Download EXAM 3 - People Server at UNCW

MAT 112-002
Name:____KEY____________
EXAM 3
Grade:___110______
2
3
3
1) Suppose that sin   , tan   0 and cos    ,    
.
3
5
2
Drawing triangle in quadrants II and III respectively, we find that
5
4
cos   
and sin   
3
5
a-5pts) Find sin(   ) .
5  4 
 2  3  
sin(   )  sin  cos   cos  sin          
  
 3   5   3   5 

4 5 6
15
b-5pts) Find cos(    ) .
5   4  2 
 3 
cos       cos  cos   sin  sin       
     
 5   3   5   3 

3 5 8
15
c-5pts) Find cos(2 ) .
2
2 1
cos(2 )  1  2sin 2   1  2   
3 9
 
d-5pts) Find sin   . Note that β/2 is in quadrant II and so has positive sine.
2
 
1  cos 
sin   

2
2
3
5  8 2 5
2
10
5
1
2) Find the exact value of the following (decimal approximations are not
acceptable):
a-5pts) sin105 (Hint: use 135o and 30o)
sin(105 )  sin(135  30 )  sin135 cos 30  cos135 sin 30

2 1 
2 3
2 6
  
 
2 2  2 2 
4
b-5pts) cos 22.5 (Hint: use the half angle identity)
2
1
 45 
1  cos 45
2  2 2
cos 22.5  cos 


2
2
2
 2 
3-10pts) Verify the following identity:
cot(   ) 
cot  cot   1
cot   cot 
cos  cos 
1
cot  cot   1 sin  sin 
sin  sin 


cos

cos

cot   cot 
sin  sin 

sin  sin 
cos  cos   sin  sin 

sin  cos   cos  sin 

cos    
 cot    
sin    
4-10pts) Verify the following identity:
   sec   1
cot 2   
 2  sec   1
1
1  cos  sec sec   1
 
cot 2   



 2  tan 2    1  cos  sec sec  1
 
2
5) Find all solutions to the following trigonometric equations for angles θ in the
range 0 ≤ θ < 2π.
a-10pts) 2 cos(5 )  1  1
cos  5   1
5  0  2n , n  0, 1, 2,...
2n
, n  0, 1, 2,...
5
2 4 6 8
  0, , , ,
5 5 5 5

b-10pts) cos(2 )  cos   0 (Hint: use a double-angle identity)
2 cos 2   1  cos   0
2 cos 2   cos   1  0
(2 cos   1)(cos   1)  0
1
or cos   1
2
2 4
  0, ,
3 3
6-10pts) At 10am, the angle of elevation of the sun is 80o. A certain building casts a
showdown that is 50 feet long. How tall is the building? (Hint: the angle of
elevation of the sun is the angle from the tip of the shadow to the top of the
building).
cos   
h  50 tan 80  283.56 ft
7-10pts) Given that the triangle with sides a, b, c and opposite angles α, β, γ has:
b  2, c  3, and   40
find the remaining side and angles for all possible triangles.
sin  sin 40

3
3
3sin 40
sin  
2
  74.62 or 180  74.62  105.38
CASE 1
  74.62
  65.38
sin 65.38
 2.83
sin 40
CASE 2
  105.38
a2
  34.62
a2
sin 34.62
 1.77
sin 40
8-10pts) Two sensors are spaced 700 feet apart along the approach to a small
airport. When an aircraft is nearing the airport, the angle of elevation from the
first sensor to the craft is 20o and the angle of elevation from the second sensor to
the craft is 15o. How high is the aircraft at this time?
d
h
sin 20
and
sin 5 sin15
700sin15

d 
700
d
sin 5
So,
h
700sin15 sin 20
 710.97 ft
sin 5
BONUS – 10pts
Verify the following identity:
sin(   ) sin(   )  cos 2   cos 2 
sin(   ) sin(   )   sin  cos   cos  sin   sin  cos   cos  sin  
 sin 2  cos 2   cos 2  sin 2 
 (1  cos 2  ) cos 2   cos 2  (1  cos 2  )
 cos 2   cos 2  cos 2   cos 2   cos 2  cos 2 
 cos 2   cos 2 