11.6: Circular Functions
- Page ID
- 122984
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7.1 Transformations of Graphs
Homework 7-1
1. amplitude 2, period \(2\pi\), midline \(y=-3\)
3. amplitude 1, period \(\dfrac{\pi}{2}\), midline \(y=0\)
5. amplitude 5, period \(6\pi\), midline \(y=0\)
7. amplitude 1, period 2, midline \(y=1\)
9. 
11. 
13. 
15. 
17. \(y=-2 \sin x\)
19. \(y=-2 \cos x\)
21. \(y=-0.75 \cos x\)
23.
a amplitude 2, period \(\dfrac{2\pi}{3}\), midline \(y=0\)
b \(y = -2\sin 3x\)
25.
a amplitude 3, period \(2\pi\), midline \(y=0\)
b \(y = 3 \sin \dfrac{x}{2}\)
27.
a amplitude 0.5, period \(4\pi\), midline \(y = 3.5\)
b \(y = 0.5 \cos \dfrac{x}{2} + 3.5\)
29.
a amplitude 2, period 4, midline \(y=-1\)
b \(y = -1 + 2 \sin \dfrac{\pi x}{2}\)
31.
a
| \(t\) | \(2t\) | \(\cos 2t\) | \(-5 \cos 2t\) | \(2 - 5\cos 2t\) |
| 0 | 0 | 1 | -5 | -3 |
| \(\dfrac{\pi}{4}\) | \(\dfrac{\pi}{2}\) | 0 | 0 | 2 |
| \(\dfrac{\pi}{2}\) | \(\pi\) | -1 | 5 | 7 |
| \(\dfrac{3\pi}{4}\) | \(\dfrac{3\pi}{2}\) | 0 | 0 | 2 |
| \(\pi\) | \(2\pi\) | 1 | -5 | -3 |
b 
33.
a
| \(t\) | \(\dfrac{t}{2}\) | \(\cos \dfrac{t}{2}\) | \(3 \cos \dfrac{t}{2}\) | \(1 + 3 \cos \dfrac{t}{2}\) |
| 0 | 0 | 1 | 3 | 4 |
| \(\pi\) | \(\dfrac{\pi}{2}\) | 0 | 0 | 1 |
| \(2\pi\) | \(\pi\) | -1 | -3 | -2 |
| \(3\pi\) | \(\dfrac{3\pi}{2}\) | 0 | 0 | 1 |
| \(4\pi\) | \(2\pi\) | 1 | 3 | 4 |
b 
35.
a
| \(t\) | \(\dfrac{t}{3}\) | \(\sin \dfrac{t}{3}\) | \(2\sin \dfrac{t}{3}\) | \(-3 + 2 \sin \dfrac{t}{3}\) |
| 0 | 0 | 0 | 0 | -3 |
| \(\dfrac{3\pi}{2}\) | \(\dfrac{\pi}{2}\) | 1 | 2 | -1 |
| \(3\pi\) | \(\pi\) | 0 | 0 | -3 |
| \(\dfrac{9\pi}{2}\) | \(\dfrac{3\pi}{2}\) | -1 | -2 | -5 |
| \(6\pi\) | \(2\pi\) | 0 | 0 | -3 |
39. 
41. 
43. 
45.
a 
b \(W(t) = 12 + 8 \cos \left( \dfrac{\pi t}{6} \right) \)
47.
a 
b \(h = 10 + 14 \cos \left( \dfrac{\pi t}{5} \right) \)
49. \(H=12-2.4 \cos \left(\dfrac{\pi t}{6}\right)\)
51. \(y=155 \cos (120 \pi t)\)
53.
a 
b period \(\dfrac{\pi}{2}\), midline \(y = 0\)
55.
a 
b period \(\dfrac{\pi}{3}\), midline \(y=4\)
57.
a 
b period \(4 \pi\), midline \(y=3\)
59. \(\dfrac{\pi}{12}, \dfrac{5\pi}{12}, \dfrac{7\pi}{12}, \dfrac{11\pi}{12}, \dfrac{13\pi}{12}, \dfrac{17\pi}{12}, \dfrac{19\pi}{12}, \dfrac{23\pi}{12}\)
61. \(\dfrac{7\pi}{12}, \frac{11\pi}{12}, \dfrac{19\pi}{12}, \dfrac{23\pi}{12}\)
63. \(\dfrac{\pi}{12}, \dfrac{5 \pi}{12}, \dfrac{3 \pi}{4}, \dfrac{13 \pi}{12}, \dfrac{17 \pi}{12}, \dfrac{7 \pi}{4}\)
65. 1.83, 2.88, 4.97, 6.02
67. 4.19
69. 0.28, 1.81, 2.37, 3.91, 4.47, 6.00
7.2 The General Sinusoidal Function
Homework 7-2
1.
a
| \(x\) | \(-\pi\) | \(\dfrac{-5\pi}{6}\) | \(\dfrac{-2\pi}{3}\) | \(\dfrac{-\pi}{2}\) | \(\dfrac{-\pi}{3}\) | \(\dfrac{-\pi}{6}\) | 0 | \(\dfrac{\pi}{6}\) | \(\dfrac{\pi}{3}\) | \(\dfrac{\pi}{2}\) | \(\dfrac{2\pi}{3}\) | \(\dfrac{5\pi}{6}\) | \(\pi\) |
| \(f(x)\) | 0 | \(\dfrac{-1}{2}\) | \(\dfrac{-\sqrt{3}}{2}\) | -1 | \(\dfrac{-\sqrt{3}}{2}\) | \(\dfrac{-1}{2}\) | 0 | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{2}\) | 1 | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) | 0 |
| \(g(x)\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) | 0 | \(\dfrac{-1}{2}\) | \(\dfrac{-\sqrt{3}}{2}\) | -1 | \(\dfrac{-\sqrt{3}}{2}\) | \(\dfrac{-1}{2}\) | 0 | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{2}\) | 1 | \(\dfrac{\sqrt{3}}{2}\) |
b 
c \(\dfrac{\pi}{3}\) to the right
d \(\dfrac{5\pi}{6}\)
e \(\dfrac{-2\pi}{3}, \dfrac{\pi}{3}\)
3.
a
| \(x\) | \(-\pi\) | \(\dfrac{-3\pi}{4}\) | \(\dfrac{-\pi}{2}\) | \(\dfrac{-\pi}{4}\) | 0 | \(\dfrac{\pi}{4}\) | \(\dfrac{\pi}{2}\) | \(\dfrac{3\pi}{4}\) | \(\pi\) |
| \(f(x)\) | 0 | 1 | undef | -1 | 0 | 1 | undef | -1 | 0 |
| \(g(x)\) | 1 | undef | -1 | 0 | 1 | undef | -1 | 0 | 1 |
b 
c \(\dfrac{\pi}{4}\) to the left
d \(-\pi, 0, \pi\)
e \(\dfrac{-\pi}{4}, \dfrac{-3\pi}{4}\)
5.
a amplitude 2, shift \(\dfrac{\pi}{6}\) to the left
b
| \(x\) | \(x + \dfrac{\pi}{6}\) | \(\cos \left( x + \dfrac{\pi}{6} \right) \) | \(-2 \cos \left( x+ \dfrac{\pi}{6} \right) \) |
| \(\dfrac{-7\pi}{6}\) | \(-\pi\) | -1 | 2 |
| \(\dfrac{-2\pi}{3}\) | \(\dfrac{-\pi}{2}\) | 0 | 0 |
| \(\dfrac{-\pi}{6}\) | 0 | 1 | -2 |
| \(\dfrac{\pi}{3}\) | \(\dfrac{\pi}{2}\) | 0 | 0 |
| \(\dfrac{5\pi}{6}\) | \(\pi\) | -1 | 2 |
| \(\dfrac{4\pi}{3}\) | \(\dfrac{3\pi}{2}\) | 0 | 0 |
| \(\dfrac{11\pi}{6}\) | \(2\pi\) | 1 | -2 |
c 
d \(\dfrac{\pi}{2}, \dfrac{7\pi}{6}\)
e \(\dfrac{\pi}{3}, \dfrac{4\pi}{3}\)
7.
a \(f(x)=\sin \left(x+\dfrac{\pi}{4}\right)\)
b \(f(x)=\cos \left(x-\dfrac{\pi}{4}\right)\)
9.
a \(f(x)=\tan \left(x-\dfrac{\pi}{3}\right)\)
b \(f(x)=\tan \left(x+\dfrac{2 \pi}{3}\right)\)
11.
a period \(\pi\), shift \(\frac{\pi}{6}\) to the right
b
| \(x\) | \(2x\) | \(2x - \dfrac{\pi}{3}\) | \(\cos \left( 2x - \dfrac{\pi}{3} \right) \) |
| \(\dfrac{\pi}{6}\) | \(\dfrac{\pi}{3}\) | 0 | 1 |
| \(\dfrac{5\pi}{12}\) | \(\dfrac{5\pi}{6}\) | \(\dfrac{\pi}{2}\) | 0 |
| \(\dfrac{2\pi}{3}\) | \(\dfrac{4\pi}{3}\) | \(\pi\) | -1 |
| \(\dfrac{11\pi}{12}\) | \(\dfrac{11\pi}{6}\) | \(\dfrac{3\pi}{2}\) | 0 |
| \(\dfrac{7\pi}{6}\) | \(\dfrac{7\pi}{3}\) | \(2\pi\) | 1 |
c 
d \(\dfrac{\pi}{6}, \dfrac{7 \pi}{6}\)
e \(\dfrac{5 \pi}{12}, \dfrac{11 \pi}{12}, \dfrac{13 \pi}{6}, \dfrac{23 \pi}{12}\)
13.
a period 2, shift \(\frac{1}{3}\) to the left
b
| \(x\) | \(\pi x\) | \(\pi x + \dfrac{\pi}{3}\) | \(\sin \left( \pi x + \dfrac{\pi}{3} \right) \) |
| \(\dfrac{-1}{3}\) | \(\dfrac{-\pi}{3}\) | 0 | 0 |
| \(\dfrac{1}{6}\) | \(\dfrac{\pi}{6}\) | \(\dfrac{\pi}{2}\) | 1 |
| \(\dfrac{2}{3}\) | \(\dfrac{2\pi}{3}\) | \(\pi\) | 0 |
| \(\dfrac{7}{6}\) | \(\dfrac{7\pi}{6}\) | \(\dfrac{3\pi}{2}\) | -1 |
| \(\dfrac{5}{3}\) | \(\dfrac{5\pi}{3}\) | \(2\pi\) | 0 |
c 
d \(\dfrac{-11}{6}, \dfrac{1}{6}\)
e \(\dfrac{-4}{3}, \dfrac{-1}{3}, \dfrac{2}{3}, \dfrac{5}{3}\)
15.
a midline \(y=4\), period \(4 \pi\), horizontal shift \(\frac{\pi}{3}\) to the right, amplitude 3
b
| \(x\) | \(\dfrac{x}{2}\) | \(\dfrac{x}{2} - \dfrac{\pi}{6}\) | \(\sin \left( \dfrac{x}{2} - \dfrac{\pi}{6} \right) \) | \(3 \sin \left( \dfrac{x}{2} - \dfrac{\pi}{6} \right) + 4\) |
| \(\dfrac{\pi}{3}\) | \(\dfrac{\pi}{6}\) | 0 | 0 | 4 |
| \(\dfrac{4\pi}{3}\) | \(\dfrac{2\pi}{3}\) | \(\dfrac{\pi}{2}\) | 1 | 7 |
| \(\dfrac{7\pi}{3}\) | \(\dfrac{7\pi}{6}\) | \(\pi\) | 0 | 3 |
| \(\dfrac{10\pi}{3}\) | \(\dfrac{5\pi}{3}\) | \(\dfrac{3\pi}{2}\) | -1 | 1 |
| \(\dfrac{13\pi}{3}\) | \(\dfrac{13\pi}{6}\) | \(2\pi\) | 0 | 4 |
c 
d no solution for \(0 \leq x \leq 2 \pi\)
e \(\dfrac{\pi}{3}\)
17. \(y=2 \sin \left(\dfrac{2 \pi}{3}(x+4)\right)+5\)
19. \(y=-5 \cos \left(\dfrac{\pi x}{180}\right)+12\)
21.
a \(f(x)=3 \sin \left(x+\dfrac{2 \pi}{3}\right)\)
b \(f(x)=3 \cos \left(x+\dfrac{\pi}{6}\right)\)
23.
a \(f(x)=2 \sin \left(2\left(x-\dfrac{\pi}{4}\right)\right)\)
b \(f(x)=-2 \cos (2 x)\)
25.
a \(f(x)=4 \sin \left[\dfrac{1}{4}\left(x-\dfrac{7 \pi}{3}\right)\right]\)
b \(f(x)=-4 \cos \left[\dfrac{1}{4}\left(x-\dfrac{\pi}{3}\right)\right]\)
27.
a midline \(T=35.35\), period 12 , amplitude 36.95
b \(T(m)=-36.95 \cos \left(\dfrac{\pi}{6} m\right)+35.35\)
c 
29.
a midline \(h=1.4\), period \(\frac{2 \pi}{0.51} \approx 12.32\), amplitude 1.4
b 
c high 11:10 am, low 5:19 pm
31.
a amplitude 3.2, period 2 , midline \(y=2\)
b \(f(t)=2+3.2 \cos (\pi t)\)
33.
a amplitude 5, period 1 , midline \(y=0\)
b \(H(x)=5 \sin (2 \pi x)+5\)
7.3 Solving Equations
Homework 7-3
1. \(\dfrac{3 \pi}{8}, \dfrac{7 \pi}{8}, \dfrac{11 \pi}{8}, \dfrac{15 \pi}{8}\)
3. \(0, \dfrac{\pi}{2}, \pi, \dfrac{3 \pi}{2}, 2 \pi\)
5. \(\dfrac{2 \pi}{9}, \dfrac{4 \pi}{9}, \dfrac{8 \pi}{9}, \dfrac{10 \pi}{9}, \dfrac{14 \pi}{9}, \dfrac{16 \pi}{9}\)
7. \(\dfrac{\pi}{12}, \dfrac{5 \pi}{12}, \dfrac{13 \pi}{12}, \dfrac{17 \pi}{12}\)
9. \(\dfrac{\pi}{18}, \dfrac{7 \pi}{18}, \dfrac{13 \pi}{18}, \dfrac{19 \pi}{18}, \dfrac{25 \pi}{18}, \dfrac{31 \pi}{18}\)
11. 0.491, 2.651, 3.632, 5.792
13. 0.540, 1.325, 2.110, 2.896, 3.681, 4.467, 5.252, 6.037
15. 1.114, 2.027, 3.209, 4.122, 5.303, 6.216
17. 0.702, 2.440, 3.843, 5.582
19. 0, 1, 2, 3, 4, 5, 6
21. \(\dfrac{\pi}{6}, \dfrac{2 \pi}{3}, \dfrac{7 \pi}{6}, \dfrac{5 \pi}{3}\)
23. \(\dfrac{5 \pi}{12}, \dfrac{7 \pi}{12}, \dfrac{13 \pi}{12}, \dfrac{5 \pi}{4}, \dfrac{7 \pi}{4}, \dfrac{23 \pi}{12}\)
25. \(\dfrac{3\pi}{2}\)
27. \(\dfrac{7}{6}, \dfrac{11}{6}, \dfrac{19}{6}, \dfrac{23}{6}, \dfrac{31}{6}, \dfrac{35}{6}\)
29. 1.14, 1.62, 3.23, 3.72, 5.24, 5.81
31. 0.44, 1.44, 2.44, 3.44, 4.44, 5.44
33. 0.01, 3.39, 6.01
35. 0.564, 1.182, 2.658, 3.276, 4.752, 5.371
37. 0.423, 2.977, 4.423
39. 1.165, 4.165
41. 2.251
43.
a \(P(t)=4000 \cos \left(\dfrac{\pi}{6} t\right)+46,000\)
b \(t=\cos ^{-1}\left(\dfrac{-1}{4}\right) \cdot \dfrac{6}{\pi} \approx 3.48\) months (Dec) or \(t=12-\cos ^{-1}\left(\dfrac{-1}{4}\right) \cdot \dfrac{6}{\pi} \approx 8.52\) months (June)
c 
\(P(t)\) is less than 45,000 between \(A\) and \(B\).
45.
a \(h(t)=11-10 \cos \left(\dfrac{\pi}{30} t\right)\)
b \(t=\cos (-0.7) \cdot \dfrac{30}{\pi} \approx 22.40 \mathrm{sec}\) or \(t=60-\cos (-0.7) \cdot \dfrac{30}{\pi} \approx 37.60 \mathrm{sec}\)
c 
Delbert is above 18 m between \(A\) and \(B\).
7.4 Chapter 7 Summary and Review
Review Problems
1. amp: 2, period: \(\frac{2 \pi}{3}\); mid: \(y=4\)
3. amp: 2.5, period: 2; mid: \(y=-2\)
5. 
7. 
9. \(y=3+2 \sin x\)
11. \(y=4-3 \sin \dfrac{x}{4}\)
13.
a period: \(4 \pi\), shift: \(\dfrac{\pi}{3}\) left
b
| \(x\) | \(\dfrac{x}{2}\) | \(\dfrac{x}{2} + \dfrac{\pi}{6}\) | \(\sin \left( \dfrac{x}{2} + \dfrac{\pi}{6} \right) \) |
| \(\dfrac{-2\pi}{3}\) | \(\dfrac{\pi}{3}\) | \(\dfrac{-\pi}{6}\) | \(\dfrac{-1}{2}\) |
| \(\dfrac{-\pi}{3}\) | \(\dfrac{-\pi}{6}\) | 0 | 0 |
| 0 | 0 | \(\dfrac{\pi}{6}\) | \(\dfrac{1}{2}\) |
| \(\dfrac{\pi}{6}\) | \(\dfrac{\pi}{12}\) | \(\dfrac{\pi}{4}\) | \(\dfrac{1}{\sqrt{2}}\) |
| \(\dfrac{\pi}{3}\) | \(\dfrac{\pi}{6}\) | \(\dfrac{\pi}{3}\) | \(\dfrac{\sqrt{3}}{2}\) |
| \(\dfrac{2\pi}{3}\) | \(\dfrac{\pi}{3}\) | \(\dfrac{\pi}{2}\) | 1 |
| \(\pi\) | \(\dfrac{\pi}{2}\) | \(\dfrac{2\pi}{3}\) | \(\dfrac{\sqrt{3}}{2}\) |
c 
d \(\dfrac{2\pi}{3}\)
e \(\dfrac{-\pi}{3}\)
15.
a mid: \(y=20\), period: 0, amp: 5
b Fill in the table of values
| \(x\) | \(\dfrac{\pi}{30} x\) | \(\cos \left( \dfrac{\pi}{30} x \right) \) | \(20 - 5 \cos \left( \dfrac{\pi}{30} x \right) \) |
| -5 | \(\dfrac{-\pi}{6}\) | \(\dfrac{\sqrt{3}}{2}\) | \(20 - \dfrac{\sqrt{3}}{2}\) |
| 0 | 0 | 1 | 15 |
| 5 | \(\dfrac{\pi}{6}\) | \(\dfrac{\sqrt{3}}{2}\) | \(20 - \dfrac{\sqrt{3}}{2}\) |
| 10 | \(\dfrac{\pi}{3}\) | \(\dfrac{1}{2}\) | 17.5 |
| 15 | \(\dfrac{\pi}{2}\) | 0 | 20 |
| 50 | \(\pi\) | -1 | 25 |
c 
d 30
e 15, 45
17. 
19.
a 
b 0.57, 3.07, 3.71
21. \(y=85.5-19.5 \cos \dfrac{\pi}{6} t\)
23.
a amp: 3, period: 12, midline: \(y = 15\)
b \(y=15-3 \cos \dfrac{\pi}{6} t\)
25. \(\dfrac{7 \pi}{12}, \dfrac{11 \pi}{12}, \dfrac{19 \pi}{12}, \dfrac{23 \pi}{12}\)
27. \(0, \dfrac{\pi}{4}, \dfrac{\pi}{2}, \dfrac{3 \pi}{4}, \pi, \dfrac{5 \pi}{4}, \dfrac{7 \pi}{4}, 2 \pi\)
29. 0.066, 1.113, 2.160, 3.207, 4.255, 5.302
31. 1.150, 1.991, 4.292, 5.133
33. \(\dfrac{\pi}{24}, \dfrac{5 \pi}{24}, \dfrac{25 \pi}{24}, \dfrac{29 \pi}{24}\)
35. No Solution
37. 0.375, 1.422, 2.470, 3.517, 4.564, 5.611
39. 2.120, 4.880