Section 4.1 Solutions

1.

3. Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

5. Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

7.

9.

11.

13.

15.

17. 240°

19. $\frac{4\pi }{3}$

21. $\frac{2\pi }{3}$

23. $\frac{7\pi }{2}\approx 11.00{\text{ in}}^{2}$

25. $\frac{9\pi }{5}\approx 5.65{\text{ cm}}^{2}$

27. 20°

29. 60°

31. −75°

33. $\frac{\pi }{2}$ radians

35. $-3\pi$ radians

37. $\pi$ radians

39. $\frac{5\pi }{6}$ radians

41. $154.795^\circ$

43. $30.23^\circ$

45. $2^\circ 55' 21''$

47. $36^\circ 52' 12''$

49. $\frac{5.02\pi }{3}\approx 5.26$ miles

51. $\frac{25\pi }{9}\approx 8.73$ centimeters

53. $\frac{21\pi }{10}\approx 6.60$ meters

55. 104.7198 cm2

57. 0.7697 in2

59. 250°

61. 320°

63. $\frac{4\pi }{3}$

65. $\frac{8\pi }{9}$

67. 1320 rad 210.085 RPM

69. 7 in./s, 4.77 RPM, 28.65 deg/s

71. $1,809,557.37\text{ mm/min}=30.16\text{ m/s}$

73. $5.76$ miles

75. $120^\circ$

77. 794 miles per hour

79. 2,234 miles per hour

81. 11.5 inches

Section 4.2 Solutions

1. The unit circle is a circle of radius 1 centered at the origin.

3. Yes, when the reference angle is $\frac{\pi }{4}$ and the terminal side of the angle is in quadrants I and III. Thus, at $x=\frac{\pi }{4},\frac{5\pi }{4}$, the sine and cosine values are equal.

5. Substitute the sine of the angle in for $y$ in the Pythagorean Theorem ${x}^{2}+{y}^{2}=1$. Solve for $x$ and take the negative solution.

7. I

9. IV

11. $\frac{\sqrt{3}}{2}\text{ , }\frac{2\sqrt{3}}{3}$

13. $\frac{1}{2}\text{ , }2$

15. $\frac{\sqrt{2}}{2}\text{ , }\sqrt{3}$

17. $0{ , }\sqrt{2}$

19. $-1\text{ , }0$

21. $1\text{ , }0$

23. $\frac{\sqrt{77}}{9}$

25. $-\frac{\sqrt{15}}{4}$

27. $\sin t=\frac{1}{2},\csc t=2,\cos t=-\frac{\sqrt{3}}{2},\sec t=-\frac{2\sqrt{3}}{3},\tan t=-\frac{\sqrt{3}}{3},\cot t=-\sqrt{3}$

29. $\sin t=-\frac{\sqrt{2}}{2},\csc t=-\sqrt{2},\cos t=-\frac{\sqrt{2}}{2},\sec t=-\sqrt{2},\tan t=1,\cot t=1$

31. $\sin t=\frac{\sqrt{3}}{2},\csc t=\frac{2\sqrt{3}}{3},\cos t=-\frac{1}{2},\sec t=-2,\tan t=\sqrt{3},\cot t=\frac{\sqrt{3}}{3}$

33. $\sin t=-\frac{\sqrt{2}}{2},\csc t=-\sqrt{2},\cos t=\frac{\sqrt{2}}{2},\sec t=\sqrt{2},\tan t=-1,\cot t=1$

35. $\sin t=0,\csc t=\varnothing,\cos t=-1,\sec t=-1,\tan t=0,\cot t=\varnothing$

37. $\sin t=-0.596,\csc t=-1.679,\cos t=0.803,\sec t=1.245,\tan t=-0.742,\cot t=-1.347$

39. −0.1736

41. 0.9511

43. −0.7071

45. −0.1392

47. −0.7660

49. –0.228

51. –2.414

53. 1.556

55. $\frac{\sqrt{2}}{4}$

57. $\frac{\sqrt{2}}{4}$

59. 0

61. $\cos\left(6t\right)-\sin\left(9t\right)$

63. even

65. even

67. 13.77 hours, period: $1000\pi$

69. 7.73 inches

Section 4.3 Solutions

1.

3. The tangent of an angle is the ratio of the opposite side to the adjacent side.

5. For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

7. $\frac{\sqrt{2}-4}{4}$

9. 5

11. $\frac{\pi }{6}$

13. $\frac{\pi }{4}$

15. $b=\frac{20\sqrt{3}}{3},c=\frac{40\sqrt{3}}{3}$

17. $a=10,000,c=10,000.5$

19. $b=\frac{5\sqrt{3}}{3},c=\frac{10\sqrt{3}}{3}$

21. $\frac{5\sqrt{29}}{29}$

23. $\frac{5}{2}$

25. $\frac{\sqrt{29}}{2}$

27. $\frac{5\sqrt{41}}{41}$

29. $\frac{5}{4}$

31. $\frac{\sqrt{41}}{4}$

33. $c=14, b=7\sqrt{3}$

35. $a=15, b=15$

37. $b=9.9970, c=12.2041$

39. $a=2.0838, b=11.8177$

41. $a=55.9808,c=57.9555$

43. $a=46.6790,b=17.9184$

45. $a=16.4662,c=16.8341$

47. 188.3159

49. 200.6737

51. 498.3471 ft

53. 1060.09 ft

55. 27.372 ft

57. 22.6506 ft

59. 368.7633 ft

61. $S 29.05^\circ W$

63. East: 13.49 inches, North: 33.38 inches

65. $18.3^\circ$

Section 4.4 Solutions

1. Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, $t$, formed by the terminal side of the angle $t$ and the horizontal axis.

3. The sine values are equal.

5. $60^\circ$

7. $80^\circ$

9. $45^\circ$

11. $\frac{\pi }{3}$

13. $\frac{\pi }{3}$

15. $\frac{\pi }{8}$

17. $60^\circ$, Quadrant IV, $\text{sin}\left(300^\circ \right)=-\frac{\sqrt{3}}{2},\cos \left(300^\circ \right)=\frac{1}{2}$

19. $45^\circ$, Quadrant II, $\text{sin}\left(135^\circ \right)=\frac{\sqrt{2}}{2}$, $\cos \left(135^\circ \right)=-\frac{\sqrt{2}}{2}$

21. $60^\circ$, Quadrant II, $\text{sin}\left(480^\circ \right)=\frac{\sqrt{3}}{2}$, $\cos \left(480^\circ \right)=-\frac{1}{2}$

23. $30^\circ$, Quadrant II, $\text{sin}\left(-210^\circ \right)=\frac{1}{2}$, $\cos \left(-210^\circ \right)=-\frac{\sqrt{3}}{2}$

25. $\frac{\pi }{6}$, Quadrant III, $\text{sin}\left(\frac{7\pi }{6}\right)=-\frac{1}{2}$, $\text{cos}\left(\frac{7\pi }{6}\right)=-\frac{\sqrt{3}}{2}$

27. $\frac{\pi }{4}$, Quadrant II, $\text{sin}\left(\frac{3\pi }{4}\right)=\frac{\sqrt{2}}{2}$, $\cos \left(\frac{4\pi }{3}\right)=-\frac{\sqrt[]{2}}{2}$

29. $\frac{\pi }{3}$, Quadrant II, $\text{sin}\left(\frac{2\pi }{3}\right)=\frac{\sqrt{3}}{2}$, $\cos \left(\frac{2\pi }{3}\right)=-\frac{1}{2}$

31. $\frac{\pi }{4}$, Quadrant IV, $\text{sin}\left(\frac{-9\pi }{4}\right)=-\frac{\sqrt{2}}{2}$, $\text{cos}\left(\frac{-9\pi }{4}\right)=\frac{\sqrt{2}}{2}$

33. $\frac{\pi }{6}$, Quadrant III, $\text{sec}\left(\frac{7\pi }{6}\right)=-\frac{2\sqrt{3}}{3}$

35. $\frac{\pi }{6}$, Quadrant I, $\text{cot}\left(\frac{13\pi }{6}\right)=\sqrt{3}$

37. $\frac{\pi }{4}$, Quadrant II, $\text{sec}\left(\frac{3\pi }{4}\right)=-\sqrt{2}$

39. $\frac{\pi }{4}$, Quadrant IV, $\text{cot}\left(\frac{11\pi }{4}\right)=-1$

41. $\frac{\pi }{3}$, Quadrant III, $\text{sec}\left(-\frac{2\pi }{3}\right)=-2$

43. $\frac{\pi }{3}$, Quadrant IV, $\text{cot}\left(-\frac{7\pi }{3}\right)=-\frac{\sqrt{3}}{3}$

45. $\text{ }60^\circ$, Quadrant IV, $\text{sec}\left(300^\circ\right)=2$

47. $\text{ }60^\circ$, Quadrant III, $\text{cot}\left(600^\circ\right)=\frac{\sqrt{3}}{3}$

49. $\text{ }30^\circ$, Quadrant II, $\text{sec}\left(-210^\circ\right)=-\frac{2\sqrt{3}}{3}$

51. $\text{ }45^\circ$, Quadrant IV, $\text{cot}\left(-405^\circ\right)=-1$

53. If $\text{ }\sin t=-\frac{2\sqrt{2}}{3},\sec t=-3,\csc t=-\frac{3\sqrt{2}}{4},\tan t=2\sqrt{2},\cot t=\frac{\sqrt{2}}{4}$

55. $\text{ }\sec t=2,\csc t=\frac{2\sqrt{3}}{3},\tan t=\sqrt{3},\cot t=\frac{\sqrt{3}}{3}$

57. $\frac{\sqrt{2}}{4}$

59. $-\frac{\sqrt{6}}{4}$

61. $\frac{\sqrt{2}}{4}$

63. $\frac{\sqrt{2}}{4}$

65. 0

Section 4.5 Solutions

1. The sine and cosine functions have the property that $f(x+P)=f(x)$ for a certain P. This means that the function values repeat for every P units on the x-axis.

3. The absolute value of the constant A (amplitude) increases the total range and the constant D (vertical shift) shifts the graph vertically.

5. At the point where the terminal side of t intersects the unit circle, you can determine that the sin t equals the y-coordinate of the point.

7. amplitude: $\frac{2}{3}$; period: 2π; midline: $y=0$; maximum: $y=23$ occurs at $x=0$; minimum: $y=−23$ occurs at $x=\pi$; for one period, the graph starts at 0 and ends at 2π

9. amplitude: 4; period: 2π; midline: $y=0$; maximum $y=4$ occurs at $x=\frac{\pi}{2}$; minimum: $y=−4$ occurs at $x=\frac{3\pi}{2}$; one full period occurs from $x=0$ to $x=2π$

11. amplitude: 1; period: π; midline: y=0; maximum: y=1 occurs at $x=\pi$; minimum: $y=−1$ occurs at $x=\frac{\pi}{2}$; one full period is graphed from $x=0$ to $x=\pi$

13. amplitude: 4; period: 2; midline: $y=0$; maximum: $y=4$ occurs at $x=0$; minimum: $y=−4$ occurs at $x=1$

15. amplitude: 3; period: $\frac{\pi}{4}$; midline: $y=5$; maximum: $y=8$ occurs at $x=0.12$; minimum: $y=2$ occurs at $x=0.516$; horizontal shift: −4; vertical translation 5; one period occurs from $x=0$ to $x=\frac{\pi}{4}$

17. amplitude: 5; period: $\frac{2\pi}{5}; midline: [latex]y=−2$; maximum: $y=3$ occurs at $x=0.08$; minimum: $y=−7$ occurs at $x=0.71$; phase shift:−4; vertical translation:−2; one full period can be graphed on $x=0$ to $x=\frac{2\pi}{5}$

19. amplitude: 1; period: 2π; midline: y=1; maximum:$y=2$ occurs at $x=2.09$; maximum:$y=2$ occurs at$t=2.09$; minimum:$y=0$ occurs at $t=5.24$; phase shift: $−\frac{\pi}{3}$; vertical translation: 1; one full period is from $t=0$ to $t=2π$

21. amplitude: 1; period: 4π; midline: $y=0$; maximum: $y=1$ occurs at $t=11.52$; minimum: $y=−1$ occurs at $t=5.24$; phase shift: −$\frac{10\pi}{3}$; vertical shift: 0

23. amplitude: 2; midline: $y=−3$; period: 4; equation: $f(x)=2\sin\left(\frac{\pi}{2}x\right)−3$

25. amplitude: 2; period: 5; midline: $y=3$; equation: $f(x)=−2\cos\left(\frac{2\pi}{5}x\right)+3$

27. amplitude: 4; period: 2; midline: $y=0$; equation: $f(x)=−4\cos\left(\pi\left(x−\frac{\pi}{2}\right)\right)$

29. amplitude: 2; period: 2; midline $y=1$; equation: $f(x)=2\cos\left(\frac{\pi}{x}\right)+1$

31. $\frac{\pi}{6},\frac{5\pi}{6}$

33. $\frac{\pi}{4},\frac{3\pi}{4}$

35. $\frac{3\pi}{2}$

37. $\frac{\pi}{2},\frac{3\pi}{2}$

39. $\frac{\pi}{2},\frac{3\pi}{2}$

41. $\frac{\pi}{6},\frac{11\pi}{6}$

43. The graph appears linear. The linear functions dominate the shape of the graph for large values of x.

45. The graph is symmetric with respect to the y-axis and there is no amplitude because the function is not periodic.

47.
a. Amplitude: 12.5; period: 10; midline: $y=13.5$;
b. $h(t)=12.5\sin\left(\frac{\pi}{5}\left(t−2.5\right)\right)+13.5;$
c. 26 ft

Section 4.6 Solutions

1.  Since $y=\csc x$ is the reciprocal function of $y=\sin x$, you can plot the reciprocal of the coordinates on the graph of $y=\sin x$ to obtain the y-coordinates of $y=\csc x$. The x-intercepts of the graph $y=\sin x$ are the vertical asymptotes for the graph of $y=\csc x$.

3. Answers will vary. Using the unit circle, one can show that $\tan(x+\pi)=\tan x$.

5. The period is the same: 2π.

7. IV

9. III

11. period: 8; horizontal shift: 1 unit to left

13. 1.5

15. 5

17. $−\cot x\cos x−\sin x$

19. stretching factor: 2; period: $\frac{\pi}{4}$; asymptotes: $x=\frac{1}{4}\left(\frac{\pi}{2}+\pi k\right)+8$, where k is an integer

21. stretching factor: 6; period: 6; asymptotes: $x=3k$, where k is an integer

23. stretching factor: 1; period: π; asymptotes: $x=πk$, where k is an integer

25. Stretching factor: 1; period: π; asymptotes: $x=\frac{\pi}{4}+{\pi}k$, where k is an integer

27. stretching factor: 2; period: 2π; asymptotes: $x=πk$, where k is an integer

29. stretching factor: 4; period: $\frac{2\pi}{3}$; asymptotes: $x=\frac{\pi}{6}k$, where k is an odd integer

31. stretching factor: 7; period: $\frac{2\pi}{5}$; asymptotes: $x=\frac{\pi}{10}k$, where k is an odd integer

33. stretching factor: 2; period: 2π; asymptotes: $x=−\frac{\pi}{4}+\pi k$, where k is an integer

35. stretching factor: $\frac{7}{5}$; period: 2π; asymptotes: $x=\frac{\pi}{4}+\pi$k, where k is an integer

37. $y=\tan\left(3\left(x−\frac{\pi}{4}\right)\right)+2$

39. $f(x)=\csc(2x)$

41. $f(x)=\csc(4x)$

43. $f(x)=2\csc x$

45. $f(x)=\frac{1}{2}\tan(100\pi x)$

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input $\csc x$ as $\frac{1}{\sin x}$.

46. $f(x)=|\csc(x)|$

47. $f(x)=|\cot(x)|$

48. $f(x)=2^{\csc(x)}$

49. $f(x)=\frac{\csc(x)}{\sec(x)}$

51.

53.

55. a. $(−\frac{\pi}{2}\text{,}\frac{\pi}{2})$;
b.

c. $x=−\frac{\pi}{2}$ and $x=\frac{\pi}{2}$; the distance grows without bound as |x| approaches $\frac{\pi}{2}$—i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;
d. 3; when $x=−\frac{\pi}{3}$, the boat is 3 km away;
e. 1.73; when $x=\frac{\pi}{6}$, the boat is about 1.73 km away;
f. 1.5 km; when $x=0$.

57. a. $h(x)=2\tan\left(\frac{\pi}{120}x\right)$;
b.

c. $h(0)=0:$ after 0 seconds, the rocket is 0 mi above the ground; $h(30)=2:$ after 30 seconds, the rockets is 2 mi high;
d. As x approaches 60 seconds, the values of $h(x)$ grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.

Section 4.7 Solutions

1. The function $y=\sin x$ is one-to-one on $\left[−\frac{\pi}{2}\text{, }\frac{\pi}{2}\right]$; thus, this interval is the range of the inverse function of $y=\sin x\text{, }f\left(x\right)=\sin^{−1}x$. The function $y=\cos x$ is one-to-one on [0,π]; thus, this interval is the range of the inverse function of $y=\cos x\text{, }f(x)=\cos^{−1}x$.

3. $\frac{\pi}{6}$ is the radian measure of an angle between $−\frac{\pi}{2}$ and $\frac{\pi}{2}$ whose sine is 0.5.

5. In order for any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval $\left[−\frac{\pi}{2}\text{, }\frac{\pi}{2}\right]$ so that it is one-to-one and possesses an inverse.

7. True. The angle, $\theta_{1}$ that equals $\arccos(−x)\text{, }x\text{>}0$, will be a second quadrant angle with reference angle, $\theta_{2}$, where $\theta_{2}$ equals $\arccos x\text{, }x\text{>}0$. Since $\theta_{2}$ is the reference angle for $\theta_{1}$, $\theta_{2}=\pi(−x)=\pi−\arccos x$

9. $−\frac{\pi}{6}$

11. $\frac{3\pi}{4}$

13. $−\frac{\pi}{3}$

15. $\frac{\pi}{3}$

17. 1.98

19. 0.93

21. 1.41

23. 0.56 radians

25. 0

27. 0.71

29. −0.71

31. $−\frac{\pi}{4}$

33. 0.8

35. $\frac{5}{13}$

37. $\frac{x}{\sqrt{1−x^{2}}}$

39. $\frac{\sqrt{x^{2}−1}}{x}$

41. $\frac{2x}{\sqrt{4x^{2}+1}}$

43. $\frac{\sqrt{2x+1}}{x+1}$

45. $\frac{\sqrt{2x+1}}{x}$

47. $\frac{x}{\sqrt{2x+1}}$

49. domain [−1,1]; range [0,π]

51. approximately $x=0.00$

53. 0.395 radians

55. 1.11 radians

57. 1.25 radians

59. 0.405 radians

61. No. The angle the ladder makes with the horizontal is 60 degrees.