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Page 1: Chapter 5 Trigonometric Functions Answer Key 5.1 The Unit ...

Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 1

5.1 The Unit Circle

Answers

1. 340°

2. 115°

3. 140°

4. 330°

5. 315°

6. Negative

7. Positive

8. Positive

9. Positive

10. −1

2

11. 2

12. 1

13. −√3

14. −1

2

15. −2√3

3

16. sin 𝜃 =1

2 and tan 𝜃 =

√3

3

17. cos 𝜃 = −√3

2 𝑎𝑛𝑑 tan 𝜃 =

√3

3

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 2

18. (Image from Wikipedia):

Page 3: Chapter 5 Trigonometric Functions Answer Key 5.1 The Unit ...

Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 3

5.2 The Sinusoidal Function Family

Answers

1.

2.

3. At multiples of 2π.

4. At values of (2𝑛 + 1)𝜋 for all integer values of n.

5. −𝜋, 0, 𝜋, 2𝜋

6. 𝜋

2,

3𝜋

2,

5𝜋

2,

7𝜋

2

7.

-3π/2 -π -π/2 π/2 π 3π/2

-2

-1

1

2

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 4

8.

-3π/2 -π -π/2 π/2 π 3π/2

-2

-1

1

2

x

y

9.

-3π/2 -π -π/2 π/2 π 3π/2

-2

-1

1

2

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 5

10.

-3π/2 -π -π/2 π/2 π 3π/2

-2

-1

1

2

x

y

Page 6: Chapter 5 Trigonometric Functions Answer Key 5.1 The Unit ...

Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 6

5.3 Amplitude of Sinusoidal Functions

Answers

1. Amplitude is the value of a (it is always positive), that appears as the coefficient of sin or cos in the

equation.

2. Amplitude is the vertical distance between the sinusoidal axis and the maximum or minimum values

of the graph.

3. 5

4. 3.5

5. ½

6. ½

7. 3.12

8. 1

9.

-3π/2 -π -π/2 π/2 π 3π/2

-4

-3

-2

-1

1

2

3

4

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 7

10.

-3π/2 -π -π/2 π/2 π 3π/2

-4

-3

-2

-1

1

2

3

4

x

y

11.

-3π/2 -π -π/2 π/2 π 3π/2

-3π/2

-π/2

π/2

π

3π/2

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 8

12.

-3π/2 -π -π/2 π/2 π 3π/2

-4

-3

-2

-1

1

2

3

4

x

y

13.

-3π/2 -π -π/2 π/2 π 3π/2

-4

-3

-2

-1

1

2

3

4

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 9

14.

-3π/2 -π -π/2 π/2 π 3π/2

-1

1

x

y

15. The dotted horizontal line represents the sinusoidal axis at 𝑦 = 2.

-3π/2 -π -π/2 π/2 π 3π/2

-1

1

2

3

4

5

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 10

5.4 Vertical Shift of Sinusoidal Functions

Answers

1.

-3π/2 -π -π/2 π/2 π 3π/2

-4

-2

2

4

6

8

x

y

2.

-3π/2 -π -π/2 π/2 π 3π/2

-1

1

x

y

3.

-3π/2 -π -π/2 π/2 π 3π/2

2

4

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 11

4.

-3π/2 -π -π/2 π/2 π 3π/2

-1

1

2

x

y

5.

-3π/2 -π -π/2 π/2 π 3π/2

-4

-3

-2

-1

x

y

6. Maximum at 4 and minimum at -2.

7. Maximum at -2 and minimum at -6.

8. Maximum at 1.5 and minimum at 0.5.

9. Maximum at 6 and minimum at 4.

10. Maximum at 0 and minimum at -2.

11. 𝑦 = −3 sin 𝑥 + 2

12. 𝑦 = −2 cos 𝑥 − 1

13. 𝑦 =1

2cos 𝑥 − 3

14. 𝑦 = 15 sin 𝑥 − 5

15. 𝑦 = −8 cos 𝑥 + 4

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 12

5.5 Frequency and Period of Sinusoidal Functions

Answers

1. The frequency is 4 and the period is 𝜋

2.

2. The frequency is 2 and the period is 𝜋.

3. The frequency is 1

2 and the period is 4𝜋.

4. The frequency is 3

4 and the period is

8𝜋

3.

5. The frequency is 3 and the period is 2𝜋

3.

6.

-3π/2 -π -π/2 π/2 π 3π/2

-2

2

4

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 13

7.

-3π/2 -π -π/2 π/2 π 3π/2

-6

-4

-2

x

y

8.

-3π/2 -π -π/2 π/2 π 3π/2

-4

-2

xy

9.

-3π/2 -π -π/2 π/2 π 3π/2

-1.5

-1

-0.5

0.5

1

1.5

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 14

10.

-3π/2 -π -π/2 π/2 π 3π/2

-4

-3

-2

-1

1x

y

11. 𝑓(𝑥) = −3 sin(3𝑥) + 2

12. 𝑓(𝑥) = 2cos (2𝑥)

13. 𝑓(𝑥) = cos (1

2𝑥) − 1

14.

3 6 9 12

-2

2

4

6

8

10

x

y

15. 𝑓(𝑥) = 4 ⋅ cos (𝜋

6𝑥) + 6

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 15

5.6 Phase Shift of Sinusoidal Functions

Answers

1.

-3π/2 -π -π/2 π/2 π 3π/2

-4-3-2-1

1234

x

y

2.

-3π/2 -π -π/2 π/2 π 3π/2

-4-3-2-1

1234

x

y

3.

-3π/2 -π -π/2 π/2 π 3π/2

-4-3-2-1

1234

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 16

4.

-3π/2 -π -π/2 π/2 π 3π/2

-4-3-2-1

1234

x

y

5.

-3π/2 -π -π/2 π/2 π 3π/2

-4-3-2-1

1234

x

y

6. Possible answer: 𝑓(𝑥) = 2 sin (x −π

2) + 1

7. Possible answer: 𝑓(𝑥) = −3 sin (𝑥 +𝜋

2)

8. Possible answer: 𝑓(𝑥) = − sin (𝑥 −𝜋

2) − 3

9. Possible answer: 𝑓(𝑥) = cos(𝑥 − 𝜋) + 3

10. Possible answer: 𝑓(𝑥) = −2 cos (𝑥 −𝜋

2) + 1

11. Possible answer: 𝑓(𝑥) = − cos (𝑥 +𝜋

2) − 3

12. 𝑓(𝑥) = −9 cos (𝜋

720(𝑥 − 180)) + 31

13. 39.69°

14. 28.67°

15. 565.5 minutes (about 9:26 AM); 1234.5 minutes (about 20:35 or 8:35 PM).

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 17

5.7 Graphs of Other Trigonometric Functions

Answers

1. You can use cos 𝑥 because sec 𝑥 =1

cos 𝑥.

2. You can use sin 𝑥 because csc 𝑥 =1

sin 𝑥.

3.

-3π/2 -π -π/2 π/2 π 3π/2

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 18

4.

-3π/2 -π -π/2 π/2 π 3π/2

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

5.

-3π/2 -π -π/2 π/2 π 3π/2

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 19

6.

-3π/2 -π -π/2 π/2 π 3π/2

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

7.

-3π/2 -π -π/2 π/2 π 3π/2

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 20

8.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

9.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 21

10.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

11.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 22

12.

-3π/2 -π -π/2 π/2 π 3π/2

-3

-2

-1

1

2

3

x

y

13.

-3π/2 -π -π/2 π/2 π 3π/2

-3

-2

-1

1

2

3

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 23

14. Possible answers: sec 𝑥 =1

cos 𝑥; sec 𝑥 = csc (𝑥 +

𝜋

2)

15. Possible answers: csc 𝑥 =1

sin 𝑥; csc 𝑥 = sec (𝑥 −

𝜋

2)

Page 24: Chapter 5 Trigonometric Functions Answer Key 5.1 The Unit ...

Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 24

5.8 Graphs of Inverse Trigonometric Functions

Answers

1.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3π/2

-π/2

π/2

π

3π/2

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 25

2.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3π/2

-π/2

π/2

π

3π/2

x

y

3. 𝑦 = sin−1(𝑥)

4. 𝑦 = tan (𝑥)

5. 𝑦 = csc (𝑥)

6. 𝑦 = sec−1(𝑥)

7. 𝑦 = cot−1(𝑥)

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 26

8.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3π/2

-π/2

π/2

π

3π/2

x

y

9.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3π/2

-π/2

π/2

π

3π/2

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 27

10.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3π/2

-π/2

π/2

π

3π/2

x

y

11.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3π/2

-π/2

π/2

π

3π/2

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 28

12.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3π/2

-π/2

π/2

π

3π/2

x

y

13.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3π/2

-π/2

π/2

π

3π/2

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 29

14.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3π/2

-π/2

π/2

π

3π/2

x

y

15.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-7π/2

-3π

-5π/2

-2π

-3π/2

-π/2

π/2

x

y

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Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 30

16.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-π/2

π/2

π

3π/2

5π/2

x

y

17.

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-3π/2

-π/2

π/2

π

3π/2

x

y

Page 31: Chapter 5 Trigonometric Functions Answer Key 5.1 The Unit ...

Chapter 5 – Trigonometric Functions Answer Key

CK-12 PreCalculus Concepts 31

18. 5

4

19. 5

12

20. 5

4