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1.4E: Exercises for Section 1.3

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

1) 240°

Answer
4π3 rad

2) 15°

3) 60°

Answer
π3 rad

4) 225°

5) 330°

Answer
11π6 rad

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

6) π2 rad

7) 7π6 rad

Answer
210°

8) 11π2 rad

9) 3π rad

Answer
540°

10) 5π12 rad

In exercises 11 - 16, evaluate the functional values.

11) cos4π3

Answer
cos4π3=0.5

12) tan19π4

13) sin(3π4)

Answer
sin(3π4)=22

14) sec(π6)

15) sin(π12)

Answer
sin(π12)=3122

16) cos(5π12)

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°=225°π180°=5π4

17) a=4,c=7

Answer
a. b=5.7
b. sinA=47,cosA=5.77,tanA=45.7,cscA=74,secA=75.7,cotA=5.74

18) a=21,c=29

19) a=85.3,b=125.5

Answer
a. c=151.7
b. sinA=0.5623,cosA=0.8273,tanA=0.6797,cscA=1.778,secA=1.209,cotA=1.471

20) b=40,c=41

21) a=84,b=13

Answer
a. c=85
b. sinA=8485,cosA=1385,tanA=8413,cscA=8584,secA=8513,cotA=1384

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 θ with a terminal side that passes through point P.

Rationalize denominators.

23) P(725,y),y>0

Answer
a. y=2425
b. sinθ=2425,cosθ=725,tanθ=247,cscθ=2524,secθ=257,cotθ=724

24) P(1517,y),y>0

25) P(x,73),x>0

Answer
a. x=23
b. sinθ=73,cosθ=23,tanθ=142,cscθ=377,secθ=322,cotθ=147

26) P(x,154),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) tan2x+sinxcscx

Answer
sec2x

28) secxsinxcotx

29)tan2xsec2x

Answer
sin2x

30) secxcosx

31) (1+tanθ)22tanθ

Answer
sec2θ

32) (sinx)(cscxsinx)

33) costsint+sint1+cost

Answer
1sint=csct

34) 1+tan2α1+cot2α

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

35) tanθcotθcscθ=sinθ

36) sec2θtanθ=secθcscθ

37) sintcsct+costsect=1

38) sinxcosx+1+cosx1sinx=0

39) cotγ+tanγ=secγcscγ

40) sin2β+tan2β+cos2β=sec2β

41) 11sinα+11+sinα=2sec2α

42)tanθcotθsinθcosθ=sec2θcsc2θ

In exercises 43 - 50, solve the trigonometric equations on the interval 0θ<2π.

43) 2sinθ1=0

Answer
{π6,5π6}

44) 1+cosθ=12

45) 2tan2θ=2

Answer
{π4,3π4,5π4,7π4}

46) 4sin2θ2=0

47) 3cotθ+1=0

Answer
{2π3,5π3}

48) 3secθ23=0

49) 2cosθsinθ=sinθ

Answer
{0,π,π3,5π3}

50) csc2θ+2cscθ+1=0

In exercises 51 - 54, each graph is of the form y=AsinBx or y=AcosBx, 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=4sin(π4x)

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=cos2πx

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(xπ4)

Answer
a. 1
b. 2π
c. π4 units to the right

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

57) y=12sin(14x)

Answer
a. 12
b. 8π
c. No phase shift

58) y=2cos(xπ3)

59) y=3sin(πx+2)

Answer
a. 3
b. 2
c. 2π units to the left

60) y=4cos(2xπ2)

61) [T] The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of 120°, how many inches does it move? Approximate to the nearest whole inch.

Answer
Approximately 42 in.

62) [T] Find the length of the arc intercepted by central angle θ in a circle of radius r. Round to the nearest hundredth.

a. r=12.8 cm, θ=5π6 rad b. r=4.378 cm, θ=7π6 rad c. r=0.964 cm, θ=50° d. r=8.55 cm, θ=325°

63) [T] 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, ω, and is given by ω=θ/t, where θ is in radians and t is time. Find the angular speed for the given data. Round to the nearest thousandth.

a. θ=7π4 rad, t=10 sec b. θ=3π5 rad, t=8 sec c. θ=2π9 rad, t=1 min d. θ=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) [T] A total of 250,000 m2 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° sector of a circle instead of a whole circle, find the length of the curved side.

65) [T] The area of an isosceles triangle with equal sides of length x is 12x2sinθ,

where θ is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle θ=5π12 rad.

Answer
30.9 in2

66) [T] A particle travels in a circular path at a constant angular speed ω. The angular speed is modeled by the function ω=9|cos(πtπ/12)|. Determine the angular speed at t=9 sec.

67) [T] An alternating current for outlets in a home has voltage given by the function V(t)=150cos368t,

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. π184; the voltage repeats every π184 sec
b. Approximately 59 periods

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

N(t)=12+3sin[2π365(t79)],

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) [T] Suppose that T=50+10sin[π12(t8)] 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°?

d) Sketch the graph of T over 0t24.

Answer

a. Amplitude = 10; Period=24
b. 47.4°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) [T] The function H(t)=8sin(π6t) 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.?

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.


This page titled 1.4E: Exercises for Section 1.3 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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