MATH 122
Final Exam Review
5.1
1. Mark the point determined by 6 on the unit circle.
5.3
2. Sketch a graph of y  sin( x) by hand.
5.3
 
 
3. Find the amplitude, period, and horizontal shift of the graph of y  2 cos  3  x    .
6 
 
5.3, 5.4 4. Graph each function by hand:
 
 
b. y  2 sin  3  x   
3 
 
a. y  4 cos(3x)
5.3
 
 
c. y  tan  2  x   
4 
 
5. Write the equation of the graph in the form y  a sin  k( x  b)  .
y

x


























5.5
1   1 
6. Evaluate sin  .
 2
5.5, 6.4
7. Evaluate exactly. a.
 3 
sin 1 sin .
2 

 1 2 
b. cos sin
.
3

6.1
8. Find one positive and one negative angle coterminal with: a. 237
6.1
9. Convert
3
a.
radians to degree measure.
4
b.
3
.
7
b. 196 to radian measure. Round to the nearest hundredth.
6.1
10. The roller on a treadmill has a diameter of 3 inches. How many revolutions per minute
must it make in order for Josh to run at a speed of 10 miles per hour? (1 mile = 5,280 ft)
6.2 11. If c represents the length of the hypotenuse, solve the right triangle with a  6
and  B  71 .
6.2 12. If sec( )  8 and  is an acute angle, find sin( ) exactly.
3
6.2
13. You are a tour guide lost in the Rockies and have just come to a gorge. You estimate
the angle of elevation of the top of the useful part of a tree that is 25 feet away from
you to be 70 . You also estimate that the gorge is 50 feet wide and want to allow an
extra 6 feet on each end for safety. Will the useable part of the tree be long enough to
bridge the gorge if you cut the tree? (Note: the tree just happens to be standing six feet
from the edge of the gorge.)
6.2
14. You are estimating the height of a lighthouse surrounded by water. When you first
observe it from the shoreline, the angle of elevation seems to be about 16 . After
walking inland 200 feet in a direct line with the first observation point, the angle of
elevation is about 10 . Find the height of the lighthouse if the ground is level.
6.3
15. Write cot  in terms of sin  given that  is in Quadrant II.
6.4, 7.2
16. Simplify so that the answer contains no trig functions.
b
2

 1 a
tan cos 1 
 cos 1 
a.
b. sin tan
2
7
3


b. a  10, B  27 , b  6
6.5
17. Solve the triangle in which a. a  4, B  27 , C  102
6.5
18. A pilot is about to land on the 8500 foot runway in Los Angeles. The angles of
depression from the plane to the ends of the runway are 16 and 17.2 . Find the
distance the plane must fly before it touches down.
6.6 19. Solve the triangle in which a  20, b  27, c  16 .
6.6 20. The distance between two planes is calculated by using LORAN to find the position of
each plane relative to the tracking station. If one plane is 30 miles from the station, a
second is 27 miles, and the angle between the two distances is 37  , find how far apart
the planes are.
cos 2 ( )
.
1  sin( )
7.1
21. Simplify
7.1
22. Verify the trig identity. sec 4 ( x)  tan 4 ( x)  1  2 tan 2 ( x).
7.2
23. Write an equivalent expression using only one trig function; evaluate exactly.
sin(61 ) cos(74 )  cos(61 ) sin(74 )
7.2
7.2
7.3
7.4
24. Find cos( 105 ) exactly. (Hint: use an addition or subtraction formula)
2 
2
3
 A   and tan( B)  ;   B 
25. Given that sin( A)  ;
, find cos( A  B)
5 2
3
2
exactly.
1 3
 
   2 , find a. sin(2 )
26. If cos( )  ;
b. cos 
3 2
2
27. Find all real solutions exactly in radians.
a.
7.5
3 sec( )  2  0
b. 2sin 2 ( )  sin( )  1
28. Find all real solutions.
c. sin 2 ( )  cos( ) 1  0
2 cos( x)  x 2  0 .
 4 
8.1 29. Convert the point  3,  from polar to rectangular coordinates.
 3 
8.1 30. Convert the point (2,2) from rectangular to polar coordinates.
8.1
5 

31. Graph the point   3,  in a polar coordinate system.
6 

32. Convert r  6 sin( ) to a rectangular equation.
8.2
33. Convert x 2  ( y  5) 2  25 to a polar equation.
8.2
34. Graph r  5 sin(3 ) .
8.4
35. (a) Graph the curve x  4cos t , y  4sin t , 0  t  2 .
(b) Find a rectangular-coordinate equation by eliminating the parameter.
8.4
36. A ball is thrown with an initial speed of 92 feet per second from a height of 7
8.1
feet at an angle of 50 to the horizontal. You may assume that the ground is level.
(a) Find the height of the ball after 2 seconds. ( b) Determine the horizontal distance
the ball travels.
Use the parametric equations x   v0 cos   t ,
y  h   v0 sin   t  16t 2
9.1
37. Given u = 1,  2 , v = 4, 3 , find 2u − 3v .
9.1
38. Find the direction of the vector 3, 4 .
9.1
39. The speed of an airplane is 400 mi/hr relative to the air. The wind is blowing due west
with a speed of 35 mi/hr. In what direction should the airplane head in order to arrive
at a point due south of its location?
9.2
40. Find the angle between 3, 4 and 7, 12 .
9.2
41. A horse pulls a wagon horizontally by exerting a force of 220 pounds on the handle.
If the handle makes an angle of 40 with the horizontal, find the work done in moving
the wagon 300 feet.
4 x  3 y  3
10.1 42. Solve the system. 
3x  2 y  1
10.1 43. A chemist needs 100 ml of a 27% solution of sulfuric acid, but only has 10% and 40%
solutions available. Find how much of each should be used.
10.3
44. Michigan State University and the University of Michigan scored a total of 50 points
during their 2015 football game. There were 14 scoring plays: touchdowns, extra point
kicks, and field goals at 6, 1, and 3 points respectively. There were twice as
many touchdowns as field goals. Find the number of each scoring play for the game.
10.3
45. Solve each system. If the system is dependent, state the complete solution.
2 x  y  3z  12

(a)  5 x  2 y  z  5
 3x  3 y  2 z  1

3x  y  z  2

(b)  x  y  2 z  1
7 x  y  5

3x  y  z  2

(c)  x  y  2 z  1
7 x  y  2

11.1
46. A parabolic reflector for a telescope is six inches in diameter. It is ground so that its
vertex is 0.15 inch below the rim (on the concave side). ( a) Insert an xy  coordinate
system with vertex at the origin and find the equation of the parabola. (b) How far is
the focus from the vertex?
11.2
47. Find the vertices, foci, and eccentricity of the ellipse with equation 4 x 2  9 y 2  36.
11.2
48. Write the equation of an ellipse centered at the origin containing the point (4,0) and
with one focus at (0,3) .
11.4
49. Find the vertex, focus, and directrix of the parabola x 2  6 x  4 y  1  0.
Draw the graph.
11.4
50. Find the center, vertices, foci, and asymptotes of 25x 2  4 y 2  200 x  8 y  296  0 .
Draw the graph.
n
.
2n  1
12.1
51. List the first four terms and the 100th term of the sequence an 
12.1
3 4 5 6
52. Find a formula for the nth term of the sequence 1, , , , , ...
5 10 17 26
12.1
53. (a) Find the fourth partial sum of the sequence 3,12, 27,48,75, …
4
(b) Evaluate the sum
 3k 2 .
k 1
12.1 54. Write sigma notation for the series
3 5 7 9 11
    .
2 3 4 5 6
12.1 55. Find the first five terms of the sequence where a1  3 , an1  2an  1 for n  1.
12.2 56. Find the 31st term of the sequence  8 ,  5,  2 ,1, 4 ,...
12.2 57. Find a23 a for the arithmetic sequence {an } which has a7  27 and a40  159. .
23
12.2 58. Evaluate the sum
2k  1
3
k 1

12.2 59. Each row in an auditorium seats 2 more people that the preceding row. There are 25
seats in the first row and 48 rows. Find the seating capacity of the auditorium.
12.3 60. Find the 10th term of the geometric sequence 12 , 18 , 27 ,
81
, ...
2
12.3 61. Find a formula for the nth (general) term of the sequence 24 , −12 , 6 , −3 , …
12
12.3 62. Find the sum.
1
 48 2 
k 1
k

12.3
63. Determine whether the infinite geometric series
k
1
 48 2  is convergent or
k 1
divergent. If it is convergent, find its sum.
12.3
64. You have a job offer with a salary of $50,000 for the first year. If you accept the job
and get a 2.5% raise every year for the next 30 years, what are your total earnings for
the entire 30 years?
12.6 65. Expand and simplify: (a) (2 x  3)5 using the binomial theorem
4
1 
(b)   2  using Pascal’s triangle.
x 
17
 2 x
12.6 66. Find the tenth term in the binomial expansion of    .
 x2 2 
12.6 67. The probability that a softball player hitting .250 will get a hit 3 times in the next eight
at bats is the term in which .25 has an exponent of 3 in the binomial expansion of
(0.25  0.75)8 . Find that probability.