C)

3-2

D) -2 -

π 12

A) 2 -

Page 24

6

5π 12

A)

3) tan

24

3

3

4) sin 15° A)

2( 3 - 1) 4

2( 3 + 1) 4

B)

5) sin 165° 2( 3 - 1) A) 4

B) -

6) tan 255° A) 3 + 2

2( 3 - 1) 4

B) - 3 + 2

7) sin 255° - 2( 3 + 1) A) 4

B)

- 2( 3 - 1) 4

2( 3 + 1) 4

C) -

2( 3 - 1) 4

D) -

C) - 2( 3 + 1)

D) - 2( 3 - 1)

C) - 3 - 2

D)

3-2

D)

2( 3 - 1) 4

2( 3 + 1) 4

C)

8) tan 165° A) -2 -

B) 2 +

3

9) sin 15° cos 45° + cos 15° sin 45° 3 1 A) B) 2 2 10) sin 250° cos 10° - cos 250° sin 10° 3 1 A) B) 2 2 11) sin 185° cos 65° - cos 185° sin 65° 3 1 A) B) 2 2 12) sin 20° cos 100° + cos 20° sin 100° 3 1 A) B) 2 2 13) cos 50° cos 10° - sin 50° sin 10° 1 3 A) B) 2 2 14) cos

3 2

5π 2π cos 18 9

A) 0

Page 25

C)

2- 3 4

D)

2+ 3 4

C)

1 4

D)

3 3

C)

25 6

D)

3 2

C)

37 12

D) -

3 2

C)

1 3

D) -

3 2

C)

1 4

D)

3

C)

1 4

D) 1

5π π 5π π cos sin + sin 12 4 12 4

A)

15) cos

3

B)

- sin

1 2

5π 2π sin 18 9 B) -1

C) 1

D)

2 2

16)

tan 20° + tan 10° 1 - tan 20° tan 10° 3 3

A)

17)

3 3

D) 2

B) - 3

C) -

1 2

D) -2

B) -

3 3

C) -

1 2

D) -2

B) -

3 3

C) -

1 2

D) -2

B)

3

C)

3 3

D) -

1 - tan 80° tan 70° tan 80° + tan 70° A) -

3

Find the exact value under the given conditions.

15 π 21 π 21) sin α = , 0 < α < ; cos β = , 0<β< 17 2 29 2 A) -

22) sin α = A)

132 493 4 π , < α < π; 5 2

A) -

cos β =

12 3π , < α < 2π; 13 2

B)

352 377

tan β = B) -

155 493

345 377

475 493

33 65

D) -

16 65

Find sin (α + β).

C) -

20 π , <β<π 21 2

D)

Find sin (α - β).

C) -

5 π , <β<π 13 2

3 3

Find cos (α + β).

C)

63 65

cos β = -

135 377

135 377

468 493

5 π , 0<β< 13 2 B)

21 3π , π<α< ; 20 2

24) sin α = A)

B)

56 65

23) tan α =

Page 26

1 2

tan 65° - tan (-55°) 1 + tan 65° tan (-55°) A) - 3

20)

C)

tan 165° - tan 45° 1 + tan 165° tan 45° A) - 3

19)

3

tan 40° + tan 110° 1 - tan 40° tan 110° A) -

18)

B)

152 377

D)

345 377

D)

352 377

Find cos (α + β).

C)

152 377

4 π , < α < π; 5 2

25) sin α = A)

-6 + 4 21 25

26) sin α = A)

A)

3 3π , π<α< ; 5 2

1 π , 0 <α< ; 3 2

30) cos α = -

sin β = -

A) -

171 140

31) cos α = A) -

12 π , < α < π; 13 2

5 π , < α < π; 13 2

21 220

2 21 π , <β<π 2 21

6 - 4 21 25

sin β = B)

9 3-8 2 5

21 3π , π<β< 5 2

-6 + 4 21 25

tan β = B)

24 π , < α < π; 25 2

8 + 3 21 25

cos β = B)

48 + 7 21 125

29) cos α = A)

4 3π , < α < 2π; 5 2

6 + 4 21 25

28) cos α = -

2 π , 0<β< 5 2 B)

6 + 4 21 25

27) sin α = A)

cos β =

21 3π , π<β< 5 2

-48 - 7 21 125

1 π , - <β<0 2 2

B)

sin β = B)

sin β = B)

9 3+8 2 5 8 π , <α<π 17 2 3 20 15 π , <α<π 17 2 401 220

Find cos (α - β).

C)

8 - 3 21 25

D)

6 - 4 21 25

D)

-8 - 3 21 25

D)

8 - 3 21 25

D)

-14 + 24 21 125

D)

9 3+8 2 3

Find sin (α - β).

C)

-8 + 3 21 25

Find cos (α + β).

C)

-8 - 3 21 25

Find cos (α + β).

C)

14 - 24 21 125

Find tan(α + β).

C)

9 3-8 2 3

Find tan(α + β).

C) -

171 221

D) -

171 220

D) -

3 20

Find tan(α - β).

C)

3 20

2 Use Sum and Difference Formulas to Establish Identities SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Establish the identity. π 1) sin x + = cos x 2

Page 27

2) cos x +

π = -sin x 2

3) cos x +

3 1 π cos x - sin x = 2 2 6

4) sin x -

π 2 (sin x - cos x) = 4 2

5) tan x -

π tan x - 1 = 4 1 + tan x

6) sin

π +x = 4

7) tan

π + x = -cot x 2

8) sec

π + u = -csc u 2

9) csc

π + u = sec u 2

2 (cos x + sin x)

10) sin (x + y) - sin (x - y) = 2 cos x sin y 11) cos (x - y) - cos (x + y) = 2 sin x sin y 12) cot ( x + y) cot (x - y) =

13)

1 - tan2 x tan2 y tan2 x - tan2 y

cos (x - y) 1 + tan x tan y = cos (x + y) 1 - tan x tan y

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Complete the identity.

14) cos (α + β) cos (α - β) = ?

A) cos2 β - sin2 α

B) cos (α2 ) cos (β 2 ) + sin (α2 ) sin (β 2 ) D) 2 - sin2 α - sin2 β

C) cos2 β - 2 sin2 α sin2 β 15) csc (α - β) = ? sec α sec β A) tan α - tan β C)

B) csc α sec β - sec α csc β

sec α sec β tan α tan β - 1

D) csc α - csc β

16) sin (α + β) cos β - cos (α + β) sin β = ?

17)

A) sin α

B) sin α cos2 β - sin α sin2 β

C) sin α cos β - cos α sin β

D) 2 sin β cos β (sin α - cos α )

cos (α - β) =? sin α cos β A) cot α + tan β

Page 28

B) cos α + sin β

C) cos α - sin β

D) cot α - 1

18)

sin (α - β) =? sin (α + β) C)

sin α - sin β sin α + sin β

D)

tan α tan β -1 tan α tan β + 1

B) sec 2 α - sec2 β

C)

1 cos2 α - cos2 β

D)

sec α sec β cos2 α - sin2 β

21) sin (π - θ) = ?

A) sin θ

B) cos θ

C) -sin θ

D) -cos θ

22) tan (π - θ) = ?

A) -tan θ

B) tan θ

C) -cot θ

D) cot θ

A)

tan α - tan β tan α + tan β

B) -1

19) sin (α + β) cos (α - β) = ?

A) sin α cos α + sin β cos β B) sin α cos α - sin α cos β + sin β cos α + sin β cos β C) sin2 α - cos2 β D) sin2 α cos2 β - sin2 β cos2 α 20) sec (α + β) sec (α - β) = ? sec 2 α sec 2 β A) 1-tan2 α tan2 β

3 Use Sum and Difference Formulas Involving Inverse Trigonometric Functions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the exact value of the expression.

3 1 1) sin cos-1 - sin-1 2 2 A) 0

B) 1

C)

2 3 2

D)

3 3

4 3 2) cos tan-1 - sin-1 3 5 A)

24 25

3) sin sin-1 A)

B)

2 6 5

C)

2 3 5

D) 1

B)

2 6 5

C)

2 3 5

D)

2 3 + 2 10 9

B)

2 6 5

C)

2 3 5

D)

2 3 + 2 10 9

B)

2 6 5

C)

2 3+1 5

D)

2 3+4 3 5

2 1 + cos-1 3 3

2 + 2 10 9

3 1 4) tan tan-1 + sin-1 4 2 A)

9+4 3 12 - 3 3

5) cos sin-1 A)

Page 29

1 1 - tan-1 3 2

4 10 + 15

5

5 4 6) cos tan-1 - cos-1 12 5 A)

63 65

B)

13 24

C)

7 13

D)

52 65

Write the trigonometric expression as an algebraic expression containing u and v.

7) cos (sin-1 u - cos-1 v) A) v 1 - u2 + u 1 - v2 C) uv - ( 1 - u2 )( 1 - v2 )

B) v 1 - u2 - u 1 - v2 D) uv + ( 1 - u2 )( 1 - v2 )

8) cos (tan-1 u + tan-1 v) A)

1 - uv u2 + 1 ·

v2 + 1

1 + uv

B)

u2 + 1 ·

v2 + 1

C)

u2 + 1 · v2 + 1 1 - uv

D)

C)

u2 + 1 · v2 + 1 1 - uv

D)

C)

u2 + 1 · v2 + 1 1 - uv

D)

u+v 2 u + 1 · v2 + 1

9) sin (tan-1 u + tan-1 v) A)

u+v u2 + 1 ·

v2 + 1

1 + uv

B)

u2 + 1 ·

v2 + 1

1 - uv u2 + 1 ·

v2 + 1

10) sin (tan-1 u - tan-1 v) A)

u-v u2 + 1 ·

v2 + 1

1 - uv

B)

u2 + 1 ·

v2 + 1

11) cos (sin-1 u + cos-1 v) A) v 1 - u2 - u 1 - v2

1 + uv u2 + 1 ·

B) v 1 - u2 + u 1 - v2 D) uv + ( 1 - u2 )( 1 - v2 )

C) uv - ( 1 - u2 )( 1 - v2 ) 4 Solve Trigonometric Equations Linear in Sine and Cosine

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the equation on the interval 0 ≤ θ < 2π.

1) cos θ - sin θ = 0 π 5π π A) , B) 4 4 4 2) cos θ = sin θ π 5π A) , 4 4 3)

3 sin θ - cos θ = -1 4π A) 0, 3

4) sin θ = - 2 - cos θ 5π A) 4

Page 30

C)

π 2

D)

π π , 6 3

3π 5π , 4 4

D)

3π 7π , 4 2

2π 3

D)

3π π , 2 6

D)

π 2

B)

π 7π , 4 4

C)

B)

π 7π , 2 6

C) 0,

B)

π 4

C)

3π 2

v2 + 1

5) sin θ + 3 cos θ = -1 3π 5π , A) 2 6

B)

π 7π , 2 6

C) 0,

2π 3

D)

3π π , 2 6

6.6 Double-angle and Half-angle Formulas 1 Use Double-angle Formulas to Find Exact Values MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Use the information given about the angle θ, 0 ≤ θ ≤ 2π, to find the exact value of the indicated trigonometric function.

12 π 1) sin θ = , 0<θ< Find cos (2θ).

13 2 A) -

2) cos θ = A) -

3) tan θ = A)

119 169

B)

5 3π , < θ < 2π 13 2

Find sin (2θ).

120 169

B)

21 3π , π<θ< 20 2

Find sin (2θ).

840 841

4) csc θ = A) -

7 25

B)

A)

Page 31

3 , tan θ > 0 2

1 9

24 25

6 11 , csc θ > 0 11

2 2 , tan θ < 0 3

-4 2 9

120 169

D) -

C) -

119 169

D)

C) -

41 841

D) -

840 841

C)

7 25

D) -

24 25

C)

4 5 9

D)

-4 5 9

7 18

D)

5 11 18

D)

4 2 9

C)

118 169

120 169

Find cos (2θ).

B) -

-5 11 18

7) sin θ =

41 841

Find cos (2θ).

6) sec θ = A)

119 169

5 π , <θ<π 4 2

5) csc θ = A)

B)

119 169

1 9

Find sin (2θ).

B)

7 18

C) -

Find sin (2θ).

B) -

7 9

C)

7 9

1 , csc θ < 0 5

8) cos θ = A) -

23 25

A)

7 25

10) cos θ = A)

B)

4 3π , < θ < 2π 5 2

9) sin θ = -

Find cos (2θ).

-4 6 25

4 5

23 25

D)

C)

24 25

D) -

C)

1 5

D)

3 5

4 6 25

Find cos (2θ).

B) -

5 π , 0<θ< 2 5

C)

7 25

24 25

Find sin (2θ).

B)

2 5

Find the exact value of the expression.

3 11) sin 2 cos-1 5 A) -

24 25

12) cos 2 sin-1 A)

12 25

C)

24 25

D)

12 25

B) -

12 13

C)

10 13

D)

2 5 + 10 13

2 6 5

C)

2 3 5

D)

2 3 + 2 10 9

D)

7 25

5 13

119 169 2 1 + 2 sin-1 3 3

13) cos sin-1 A)

B) -

7 5+8 2 27

B)

Solve the problem.

14) If sin θ = A) -

24 25

15) If tan θ = A)

B) -

16) If cos θ = -

Page 32

7 25

C)

24 25

7 , and θ terminates in quadrant III, then find cos 2θ.

24

527 625

A) -

4 , and θ terminates in quadrant IV, then find sin 2θ.

5

119 169

B)

336 625

C) -

336 625

D) -

527 625

5 , and θ terminates in quadrant II, then find cos 2θ.

13 B) -

120 169

C)

120 169

D)

119 169

17) If sin θ = A)

24 7

B)

336 527

B) -

19) If cos θ = A)

24 7

D) -

7 24

C)

527 336

D) -

527 336

C)

169 119

D)

C)

7 5

D) -

C) -

336 527

5 π , and < θ < π, then find tan 2θ.

13 2

120 119

B)

20) If cos 2θ = A)

7 24

7 3π , and π < θ < , then find tan 2θ.

24 2

18) If tan θ = A)

4 3π , and < θ < 2π, then find tan 2θ.

5 2

119 120

169 120

24 π , and < 2θ < π, then find sin θ.

25 2

7 2 10

B) -

7 2 10

7 5

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

21) The path of a projectile fired at an inclination θ (in degrees) to the horizontal with an initial speed v0 is a parabola. The range R of the projectile, that is, the horizontal distance that the projectile travels, is found by using the formula

R=

2 v0 g

sin (2θ)

where g is the acceleration due to gravity. The maximum height H of the projectile is

H=

2 v0 4g

(1 - cos (2θ))

Find the range R and the maximum height H in terms of g if the projectile is fired with an initial speed of 200 meters per second at an angle of 15° and then at an angle of 22.5 °. Do not use a calculator, but simplify the answers.

2 Use Double-angle Formulas to Establish Identities SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Establish the identity.

1) tan 2 u (1 + cos 2u) = 1 - cos 2u 1 2) cos4 x = ( 3 + 4 cos 2x + cos 4x) 8 3) cos 3x = cos3 x - 3 sin2 x cos x 4) sin 4u = 2 sin 2u cos 2u

Page 33

5) sin 4x = (4 sin x cos x)(1 - 2 sin2 x) 6) sin3 3x =

1 (sin 3x)(1 - cos 6x) 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Complete the identity.

7) cos (4θ) = ?

A) cos4 θ - 6 sin2 θ cos2 θ + sin4 θ C) cos3 θ - 3 sin2 θ cos θ

B) 3 sin θ - 4 sin3 θ D) 4 sin θ cos3 θ - 4 sin3 θ cos θ

8) sin (4θ) = ?

A) 4 sin θ cos3 θ - 4 sin3 θ cos θ C) 1 - 8 sin2 θ cos2 θ 9) csc (2θ) - sec (2θ)(tan θ - 1) = ? cos θ - sin θ A) sin (2θ)(sin θ + cos θ) C)

10)

2 tan θ 1 - tan θ

D)

csc θ + sec θ sin (2θ)(sin θ - cos θ)

B) tan (2θ) + 2 tan2 θ

C)

tan (2θ) + tan2 θ sec (2θ)

D) 2 tan θ + 2

1 sin (3θ) sin (2θ) 6

C)

1 sin (2θ) cos (2θ) 2

D)

1 sin2 (2θ) 4

C)

tan (2θ) sin θ

D)

cot (2θ) cos θ

B)

1 sec θ csc θ = ?

2 A) csc (2θ)

13) 1 -

Page 34

sin θ + cos θ sin (2θ)(sin θ - cos θ)

csc θ - sec θ sin (2θ)(sin θ + cos θ)

11) sin θ cos3 θ + sin3 θ cos θ = ?

1 A) sin (2θ) 2

14)

B)

sin (2θ) + 2 sin2 θ =? cos (2θ) A)

12)

B) 3 sin θ - 4 sin3 θ D) cos3 θ - 3 sin2 θ cos θ

B) sec (2θ)

1 sin (2θ) = ?

2

A)

sin3 θ + cos3 θ sin θ + cos θ

B)

sin θ cos θ cos (2θ) sin θ + cos θ

C)

sin θ cos (2θ) sin θ + cos θ

D)

sin3 θ - cos3 θ sin θ + cos θ

sin (3θ) =? sin θ A) 4 cos2 θ - 1

B) 2 sin θ cos θ

C) 2 cos (2θ)

D) cos (2θ) + sin θ cos θ

15) (4 sin θ cos θ)(1 - 2 sin2 θ) = ?

A) sin (4θ) 16) sin (2θ) tan θ + cos (2θ) = ?

A) 1 17) csc2 A)

cos (4θ) sin (2θ)

B) sin (2θ) cos (2θ)

C) sin (2θ) + cos (2θ)

D)

B) 2 cos (2θ)

C) sec (2θ)

D) cos (3θ)

θ =?

2 2 1 - cos θ

B)

2 1 + sin θ

1 - sec θ 2

D)

1 + sec θ 2

C) 2 sin2 θ sec θ

D)

2 1 - sec θ

1 + sin θ 2

C)

θ θ 18) csc2 - sec 2 = ?

2 2 A) 4 cot θ csc θ

19) sin

θ θ cos = ?

2 2

A)

sin2 20)

B) 0

1 sin θ 2 θ 2

sin2 θ A)

1 - cos θ 2

C)

1 cos θ 2

D)

B) sin2

θ 2

C)

1 2 - 2 cos θ

θ D) cos2 2

B) tan

θ 2

C) cot

θ 2

D) tan θ

B)

=?

1 2 + 2 cos θ

21) csc (2θ) + cot (2θ) = ?

A) cot θ

Simplify the expression.

22) 4 sin θ cos3 θ - 4 sin3 θ cos θ A) sin 4θ C) 4 sin θ cos θ cos 2θ 23) sin 8x cos 8x 1 A) sin 16x 2

B) sin 2θ cos 2θ D) 4 sin θ cos θ

B) 2 sin 4x

C) cos 8x

D) cos 4x

C) 2 sin 4x

D) cos 4x

24) cos2 4x - sin2 4x A) cos 8x

B)

1 sin 16x 2

25) 4 sin 2x cos 2x A) 2 sin 4x

Page 35

B) cos 8x

C)

1 sin 16x 2

D) cos 4x

26) 1 - 2 sin2 2x A) cos 4x

B) 2 sin 4x

1 sin 16x 2

C) cos 8x

D)

C) 2 sin 4x

D) cos 4x

27) 2 cos2 4x - 1 A) cos 8x

B)

1 sin 16x 2

3 Use Half-angle Formulas to Find Exact Values MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Use the information given about the angle θ, 0 ≤ θ ≤ 2π, to find the exact value of the indicated trigonometric function.

1 π θ 1) sin θ = , 0 < θ < Find sin .

4 2 2 8 - 2 15 4

B)

1 , tan θ > 0 4

Find cos

8 + 2 15 4

B)

1 , csc θ > 0 4

Find sin

6 4

B)

A)

2) sin θ = A)

3) cos θ = A)

4) sec θ = 4, 0 < θ < A)

10 4

A) -

5 5

B)

3 , sin θ > 0 5

Find cos

5 5

A) -

Page 36

B) -

10 10

B)

5 5

C)

6 4

D)

8 - 2 15 4

C)

10 4

D)

8 - 2 15 4

C)

8 - 2 15 4

D)

8 + 2 15 4

C) -

30 10

D)

30 10

C) -

30 10

D)

30 10

C) -

5 5

D) -

θ .

2

5 5

Find sin

8 + 2 15 4

θ .

2

5 5

3 3π , < θ < 2π 5 2

D)

θ .

2

6 4

Find cos

6) cos θ = -

7) sin θ = -

B)

6 4

θ .

2

8 + 2 15 4

Find cos

C)

θ .

2

10 4

3 3π , π<θ< 5 2

5) cos θ = -

A)

π 2

10 4

θ .

2 30 10

8) sec θ = A)

2 5 5

9) sin θ = A)

5 π , <θ<π 3 2

12 3π , < θ < 2π 13 2

Find cos

18 - 6 5 6

B)

4 3π , ≤ θ ≤ 2π 5 2

Find cos

3 10 10

B)

A) -

A) -

D)

5 5

C)

5 26

D) -

2 13 13

C)

5 6

D) -

3+ 6

C)

3 10 10

D) -

2 5

D) -

1 2

2+

2

θ .

2

3 13 13

3 , tan θ > 0 2

10) csc θ = -

2 5

C) -

Find cos B) -

θ .

2

5 5

B) -

3 13 13

11) cos θ =

Find sin

θ .

2

18 + 6 5 6 θ .

2

2 5

Use the Half-angle Formulas to find the exact value of the trigonometric function.

12) sin 22.5° 1 1 1 A) 2- 2 B) 2+ 2 C) 2- 2 2 2 2 13) cos 22.5° 1 2+ A) 2

5

2

B)

1 2

2-

2

C) -

1 2

2-

2

D) -

1 2

2+

2

14) sin 75° 1 A) 2

2+

3

B)

1 2

2-

3

C) -

1 2

2+

3

D) -

1 2

2-

3

15) cos 75° 1 A) 2

2-

3

B)

1 2

2+

3

C) -

1 2

2+

3

D) -

1 2

2-

3

B) 2 -

3

C) -2 -

3

D) -2 +

3

16) tan 75° A) 2 + 17) sin

5π 12

A)

18) cos

1 2

2+

3

B)

1 2

2-

3

C) -

1 2

2+

3

D) -

1 2

2-

3

2-

3

B)

1 2

2+

3

C) -

1 2

2+

3

D) -

1 2

2-

3

5π 12

A)

Page 37

3

1 2

19) sin 165° 1 2A) 2 20) cos 165° 1 A) 2

2+

21) tan 165° A) -2 +

3

1 2

B) -

3

3

2+

1 2

2-

B) 2 +

3

B)

3

3

C)

1 2

2+

3

D) -

1 2

2-

3

C)

1 2

2+

3

D) -

1 2

2-

3

C) 2 -

3

D) -2 -

3

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

22) sin 15° MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the problem.

23) Find cos A)

10 4

24) Find sin A)

A)

A)

Page 38

8-2 4

C)

15

D)

8+2 4

15

8+2 4

15

C)

10 4

D)

8-2 4

15

B)

6 4

C)

8-2 4

15

D)

8+2 4

15

D)

30 10

θ 3 , given that cos θ = - and θ terminates in 90° < θ < 180°.

2 5 B) -

5 5

C) -

30 10

θ 3 , given that sin θ = - and θ terminates in 270° < θ <360°.

2 5

10 10

28) Find tan

6 4

θ , given that sec θ = 4 and θ terminates in 0 <θ < π/2.

2

5 5

27) Find sin A)

B)

10 4

26) Find cos

B)

θ 1 , given that cos θ = and θ terminates in 0 < θ < 90°.

2 4

6 4

25) Find cos A)

θ 1 , given that cos θ = and θ terminates in 0 < θ < π/2.

2 4

B)

5 5

C) -

5 5

D) -

30 10

10 - 1 -3

D)

10 - 1 3

θ , given that tan θ = 3 and θ terminates in π < θ < 3π/2.

2

10 + 1 -3

B)

10 + 1 3

C)

29) Find cos θ, given that cos 2θ = A)

10 4

30) Find sin θ, given that cos 2θ = A)

6 4

1 and θ terminates in quadrant I. 4 B)

6 4

C)

8 - 2 10 4

D)

8-2 5 2

C)

8 - 2 10 4

D)

10 - 2 6 4

1 π and 0 < θ < .

4 2 B)

10 4

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

31) The two equal sides of an isosceles triangle measure three feet. Let the angle between the sides measure θ. Find θ the area A of the triangle as a function of . The answer may include more than one trigonometric function.

2

6.7 Product-to-Sum and Sum-to-Product Formulas 1 Express Products as Sums MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Express the product as a sum containing only sines or cosines.

1) sin (6θ) cos (4θ) 1 A) [sin (10θ) + sin (2θ)] 2 C)

1 [cos (10θ) - cos (2θ)] 2

D)

2) sin (8θ) sin (4θ) 1 A) [ cos (4θ) - cos (12θ)] 2 C)

1 [cos (12θ) - cos (4θ)] 2

1 [cos (11θ) - cos θ] 2

Page 39

1 [sin (12θ) + cos (4θ)] 2

B)

1 [cos (11θ) - sin θ] 2

B)

1 [- cos(3θ) - cos (11θ)] 2

5) sin (3θ) cos (4θ) 1 A) [sin (7θ) - sin θ] 2

D)

D) cos2 (30θ2 )

4) sin (4θ) sin (7θ) 1 A) [cos (3θ) - cos (11θ)] 2 C)

1 [sin (10θ) + cos (2θ)] 2

B) sin2 (32θ2 )

3) cos (6θ) cos (5θ) 1 A) [ cos θ + cos (11θ)] 2 C)

B) sin cos (24θ2 )

1 [cos (11θ) - sin (3θ)] 2

D) sin2 (28θ2 )

B)

1 [cos (7θ) + sin θ] 2

C)

1 [cos (7θ) - cos θ] 2

D) sin cos (12θ2 )

6) cos (4θ) cos (7θ) 1 A) [cos (3θ) + cos (11θ)] 2 C)

7) cos

B)

1 [cos (11θ) - cos (3θ)] 2

1 [cos (11θ) - sin (3θ)] 2

D) cos2 (19θ2 )

11θ θ cos 2 2

A)

1 [cos (5θ) + cos (6θ)] 2

B)

1 [cos (12θ) - sin (10θ)] 4

C)

1 cos2 (11θ) 4

D)

1 [cos (6θ) - sin (5θ)] 2

8) sin

θ 11θ cos 2 2

A)

1 [sin (6θ) - sin (5θ)] 2

B)

1 [cos (12θ) - sin (10θ)] 4

C)

1 sin cos (11θ) 4

D)

1 [cos (6θ) + sin (5θ)] 2

9) -2 sin (5θ) sin θ A) cos (6θ) - cos (4θ)

B) cos (6θ) + cos (4θ)

10) 2 cos (7θ) cos θ A) cos (8θ) + cos (6θ) C) cos (14θ) + cos (2θ)

C) cos (7θ) + cos (3θ)

D) cos (7θ) - cos (3θ)

B) cos (8θ) + sin (6θ) D) cos (10θ) + sin (4θ)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Solve the problem.

11) A product of two oscillations with different frequencies such as f(t) = sin (10t) sin(t) is important in acoustics. The result is an oscillation with ʺoscillating amplitude.ʺ (i) Write the product f(t) of the two oscillations as a sum of two cosines and call it g(t). (ii) Using a graphing utility, graph the function g(t) on the interval 0 ≤ t ≤ 2π. (iii) On the same system as your graph, graph y = sin t and y = -sin t. (iv) The last two functions constitute an ʺenvelopeʺ for the function g(t). For certain values of t, the two cosine functions in g(t) cancel each other out and near-silence occurs; between these values, the two functions combine in varying degrees. The phenomenon is known (and heard) as ʺbeats.ʺ For what values of t do the functions cancel each other?

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Complete the identity.

12) sin (5θ) sin (7θ) cos (5θ) cos (7θ) = ? cos2 (2θ) - cos2 (12θ) A) 4 C)

Page 40

cos2 (12θ) + cos2 (2θ) 4

B)

sin2 (70θ) 4

D) cos2 (70θ)

13) sin θ [sin (5θ) + sin (7θ)] = ?

A) cos θ [cos (5θ) - cos (7θ)] C)

B) cos θ [cos (5θ) + cos (7θ)]

1 cos θ [cos (5θ) - cos (7θ)] 2

D)

1 cos θ [cos (5θ) + cos (7θ)] 2

2 Express Sums as Products MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Express the sum or difference as a product of sines and/or cosines.

1) sin (8θ) + sin (4θ) A) 2 sin (6θ) cos (2θ) B) 2 cos (6θ) sin (2θ)

C) 2 sin (6θ) sin (2θ)

D) 2 sin (12θ)

2) cos (8θ) - cos (4θ) A) -2 sin (6θ) sin (2θ)

B) 2 cos (6θ) cos (2θ)

C) -2 cos (6θ) sin (2θ)

D) 2 cos (2θ)

3) cos (5θ) + cos (3θ) A) 2 cos (4θ) cos θ

B) 2 sin (4θ) sin θ

C) 2 cos (4θ) sin θ

D) 2 cos (4θ)

4) cos (4θ) - cos (6θ) A) 2 sin (5θ) sin θ

B) -2 sin (5θ) sin θ

C) -2 cos (5θ) sin θ

D) cos (-2θ)

Complete the identity. sin (5θ) + sin (11θ) 5) =? cos (5θ) + cos (11θ) B) tan (5θ) + tan (11θ) D) tan (8θ) cot (3θ)

A) tan (8θ) C) 2 tan (8θ) tan (3θ) 6)

cos (3θ) - cos (9θ) =? cos (3θ) + cos (9θ) A) tan (3θ) tan (6θ)

7)

C) cot (6θ)

B) tan (5θ) + tan (3θ)

C)

B) tan α + tan β

C) tan

sin (5θ) cos (3θ)

sin (10θ) cos (6θ)

D)

cos (10θ) sin (6θ)

sin α + sin β =? cos α + cos β A) tan

α+ β 2

9) tan α + cot β = ? cos (α - β) A) cos α sin β

B)

cos (α + β) cos α sin β

10) 1 - cos (2θ) + cos (4θ) - cos (6θ) = ?

A) 4 sin θ cos (2θ) sin (3θ) C) 4 cos θ cos (2θ) sin (3θ)

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D) 0

sin (4θ) + sin (6θ) =? cos (2θ) + cos (4θ) A)

8)

B) -tan (6θ)

C)

α- β 2

sin (α + β) cos α sin β

B) 4 sin θ sin (2θ) sin (3θ) D) 4 cos θ cos (2θ) cos (3θ)

D) cot

D)

α-β 2

sin (α - β) cos α sin β

Express the sum or difference as a product of sines and/or cosines.

11) sin (8θ) - sin (4θ) A) 2 sin (2θ) cos (6θ) B) 2 cos (4θ) cos (6θ) 12) sin (4θ) - sin (6θ) A) -2 sin θ cos (5θ) 13) cos

C) 2 sin (6θ) cos (2θ)

D) 2 sin (2θ)

B) 2 cos (4θ) cos (5θ)

C) 2 sin (5θ) cos θ

D) -2 sin θ

B) 2 sin (2θ) sin θ

C) 2 sin (2θ) sin

3θ 5θ + cos 2 2 θ 2

D) 2 cos (2θ)

5θ sin θ 2

C) 2 cos (5θ) sin θ

D) 2 sin (5θ)

15) sin (6θ) - sin (4θ) A) 2 sin θ cos (5θ)

B) 2 sin (5θ) cos θ

C) -2 sin θ cos (5θ)

D) -2 sin (5θ) cos θ

16) sin (4θ) - sin (2θ) A) 2 sin θ cos (3θ)

B) sin θ cos (3θ)

C) 2 sin (3θ) cos θ

D) sin (3θ) cos θ

A) 2 cos (2θ) cos

14) sin

θ 2

7θ 3θ + sin 2 2

A) 2 sin

5θ cos θ 2

B) 2 sin

Solve the problem.

17) On a Touch-Tone phone, each button produces a unique sound. The sound produced is the sum of two tones, given by y = sin (2πlt) and y = sin (2πht) where l and h are the low and high frequencies (cycles per second) shown on the illustration.

The sound produced is thus given by y = sin (2πlt) + sin (2πht) Write the sound emitted by touching the 8 key as a product of sines and cosines.

A) y = 2 sin (2188πt) cos (484πt) B) y = 2 sin (2329πt) cos (625πt) D) y = 2 sin (625πt) cos (2329πt) C) y = 2 sin (484πt) cos (2188πt) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

18) If two sound sources at the same volume are equidistant from a microphone, the pressure on the microphone is given by p = a cos ω1 t + a cos ω2 t where a, ω1 , ω2 are constants and t is time. Write p as a product of cosine functions.

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Ch. 6 Analytic Trigonometry Answer Key 6.1 The Inverse Sine, Cosine, and Tangent Functions 1 Find the Exact Value of the Inverse Sine, Cosine, or Tangent Function 1) A 2) A 3) A 4) A 5) A 6) A 2 Find an Approximate Value of an Inverse Sine, Cosine, or Tangent Function 1) A 2) A 3) A 4) A 5) A 6) A 7) A 3 Use Properties of Inverse Functions to Find Exact Values of Certain Composite Functions 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 4 Find the Inverse Function of a Trigonometric Function 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 15) A 16) A 5 Solve Equations Involving Inverse Trigonometric Functions 1) A 2) A 3) A 4) A 5) A 6) A Page 43