Download Math 112: Winter 2006

Math 112: Winter 2006
Midterm Review
1. Convert to radians and find the reference angle in radians:
5
a. 75  
; angle is in first quadrant so it is its own reference angle
12
b. 1 

180
; angle is in first quadrant so it is its own reference angle
5

; R.A.=
6
6
2. Covert to degrees and find the reference angle in degrees:
c.  150   

 180 

a. 1  
  57.3 ; angle is in first quadrant so it is its own reference
  
angle
11
 660  ; R.A.= 60˚
3
c.  13  2340  ; R.A. = 180˚
b.
3. Given the value of one trigonometric function, and the quadrant in which α lies,
find the values of the other five trigonometric functions:
2

a. sin   , 0   
7
2
3 5
3 5
7
2
7
cos  
, tan  
, csc   , sec  
, cot  
2
7
2
3 5
3 5
3 
b. cos   
,  
13 2
sin  
2
13
, tan   
c. tan   5 ,    
sin   
5
26
13
13
2
3
, sec   
, csc  
, cot   
3
2
3
2
3
2
, cos   
1
26
, cot  
26
1
, csc   
, sec    26
5
5
4. Find exact values of sin t , cos t , and tan t for the following:
a. t  0 
sin t  0 , cos t  1, tan t  0
b. t 
2
3
sin t 
3
1
, cos t   , tan t   3
2
2
c. t  300 
3
1
sin t 
, cos t  , tan t  3
2
2
5
6
3
1
1
sin t  , cos t 
, tan t 
2
2
3
d. t  
e. t  9
sin t  0 , cos t  1 , tan t  0
5. Find ALL solutions of the following equations. (You may give your solutions in
radians or degrees, but make sure you know which you are giving). Please round
your solutions to the nearest hundredth.
a. cos t  1
RADIANS: t    2k , where k is any integer
DEGREES: t  180  360k , where k is any integer
b. sin x  .5624
RADIANS: t  .60  2k or t  2.54  2k , where k is any integer
DEGREES: t  34.22  360k or t  145.78  360k , where k is any
integer
c. tan t  3
4
 2k , where k is any integer
3
3
DEGREES: t  60  360k or t  240  360k , where k is any integer
RADIANS: t 

 2k or t 
6. Give the period, amplitude and midline of the following functions, and sketch
their graphs:
a. f (t )   sin t
Period: 2
Amplitude: 1
Midline: y  0
b.
c.
 
f (t )  3 cos t  
2

Period: 2
Amplitude: 3
Midline: y  0
f (t )  cos2t     2
Period: 
Amplitude: 1
Midline: -2
7. A ferris wheel has diameter 200 feet and one complete revolution takes an hour.
Let t , the time in minutes, be 0 when you are at the 6 o’clock position.
ANSWERS ARE GIVEN USING DEGREE MEASUREMENTS
a. Write  , measured from the 3 o’clock position, as a function of t .
  6t  90
b. Find a formula for h , your height in feet above the ground, in terms of  .
h  100  100 sin 
c. Find a formula for h , your height in feet about the ground, in terms of t .
h  100  100 sin( 6t  90)
d. Graph h  f (t )
8. Prove the following identities:
tan 2 x  1
a.
 tan 2 x
2
cot x  1
sin 2 x
1
tan x  1 cos 2 x


cot 2 x  1 cos 2 x
1
sin 2 x
2
b. 1  sin x 
sin 2 x cos 2 x sin 2 x  cos 2 x
1

2
2
2
2
cos x cos x 
cos x
cos 2 x  sin x  tan 2 x

1
cos 2 x sin 2 x cos 2 x  sin 2 x
cos 2 x

2
sin x
sin 2 x sin 2 x
sin 2 x
cos x
sec x  tan x
cos x

sec x  tan x
cos x
cos x
cos 2 x 1  sin 2 x (1  sin x)(1  sin x)




 1  sin x
1
sin x 1  sin x 1  sin x 1  sin x
1  sin x

cos x cos x
cos x
1
1  2 sin 2 x
1
1
sec 2 x 

cos 2 x 1  2 sin 2 x
c. sec 2 x 
d. tan
x 1  cos x

2
sin x
9. Find sin 2 x , cos 2x , tan 2x , sin
a. 0  x 

2
and cos x 
x
x
x
, cos , and tan :
2
2
2
3
5
24
7
24
, cos 2 x   , tan 2 x  
25
25
7
x 1
x
2
x
8
, cos 
, tan 
sin 
2 2
2
10
2
10
sin 2 x 

3
2
5
2 66
2 66
19
sin 2 x  
, cos 2 x 
, tan 2 x  
19
25
25
b.
sin
 x   and sin x 
x

2
5 3
x
, cos 
10
2
5 3
x
5 3
, tan 
10
2
5 3
3
and tan x  5
2
2 5
2
5
sin 2 x 
, cos 2 x   , tan 2 x  
3
6
6
c.   x 
10. Find exact values for the following(Assume all angles are in the first quadrant):
5
5
a. sin(tan 1 ) 
3
34
2
5
b. tan(cos 1 ) 
3
2
1
15
c. cos(sin 1 ) 
4
4
11. Solve the triangles (Angle A is the angle opposite side a, angle B is opposite side
b, angle C is opposite side c):
a. a = 15, b = 3, A = 30˚
c = 17.5, B = 5.74˚, C = 144.3˚
b. c = 10, C = 45˚, B= 30˚
A = 105˚, a = 13.67, b = 7.07
c. a = 10, b = 12, c = 18
A = 31.6˚, B = 38.9˚, C = 109.5˚
12. Use double angle and half angle formulas to find exact values for the following:
a. cos
7
2 3

12
2
b. tan 105  
c. sin
7

24
2 3
2 3
2 2 3
2