Math 1316 General Review for Trigonometry Last Updated …

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Math 1316

General Review for Trigonometry

Last Updated 08/15/2014

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the measure of each angle in the problem.

1. Supplementary angles with measures 3x + 8

and 2x - 3 degrees

a. 83? and 97?

b. 98? and 82?

c. 128? and 52?

d. 113? and 67?

2. Complementary angles with measures 4x and

5x - 9 degrees

a. 84? and 96?

b. 44? and 46?

c. 11? and 79?

d. 46? and 44?

7. 4x + 5y = 0, x K 0; Find csc .

a.

5 4

b.

-

5 4

c.

41 4

d.

-

4 5

Evaluate the expression. 8. sin 450? a. 0

c. 1

b.

1 2

d. Undefined

3. A wheel makes 396 revolutions per minute. How many revolutions does it make per second? a. 13.2 revolutions per second b. 2376 revolutions per second c. 3.96 revolutions per second d. 6.6 revolutions per second

4. A wheel is rotating 720 times per minute.

Through how many degrees does a point on

the

edge

of

the

wheel

move

in

1 4

seconds?

a. 72? c. 1080?

b. 270? d. 810?

9. cos(-90?)

a.

3 2

c. -1

b. 0 d. Undefined

Use the appropriate identity to find the indicated function

value. Rationalize the denominator, if applicable. If the

given value is a decimal, round your answer to three

decimal places.

10. cot , given that tan = 0.3474

a. 2.879

b. 2.872

c. 2.893

d. 2.886

Suppose that is in standard position and the given point

is on the terminal side of . Give the exact value of the

indicated trig function for .

5. (18, 24); Find csc .

a.

5 4

b.

3 4

c.

5 3

d.

4 3

An equation of the terminal side of an angle in standard

position is given along with a restriction on x. Find the

indicated trigonometric function value of . Do not use a

calculator.

6. -3x + y = 0, x L 0; Find sin .

a.

10 10

b.

3 10 10

c. 3

d.

1 3

11. tan , given that cot =

5 8

a.

55 8

b.

8 5

c.

5 5

d.

85 5

Use the fundamental identities to find the value of the

trigonometric function.

12.

Find tan , given that

sin =

3 4

and is in

quadrant II.

a. -

7 9

b.

-

37 7

c.

-

3 2

d.

5 4

1

13.

Find sec , given that

tan =

3 4

and is in

quadrant I.

a.

-

3 2

b.

3 7

7

c.

5 4

d. -

7 9

14. Find sec , given that tan = 0.57735027 and

is in quadrant I.

a. -1.1547005

b. -1.2559261

c. 1.2559261

d. 1.1547005

15. Find tan , given that cos = -0.58778525 and

is in quadrant II.

a. 1.9734303

b. 1.3763819

c. -1.9734303

d. -1.3763819

16. Find the exact value of x in the figure.

14

a. 5 3 c. 7 6

b. 7 3 d. 8 3

17. Find the exact value of x in the figure.

28

x

a. 14 3

c.

28 3

6

b.

28 3

3

d. 14 6

Find the exact value of the expression.

18. cos 150?

a.

3 2

b.

2 2

c. -

3 2

d. -

2 2

19. sec 210? a. - 2

c.

23 3

b. 2

d.

-

23 3

20. cos (-2190?)

a. -

3 2

c.

1 2

b.

3 2

d.

-

1 2

Solve the problem.

21. On a sunny day, a flag pole and its shadow

form the sides of a right triangle. If the

hypotenuse is 40 meters long and the shadow

is 32 meters, how tall is the flag pole?

a. 72 m

b. 64 m

c. 51 m

d. 24 m

2

22. To measure the width of a river, a surveyor starts at point A on one bank and walks 74 feet down the river to point B. He then measures the angle ABC to be 23?32'12''. Estimate the width of the river to the nearest foot. See the figure below.

C

A

a. 32 ft c. 30 ft

74 ft

B

b. 68 ft d. 170 ft

23. An airplane travels at 180 km/h for 5 hr in a

direction of 289? from Greenville. At the end

of this time, how far west of Greenville is the

plane (to the nearest kilometer)?

a. 310 km

b. 952 km

c. 293 km

d. 851 km

24. A ship travels 99 km on a bearing of 35?, and

then travels on a bearing of 125? for 129 km.

Find the distance from the starting point to the

end of the trip, to the nearest kilometer.

a. 81 km

b. 57 km

c. 163 km

d. 228 km

25. Find h as indicated in the figure. Round to the nearest foot.

24.9?

102 ft

a. 68 ft c. 65 ft

59.3?

b. 70 ft d. 62 ft

26. The angle of elevation from a point on the

ground to the top of a tower is 38? 19. The

angle of elevation from a point 145 feet farther

back from the tower is 26? 41. Find the height

of the tower. Round to the nearest foot.

a. 2002 ft

b. 196 ft

c. 211 ft

d. 200 ft

Find the exact value without using a calculator.

27.

csc

5 3

a. - 3

b. - 2

c.

-

1 2

d.

-

23 3

28.

sec

-5 4

a. -2

c.

-

23 3

b. - 2

d.

2 2

Find the length of an arc intercepted by a central angle

in a circle of radius r. Round your answer to 1 decimal

place.

29.

r

=

15.95

ft;

=

29

radians

a. 3.5 ft c. 0.9 ft

b. 1.7 ft d. 5.2 ft

30. r = 116.15 in.; = 162? a. 328.4 in. c. 656.8 in.

b. 164.2 in. d. 104.5 in.

31. Find the distance between City E, 43? N and

City F, 74? S. (Round to the nearest kilometer.)

a. 3455 km

b. 13,069 km

c. 13,077 km

d. 3463 km

Assume that the cities lie on the same north-south line

and that the radius of the earth is 6400 km.

32. Find the latitude of Winnipeg, Canada if

Winnipeg and Austin, TX, 30?N, are 2234 km

apart.

a. 20?N

b. 70?N

c. 50?N

d. 60?N

3

33. A wheel with a 38-inch radius is marked at

two points on the rim. The distance between

the marks along the wheel is found to be 14

inches. What is the angle (to the nearest tenth

of a degree) between the radii to the two

marks?

a. 19.1?

b. 23.1?

c. 17.1?

d. 21.1?

34. Two wheels are rotating in such a way that the

rotation of the smaller wheel causes the larger

wheel to rotate. The radius of the smaller

wheel is 3.1 centimeters and the radius of the

larger wheel is 17.4 centimeters. Through how

many degrees (to the nearest hundredth of a

degree) will the larger wheel rotate if the

smaller one rotates 140??

a. 24.94?

b. 25.94?

c. 26.94?

d. 24.84?

Find the area of a sector of a circle having radius r and

central angle . If necessary, express the answer to the

nearest tenth.

35.

r

=

23.0

ft,

=

3

radians

a. 1107.9 ft2 c. 554.0 ft2

b. 50.4 ft2 d. 24.1 ft2

36. r = 19.5 mi, = 151? a. 1002.1 mi2 c. 501.1 mi2

b. 25.7 mi2 d. 33.9 mi2

37. Find the measure (in radians) of a central angle

of a sector of area 46 square inches in a circle of

radius 7 inches. Round to the nearest

hundredth.

a. 2.82 radians

b. 0.94 radians

c. 3.76 radians

d. 1.88 radians

38. A pendulum swings through an angle of 19?

each second. If the pendulum is 17 cm in

length and the complete swing from right to

left lasts 2 seconds, what area is covered by

each complete swing? Round to the nearest

hundredth.

a. 47.92 cm2

b. 191.67 cm2

c. 95.84 cm2

d. 5.64 cm2

Find the exact circular function value.

39.

tan

-5 6

a. 3

b. - 3

c.

3 2

d.

3 3

40.

csc

-2 3

a.

-

1 2

c. - 2

b.

-

23 3

d. - 3

The figure shows an angle in standard position with its terminal side intersecting the unit circle. Evaluate the indicated circular function value of .

41. Find sin .

-

5 13

,

12 13

a.

5 12

c.

-

5 13

b.

12 13

d.

-

12 13

4

42. Find csc .

Suppose an arc of length s lies on the unit circle x2 + y2 = 1, starting at point (1, 0) and terminating at the point (x, y). Use a calculator to find the approximate coordinates (x, y). Round coordinates to four decimal places when appropriate.

a.

25 24

c.

-

25 24

7 25

,

-

24 25

b.

24 7

d.

-

25 7

43. sec 0.1943 a. 0.1931 c. 0.9812

b. 0.1968 d. 1.0192

Use a table or a calculator to evaluate the function. Round

to four decimal places.

44. csc 0.2391

a. 0.2368

b. 0.9716

c. 4.2225

d. 1.0293

45. s = 7.6 a. (0.9679, 0.2513) b. (-0.2513, -0.9679) c. (-0.2513, 0.9679) d. (0.2513, 0.9679)

Find the exact values of s in the given interval that satisfy

the given condition.

46.

[0,

2);

tan2

s

=

1 3

a.

6

,

7 6

b.

3

,

2 3

,

4 3

,

5 3

c.

3

,

4 3

d.

6

,

5 6

,

7 6

,

11 6

47. [-, ); 2 cos2 s = 1

a.

4

,

4

,

4

,

7 4

b.

-

2 3

,

-

3

,

3

,

2 3

c.

-

7 4

,

-

4

,

-

4

,

-

4

d.

-

3 4

,

-

4

,

4

,

4

5

48. Let angle POQ be designated . Angles PQR and VRQ are right angles. If = 45?, find the exact length of OQ.

50. Let angle POQ be designated . Angles PQR and VRQ are right angles. If = 27?, find the length of OU accurate to four decimal places.

a. 2 c. 0

b. 1

d.

2 2

49. Let angle POQ be designated . Angles PQR and VRQ are right angles. If = 80?, find the length of OQ accurate to four decimal places.

a. 0.1736 c. 5.7588

b. 0.9848 d. 5.6713

a. 2.2027 c. 1.1223

b. 0.8910 d. 0.4540

51.

=

6

radian

per

min,

t

=

13

min

a.

78

radians

b.

13 6

radians

c.

78

radian

d.

6 13

radians

Use

the

formula

=

t

to

find

the

value

of

the

missing

variable. Give an exact answer unless otherwise indicated.

52. = 9.2302 radians per min, = 13.09 radians

(Round to four decimal places when

necessary.)

a. 1.4182 min

b. 0.7051 min

c. 22.3202 min

d. 120.8233 min

Use the formula v = r to find the value of the missing variable. Give an exact answer unless otherwise indicated.

53. v = 16 ft per sec, r = 3.3 ft (Round to four decimal places when necessary.) a. 0.952 radian per sec b. 5.0929 radians per sec c. 0.2063 radian per sec d. 4.8485 radians per sec

6

Use the formula s = rt to find the value of the missing

a.

variable. Give an exact answer.

54.

s

=

3

m,

r

=

7

m,

t

=

32

sec

a.

672

radian per sec

b. 672 radians per sec

c.

21 32

radians per sec

d.

32 21

radian per sec

55. A wheel is rotating at 8 radians/sec, and the

wheel has a 38-inch diameter. To the nearest

b.

foot, what is the speed of a point on the rim in

ft/min?

a. 755 ft/min

b. 765 ft/min

c. 760 ft/min

d. 750 ft/min

56. A wheel with a 22-inch diameter is turning at

the rate of 58 revolutions per minute. To the

nearest inch, what is the speed of a point on

the rim in in./min?

a. 4055 in./min

b. 4009 in./min

c. 4062 in./min

d. 4016 in./min

Graph the function.

57.

y =

sin

1 4

x

c.

7

d.

b.

c.

58.

y

=

cos

1 3

x

d. a.

8

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