1.6E: Exercises

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In exercises 1 - 5, convert each angle in degrees to radians. Write the answer as a multiple of $ \pi $.

1) $240^{\circ} $

Answer: $\frac{4 \pi }{3}$ rad

2) $15^{\circ} $

3) $60^{\circ} $

Answer: $\frac{ \pi }{3}$ rad

4) $-225^{\circ} $

5) $330^{\circ} $

Answer: $\frac{11 \pi }{6}$ rad

In exercises 6 - 10, convert each angle in radians to degrees.

6) $\frac{ \pi }{2}$ rad

7) $\frac{7 \pi }{6}$ rad

Answer: $210^{\circ} $

8) $\frac{11 \pi }{2}$ rad

9) $-3 \pi $ rad

Answer: $-540^{\circ} $

10) $\frac{5 \pi }{12}$ rad

In exercises 11 - 16, evaluate the functional values.

11) $\cos \frac{4 \pi }{3}$

Answer: $\cos \frac{4 \pi }{3}=-0.5$

12) $\tan \frac{19 \pi }{4}$

13) $\sin \left(-\frac{3 \pi }{4}\right)$

Answer: $\sin \left(-\frac{3 \pi }{4}\right) = -\frac{\sqrt{2}}{2}$

14) $\sec \left(-\frac{ \pi }{6}\right)$

15) $\sin \left(-\frac{ \pi }{12}\right)$

Answer: $\sin \left(-\frac{ \pi }{12}\right) = \dfrac{\sqrt{3}-1}{2\sqrt{2}}$

16) $\cos \left(-\frac{5 \pi }{12}\right)$

In exercises 17 - 22, consider triangle $ABC,$ a right triangle with a right angle at $C.$

a. Find the missing side of the triangle.

b. Find the six trigonometric function values for the angle at $A.$

Where necessary, round to one decimal place.

$An image of a triangle. The three corners of the triangle are labeled “A”, “B”, and “C”. Between the corner A and corner C is the side b. Between corner C and corner B is the side a. Between corner B and corner A is the side c. The angle of corner C is marked with a right triangle symbol. The angle of corner A is marked with an angle symbol.$ 225^{\circ} =225^{\circ} \cdot \pi 180^{\circ} =5 \pi 4

17) $a=4, \; c=7$

Answer: a. $b=5.7$
b. $\sin A=\frac{4}{7},\; \cos A=\frac{5.7}{7},\; \tan A=\frac{4}{5.7},\; \csc A=\frac{7}{4},\; \sec A=\frac{7}{5.7},\; \cot A=\frac{5.7}{4}$

18) $a=21, \; c=29$

19) $a=85.3, \; b=125.5$

Answer: a. $c=151.7$
b. $\sin A=0.5623,\; \cos A=0.8273,\; \tan A=0.6797,\;\csc A=1.778,\;\sec A=1.209,\;\cot A=1.471$

20) $b=40, \; c=41$

21) $a=84, \; b=13$

Answer: a. $c=85$
b. $\sin A=\frac{84}{85},\;\cos A=\frac{13}{85},\; \tan A=\frac{84}{13},\; \csc A=\frac{85}{84},\; \sec A=\frac{85}{13},\;\cot A=\frac{13}{84}$

22) $b=28, \; c=35$

In exercises 23 - 26, $P$ is a point on the unit circle.

a. Find the (exact) missing coordinate value of each point and

b. find the values of the six trigonometric functions for the angle $ \theta $ with a terminal side that passes through point $P$.

Rationalize denominators.

23) $P\left(\frac{7}{25},\, y\right), \quad y>0$

Answer: a. $y=\frac{24}{25}$
b. $\sin \theta =\frac{24}{25},\; \cos \theta =\frac{7}{25},\; \tan \theta =\frac{24}{7},\; \csc \theta =\frac{25}{24},\; \sec \theta =\frac{25}{7},\;\cot \theta =\frac{7}{24}$

24) $P\left(-\frac{15}{17},\, y\right), \quad y>0$

25) $P\left(x,\, \frac{\sqrt{7}}{3}\right), \quad x>0$

Answer: a. $x=−\frac{\sqrt{2}}{3}$
b. $\sin \theta =\frac{\sqrt{7}}{3},\;\cos \theta =-\frac{\sqrt{2}}{3},\; \tan \theta =-\frac{\sqrt{14}}{2},\;\csc \theta =\frac{3\sqrt{7}}{7},\;\sec \theta =-\frac{3\sqrt{2}}{2},\;\cot \theta =-\frac{\sqrt{14}}{7}$

26) $P\left(x,\, -\frac{\sqrt{15}}{4}\right), \quad y>0$

In exercises 27 - 34, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

27) $\tan^2x+\sin x\csc x$

Answer: $\sec^2x$

28) $\sec x\sin x\cot x$

29)$\dfrac{\tan^2x}{\sec^2x}$

Answer: $\sin^2x$

30) $\sec x-\cos x$

31) $(1+\tan \theta )^2-2\tan \theta $

Answer: $\sec^2 \theta $

32) $(\sin x)(\csc x-\sin x)$

33) $\dfrac{\cos t}{\sin t}+\dfrac{\sin t}{1+\cos t}$

Answer: $\dfrac{1}{\sin t} = \csc t$

34) $\dfrac{1+\tan^2 \alpha }{1+\cot^2 \alpha }$

In exercises 35 - 42, verify that each equation is an identity.

35) $\dfrac{\tan \theta \cot \theta }{\csc \theta }=\sin \theta $

36) $\dfrac{\sec^2 \theta }{\tan \theta }=\sec \theta \csc \theta $

37) $\dfrac{\sin t}{\csc t} + \dfrac{\cos t}{\sec t} = 1$

38) $\dfrac{\sin x}{\cos x+1}+\dfrac{\cos x−1}{\sin x}=0$

39) $\cot \gamma +\tan \gamma =\sec \gamma \csc \gamma $

40) $\sin^2 \beta +\tan^2 \beta +\cos^2 \beta =\sec^2 \beta $

41) $\dfrac{1}{1−\sin \alpha }+\dfrac{1}{1+\sin \alpha }=2\sec^2 \alpha $

42)$\dfrac{\tan \theta −\cot \theta }{\sin \theta \cos \theta }=\sec^2 \theta −\csc^2 \theta $

In exercises 43 - 50, solve the trigonometric equations on the interval $0 \leq \theta <2 \pi .$

43) $2\sin \theta −1=0$

Answer: $\big\{\frac{ \pi }{6},\frac{5 \pi }{6}\big\}$

44) $1+\cos \theta =\frac{1}{2}$

45) $2\tan^2 \theta =2$

Answer: $\big\{\frac{ \pi }{4},\, \frac{3 \pi }{4},\, \frac{5 \pi }{4},\, \frac{7 \pi }{4}\big\}$

46) $4\sin^2 \theta −2=0$

47) $\sqrt{3}\cot \theta +1=0$

Answer: $\big\{\frac{2 \pi }{3},\, \frac{5 \pi }{3}\big\}$

48) $3\sec \theta −2\sqrt{3}=0$

49) $2\cos \theta \sin \theta =\sin \theta $

Answer: $\big\{0,\, \pi ,\, \frac{ \pi }{3},\, \frac{5 \pi }{3}\big\}$

50) $\csc^2 \theta +2\csc \theta +1=0$

In exercises 51 - 54, each graph is of the form $y=A\sin Bx$ or $y=A\cos Bx$, where $B>0.$ Write the equation of the graph.

51)

$An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function that starts at the point (-4, 0) and decreases until the point (-2, 4). After this point the function begins increasing until it hits the point (2, 4). After this point the function begins decreasing again. The x intercepts of the function on this graph are at (-4, 0), (0, 0), and (4, 0). The y intercept is at the origin.$

Answer: $y=4\sin\left(\frac{ \pi }{4}x\right)$

52)

$An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function that starts at the point (-4, -2) and increases until the point (-3, 2). After this point the function decreases until it hits the point (-2, -2). After this point the function increases until it hits the point (-1, 2). After this point the function decreases until it hits the point (0, -2). After this point the function increases until it hits the point (1, 2). After this point the function decreases until it hits the point (2, -2). After this point the function increases until it hits the point (3, 2). After this point the function begins decreasing again. The x intercepts of the function on this graph are at (-3.5, 0), (-2.5, 0), (-1.5, 0), (-0.5, 0), (0.5, 0), (1.5, 0), (2.5, 0), and (3.5, 0). The y intercept is at the (0, -2).$

53)

$An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function. There are many periods and only a few will be explained. The function begins decreasing at the point (-1, 1) and decreases until the point (-0.5, -1). After this point the function increases until it hits the point (0, 1). After this point the function decreases until it hits the point (0.5, -1). After this point the function increases until it hits the point (1, 1). After this point the function decreases again. The x intercepts of the function on this graph are at (-0.75, 0), (-0.25, 0), (0.25, 0), and (0.75, 0). The y intercept is at (0, 1).$

Answer: $y=\cos 2 \pi x$

54)

$An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function. There are many periods and only a few will be explained. The function begins decreasing at the point (-1.25, 0.75) and decreases until the point (-0.75, -0.75). After this point the function increases until it hits the point (0.25, 0.75). After this point the function decreases until it hits the point (0.25, -0.75). After this point the function increases until it hits the point (0.75, 0.75). After this point the function decreases again. The x intercepts of the function on this graph are at (-1, 0), (-0.5, 0), (0, 0), and (0.5, 0). The y intercept is at the origin.$

In exercises 55 - 60, find

a. the amplitude,

b. the period, and

c. the phase shift with direction for each function.

55) $y=\sin\left(x−\frac{ \pi }{4}\right)$

Answer: a. $1$
b. $2 \pi $
c. $\frac{ \pi }{4}$ units to the right

56) $y=3\cos(2x+3)$

57) $y=−\frac{1}{2}\sin\left(\frac{1}{4}x\right)$

Answer: a. $\frac{1}{2}$
b. $8 \pi $
c. No phase shift

58) $y=2\cos\left(x−\frac{ \pi }{3}\right)$

59) $y=−3\sin( \pi x+2)$

Answer: a. $3$
b. $2$
c. $\frac{2}{ \pi }$ units to the left

60) $y=4\cos\left(2x−\frac{ \pi }{2}\right)$

61) [Technology Required] The diameter of a wheel rolling on the ground is $40$ in. If the wheel rotates through an angle of $120^{\circ}$ , how many inches does it move? Approximate to the nearest whole inch.

Answer: Approximately $42$ in.

62) [Technology Required] Find the length of the arc intercepted by central angle $ \theta $ in a circle of radius $r$. Round to the nearest hundredth.

a. $r=12.8$ cm, $ \theta =\frac{5 \pi }{6}$ rad b. $r=4.378$ cm, $ \theta =\frac{7 \pi }{6}$ rad c. $r=0.964$ cm, $ \theta =50^{\circ}$ d. $r=8.55$ cm, $ \theta =325^{\circ}$

63) [Technology Required] As a point $P$ moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, $ \omega $, and is given by $ \omega = \theta /t$, where $ \theta $ is in radians and $t$ is time. Find the angular speed for the given data. Round to the nearest thousandth.

a. $ \theta =\frac{7 \pi }{4}$ rad, $t=10$ sec b. $ \theta =\frac{3 \pi }{5}$ rad, $t=8$ sec c. $ \theta =\frac{2 \pi }{9}$ rad, $t=1$ min d. $ \theta =23.76$ rad, $t=14$ min

Answer: a. $0.550$ rad/sec
b. $0.236$ rad/sec
c. $0.698$ rad/min
d. $1.697$ rad/min

64) [Technology Required] A total of $250,000\text{ m}^2$ of land is needed to build a nuclear power plant. Suppose it is decided that the area on which the power plant is to be built should be circular.

a)Find the radius of the circular land area.

b)If the land area is to form a $45^{\circ}$ sector of a circle instead of a whole circle, find the length of the curved side.

65) [Technology Required] The area of an isosceles triangle with equal sides of length $x$ is $\frac{1}{2}x^2\sin \theta $, where $ \theta $ is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length $8$ in. and angle $ \theta =\frac{5 \pi }{12}$ rad.

Answer: $ \approx 30.9\text{ in}^2$

66) [Technology Required] A particle travels in a circular path at a constant angular speed $ \omega $. The angular speed is modeled by the function $ \omega =9 \left| \cos( \pi t− \pi /12) \right|$. Determine the angular speed at $t=9$ sec.

67) [Technology Required] An alternating current for outlets in a home has voltage given by the function $V(t)=150\cos 368t$,

where $V$ is the voltage in volts at time $t$ in seconds.

a) Find the period of the function and interpret its meaning.

b) Determine the number of periods that occur when $1$ sec has passed.

Answer: a. $\frac{ \pi }{184}$; the voltage repeats every $\frac{ \pi }{184}$ sec
b. Approximately $59$ periods

68) [Technology Required] The number of hours of daylight in a northeast city is modeled by the function

\[N(t)=12+3\sin\left[\frac{2 \pi }{365}(t−79)\right], \nonumber \]

where $t$ is the number of days after January 1.

a) Find the amplitude and period.

b) Determine the number of hours of daylight on the longest day of the year.

c) Determine the number of hours of daylight on the shortest day of the year.

d) Determine the number of hours of daylight 90 days after January 1.

e) Sketch the graph of the function for one period starting on January 1.

69) [Technology Required] Suppose that $T=50+10\sin\left[\frac{ \pi }{12}(t−8)\right]$ is a mathematical model of the temperature (in degrees Fahrenheit) at $t$ hours after midnight on a certain day of the week.

a) Determine the amplitude and period.

b) Find the temperature 7 hours after midnight.

c) At what time does $T=60^{\circ}$ ?

d) Sketch the graph of $T$ over $0 \leq t \leq 24$.

Answer

a. Amplitude = $10$; Period=$24$
b. $47.4^{\circ}$ F
c. $14$ hours later, or 2 p.m.
d.

$An image of a graph. The x axis runs from 0 to 365 and is labeled “t, hours after midnight”. The y axis runs from 0 to 20 and is labeled “T, degrees in Fahrenheit”. The graph is of a curved wave function that starts at the approximate point (0, 41.3) and begins decreasing until the point (2, 40). After this point, the function increases until the point (14, 60). After this point, the function begins decreasing again.$

70) [Technology Required] The function $H(t)=8\sin\left(\frac{ \pi }{6}t\right)$ models the height $H$ (in feet) of the tide $t$ hours after midnight. Assume that $t=0$ is midnight.

a) Find the amplitude and period.

b) Graph the function over one period.

c) What is the height of the tide at 4:30 a.m.?

In exercises 71 - 79, evaluate the functions. Give the exact value.

71) $\tan^{−1}\left(\frac{\sqrt{3}}{3}\right)$

Answer: $\frac{ \pi }{6}$

72) $\cos^{−1}\left(−\frac{\sqrt{2}}{2}\right)$

73) $\cot^{−1}(1)$

Answer: $\frac{ \pi }{4}$

74) $\sin^{−1}(−1)$

75) $\cos^{−1}\left(\frac{\sqrt{3}}{2}\right)$

Answer: $\frac{ \pi }{6}$

76) $\cos\big(\tan^{−1}(\sqrt{3})\big)$

77) $\sin\left(\cos^{−1}\left(\frac{\sqrt{2}}{2}\right)\right)$

Answer: $\frac{\sqrt{2}}{2}$

78) $\sin^{−1}\left(\sin\left(\frac{ \pi }{3}\right)\right)$

79) $\tan^{−1}\left(\tan\left(−\frac{ \pi }{6}\right)\right)$

Answer: $-\frac{ \pi }{6}$

80) [Technology Required] An airplane’s Mach number $M$ is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by $ \mu =2\sin^{−1}\left(\frac{1}{M}\right).$

Find the Mach angle (to the nearest degree) for the following Mach numbers.

$An image of a birds eye view of an airplane. Directly in front of the airplane is a sideways “V” shape, with the airplane flying directly into the opening of the “V” shape. The “V” shape is labeled “mach wave”. There are two arrows with labels. The first arrow points from the nose of the airplane to the corner of the “V” shape. This arrow has the label “velocity = v”. The second arrow points diagonally from the nose of the airplane to the edge of the upper portion of the “V” shape. This arrow has the label “speed of sound = a”. Between these two arrows is an angle labeled “Mach angle”. There is also text in the image that reads “mach = M > 1.0”.$

a. $ \mu =1.4$

b. $ \mu =2.8$

c. $ \mu =4.3$

Answer: a. $\sim 92^{\circ} $ b. $\sim 42^{\circ} $ c. $\sim 27^{\circ} $

81) [Technology Required] Using $ \mu =2\sin^{−1}\left(\frac{1}{M}\right)$, find the Mach number M for the following angles.

a. $ \mu =\frac{ \pi }{6}$

b. $ \mu =\frac{2 \pi }{7}$

c. $ \mu =\frac{3 \pi }{8}$

82) [Technology Required] The temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function

$T(x)=5+18\sin\left[\frac{ \pi }{6}(x−4.6)\right],$

where $x$ is time in months and $x=1.00$ corresponds to January 1. Determine the month and day when the temperature is $21^{\circ} C.$

Answer: $x \approx 6.69,\, 8.51$; so, the temperature occurs on June 21 and August 15

83) [Technology Required] The depth (in feet) of water at a dock changes with the rise and fall of tides. It is modeled by the function $D(t)=5\sin\left(\frac{ \pi }{6}t−\frac{7 \pi }{6}\right)+8,$ where $t$ is the number of hours after midnight. Determine the first time after midnight when the depth is $11.75$ ft.

84) [Technology Required] An object moving in simple harmonic motion is modeled by the function $s(t)=−6\cos\left(\dfrac{ \pi t}{2}\right),$ where $s$ is measured in inches and $t$ is measured in seconds. Determine the first time when the distance moved is $4.5$ in.

Answer: $\sim 1.5$ sec

85) [Technology Required] A local art gallery has a portrait 3 ft in height that is hung 2.5 ft above the eye level of an average person. The viewing angle $ \theta $ can be modeled by the function $ \theta =\tan^{−1}\frac{5.5}{x}−\tan^{−1}\frac{2.5}{x}$, where $x$ is the distance (in feet) from the portrait. Find the viewing angle when a person is 4 ft from the portrait.

86) [Technology Required] Use a calculator to evaluate $\tan^{−1}(\tan(2.1))$ and $\cos^{−1}(\cos(2.1))$. Explain the results of each.

Answer: $\tan^{−1}(\tan(2.1)) \approx −1.0416$; the expression does not equal $2.1$ since $2.1>1.57=\frac{ \pi }{2}$—in other words, it is not in the restricted domain of $\tan x$. $\cos^{−1}(\cos(2.1))=2.1$, since $2.1$ is in the restricted domain of $\cos x$.

87) [Technology Required] Use a calculator to evaluate $\sin(\sin^{−1}(−2))$ and $\tan(\tan^{−1}(−2))$. Explain the results of each.

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

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