Post on 16-Oct-2021
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
Chapter 5 – Trigonometric Functions Answer Key
CK-12 PreCalculus Concepts 2
18. (Image from Wikipedia):
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
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
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
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
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
-π/2
π/2
π
3π/2
x
y
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
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
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
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
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
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
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
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
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).
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
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
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
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
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
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
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)
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
-π/2
π/2
π
3π/2
x
y
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
-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)
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
-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
-π/2
π/2
π
3π/2
x
y
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
-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
-π/2
π/2
π
3π/2
x
y
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
-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
-π/2
π/2
π
3π/2
x
y
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
-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
-π/2
π/2
x
y
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
2π
5π/2
x
y
17.
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-3π/2
-π
-π/2
π/2
π
3π/2
x
y
Chapter 5 – Trigonometric Functions Answer Key
CK-12 PreCalculus Concepts 31
18. 5
4
19. 5
12
20. 5
4