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Quarterly 3 Review (Chapter 4 & 5.1 – 5.3)
Reminder: You should study this review alongside your notes, and any quizzes, tests or graded
assignments you have received. This review is not a standalone ‘study-guide’ for the test.
Chapter 4:
Given an angle that measures
7
20
radians and
radians :
16
23
a) sketch the angle in standard position
b) Determine a positive and negative coterminal angle
c) If possible find its complement and supplement
d) Convert the angle to degree measure
2. A sports car is moving at the rate of 100 miles per hour and the diameter of its wheel is 16 inches. Find
the angular speed of the wheel in radians per minute.
3. Find sin  ,sec , and tan  for the diagram to the right.
24
and sin   0 find the value
7
of the other 5 trigonometric functions.
4. Given that cot  
5. Determine the reference angle given:
a) 217
17
b)
30
(-6, -5)
6. Determine the quadrant that  lies in if:
a) sin  0 and tan  0
b) sec  0 and csc  0
7. Find two exact values for  in radians ( 0    2 ) and in degrees ( 0    360 )
a) cot  
 3
3
b) cos  
2
2
c) csc  2
#8-13 Graph the functions below: Be sure you can identify (when appropriate) the period, amplitude,
domain, range, x-intercepts, vertical asymptotes, and any key points.


8. f  x   2sin  x  
4



9. f  x   1  cos  2 x  
3

x
10. f  x    tan    2
3
1
2 

11. f  x   cot  x 

2 
3 
x 
12. f  x   3csc     1
2 4
5 

13. f  x   2  sec  x 

6 

#14-16 Graph the functions below: For the functions below, sketch a graph; be sure you can identify the
domain, range, period, and 3 key points.
x
14. f  x    arcsin  
15. f  x   2arccos  x  1
16. f  x     arctan  2x 
2
#17-25 Find the exact value of the following without using a calculator.
17. tan  arctan  21
18. sin arcsin  0.1 
7 

19. arccos  cos

2 

 7 
20. sin 1  sin

4 

5 

21. arctan  tan

6 

22. cos  arccos  
23. sin  arctan x 
7

24. csc  arcsin 
8

8 

25. cot  arccos 
15 

26. Write an algebraic expression equivalent to the expression:
a. sec  arcsin  x 1 

 x  3 
b. tan  arccos 

 5 

27. A 40 inch pendulum swings through an angle of 18°. Find the length of the arc in inches through which
the tip of the pendulum swings.
#28-33 Solve the triangles below, if two solutions exist, find both. (these are oblique triangles)
28. B  110 , C  30 , c  11
29. B  150 , a  64, b  10
30. C  50 , a  25, c  22
31. a  7, b  15, c  19
32. A  21 , B  42 , a  6
33. B  12 , a  32, c  36
35. From a distance of 70meters, the angle of elevation to the top of a building is 23°. Approximate the
height of the building.
36. A cowboy is traveling due west on a road that passes the local jail. At a given time the bearing to the
jail is N 52° W, and after he travels 8 more miles the bearing is N 18ᵒ W. What is the closest the cowboy
will come to the jail while on this road?
37. Find the area of the given triangles:
a. B  80 , a  4, c  8
b. a  15, b  8, c  10
38. Two spaceships leave a space station at the same time. One is flying 425 miles per hour at a bearing of
N 5° W, and the other is flying at 530 miles per hour at a bearing of N 67° E. Determine the distance
between the two ships after they have flown for 2 hours.
General Review:
f 
39. Given the functions below find: a)  f  g  x  , b)  f  g  x  , c)  fg  ( x), d )    x 
g
x2
f  x   x  4, g  x   2
x 1
40. Find the domain, range and x-intercepts for f  x   x2  4x  32
2
41. Algebraically show that the following are inverse functions:
3 x
a. f  x   3  4 x, g  x  
b. f  x   x  1, g  x   x 2  1, x  0
4
42. Find all the zeros of the function f  x   6 x3  5x2  24 x  20
43. Find the domain of the function and identify and horizontal or vertical asymptotes:
5x
a. f  x  
x  12
6x
b. f  x   2
x 1
x3  4 x 2
c. f  x   2
x  3x  2
44. Find the domain, vertical asymptote, and x-intercept of f  x   log5  x  2  3
45. Solve the logarithmic expression algebraically: log  x 1  log  x  2  log  x  2 .
46. Solve for x: 14e3 x2  560
Chapter 5:
1
47. Find cot 𝑥 given cos 𝑥 = 6 and sin 𝑥 =
√35
.
6
48. Find sec 𝑥 given tan 𝑥 = −5 and cos 𝑥 > 0.
𝜋
49. If cos 𝑥 = 0.61, find sin (𝑥 − 2 ).
# 50-53 Simplify each expression.
50. csc 𝑥 sec 𝑥 − tan 𝑥
51.
1−sin2 𝑥
csc2 𝑥−1
52. cot 𝑥 − cos3 𝑥 csc 𝑥
53.
1−cos 𝑥
tan 𝑥
sin 𝑥
+ 1+cos 𝑥
# 54-57 Verify each identity.
54. sec 2 𝑥 (1 − cos2 𝑥) = tan2 𝑥
55. tan 𝑥 csc 2 𝑥 − tan 𝑥 = cot 𝑥
sin 𝑥
cos 𝑥
56. 1−cot 𝑥 + 1−tan 𝑥 = sin 𝑥 + cos 𝑥
57.
csc2 𝑥+2 csc 𝑥−3
csc2 𝑥−1
csc 𝑥+3
= csc 𝑥+1
#58-63 Solve for x over [0, 2π), then find all solutions.
58. 3 csc 𝑥 = 2 csc 𝑥 + √2
59. 6 tan2 𝑥 − 2 = 4
60. 2 sin2 𝑥 = sin 𝑥 + 1
61. cot 2 𝑥 csc 2 𝑥 − cot 2 𝑥 = 9
62. csc 𝑥 + cot 𝑥 = 1