Download MAC 1114FinalExamREVIEW

MAC 1114 – Trigonometry – Final Exam Review
Find the least positive coterminal angle.
750
34
2.
15
13
3.
3
1.
Convert from radians to degrees or degrees to radians without a calculator.
115
5. 240
7
6.
6
4.
7.

9
Find the exact value of each expression.
8
3
9. csc120
10. arccos0
8.
cos



 2 
 2  


11. cos  sin 1 
 

 3
12. tan  
 2 

 3
13. csc 1 
 1 

 3
14. cot 1 
Solve.
15. Find the length of the arc on a circle of radius 9 ft and central angle 60 .
16. Find the central angle in radians of a circle of radius 14.5 centimeters that intercepts an arc of
length 25 centimeters.
17. A lawn roller with a 10-inch radius makes 1.2 revolutions per second. Find the angular velocity
of the roller in radians per second. Also, find the linear velocity of the tractor pulling the roller.
18. Let  
5
and evaluate the six trigonometric functions without using a calculator.
3


19. Let  be an angle in standard position whose terminal side passes through the point 1,  3 .
Find the exact values of all six trigonometric functions for  .
1
and assume that  is in quadrant 4. Determine the exact value of csc .
3


21. Graph the function f  x   2 cos  6 x   . What is the amplitude, period, and phase shift?
2

20. Let cos  
What is the range of f  x  ?
22. Graph the function f  x  
1 

sin  x   . What is the amplitude, period, and phase shift?
2 
3
23. A 15-foot ladder leans against a wall at an angle of 60 degrees. How high up the wall does the
ladder rest? How far is the base of the ladder from the wall?
24. At some distance from a tower, the angle of elevation to the top of the tower is 36.5⁰. Sixty feet
farther away from that point (in the same direction), the angle of elevation is 21.6⁰. How tall is
the tower?
3

2
3
with     and cos   with
   2 . Determine the exact
7
2
5
2
 


value of cos ,sin  ,cos     ,sin     , tan  , tan  2  ,cos   ,sin     .
2
2

25. Given sin  
Verify/Prove Each Identity
26.
1
1

 2sec2 x
1  sin x 1  sin x
27. tan 4   tan 2  sec2   tan 2 
28.
sin 3 x  cos3 x
1
 1  sin  2 x 
sin x  cos x
2
Find all real numbers that satisfy the equation.
29. 3cot 2 x  1  0
Find all values of x in [0, 2 ) that satisfy the equation.
30. 2sin x  3cos x  1
31. 2sin 2 x  cos x  1  0
Solve each triangle. If no triangle exists explain why. If two different triangles are possible, find
both.
32.  
102.3 ,   28.7 , b  27.4
33. a  12, b  31,  20.5
34. a  18, b  20,  76
35. a  8, b  19, c  14
Find the area of the triangle.
36. a  8, b  19, c  14
37.  
102.3 ,   28.7 , b  27.4
Let v=<-2,5> and w=<3,4>.
38.
39.
40.
41.
42.
43.
v-2w
|v+w|
Write v as a linear combination of the unit vectors i and j.
Find the magnitude and direction angle for v.
Find the dot product of v and w.
Determine the angle between v and w.
1
2
44. Show that the vectors  ,8 and 10,
5
are perpendicular.
8
45. An airplane is flying with airspeed 186 mph is at a compass heading of 136⁰. If the wind speed at
that elevation is 18mph directly out of the west, then what is the groundspeed and actual compass
heading of the plane?
Complex Numbers
46. Write z  2  2 3i in trigonometric form.

47. Let z1  24 cos300  i sin 300
 and z
2
 8  cos75  i sin 75  and find
answer in standard form, a  bi.

48. Determine 2  2 3i

9
Polar Coordinates


49. Convert the polar coordinates  3 3,

 to rectangular coordinates.
6
50. Convert the rectangular coordinates  3, 4  to polar coordinates.
z1
. Write final
z2
51. Convert the polar equation 5r  sec into rectangular form. What does the graph look like?
Parametric Equations
52. Sketch the curve given by x  t 2 , y  t  3 for 1  t  2 .
53. Using the parametric equations from (#50) eliminate the parameter and state an equivalent
rectangular equation. State the domain.
54. The center field fence in a baseball park is 10 feet high and 400 feet from home plate. The
baseball is hit 3 feet from above the ground. It leaves the bat at an angle of  with a horizontal
speed of 146 feet per second.
a. Write a set of parametric equations for the path of the baseball.
b. Use your calculator to graph to sketch the path of the baseball for  = 15 and then for
  23 .
c. Of the two angles in part b, which one will be a home run (ball goes over the fence)?