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MAT 115 – Section 001
Name: ____KEY________________
Exam 3
Grade: ___110__________
Directions: Answer each of the following questions. Be sure to show all work in order
to receive partial credit. If you need additional space, you may use the back of the test
paper.
1) Convert the following angles from degrees into radians:
a – 5pts) 20 20/180 π = π/9
b – 5pts) 240 240/180 π = 4π/3
2) In a circle of radius 3 feet,
a – 5pts) find the arc length subtended by a central angle of 40 .
First, convert the angle into RADIANS! 40/180 π = 2π/9
s = rθ = 3(2π/9) = 2π/3 feet
B – 5pts) What is the area of the sector created by this central angle?
A = ½ r2θ = ½ (3)22π/9 = π ft2
3
3
   2 , find the values of the
where
4
2
remaining five trigonometric functions.
3 – 10pts) Given that sin   
sinθ = -3/4
cosθ = √7/4
tanθ = -3/√7
cscθ = -4/3
secθ = 4/√7
cotθ = -√7/3
4) Find the exact values of the following (decimal approximations are not
acceptable):
Consult the Unit Circle to find the appropriate triangles!
 4 
a – 5pts) sin 
 = -√3 / 2
 3 
 7 
b – 5pts) cos 
 = √2 / 2
 4 
5) Consider the graph of


y  3cos  2 x  
3

a – 2pts) What is the amplitude? ___3_______
b – 2pts) What is the period?
___π_______
c – 2pts) What is the phase shift? ___π/6_______
d – 4pts) Sketch the graph of one complete period of this function
(including at least five labeled points) below.
6) Find the exact value of the following (decimal approximations are not
acceptable):
 2
a – 5pts) sin 1 
 = π/4
 2 

 2 
b – 5pts) sin sin 1     = -2/3
 3 

7) Verify the following trigonometric identities:
a – 5pts) tan   cot   sec csc  0
sin  cos 
1


cos  sin  sin  cos 
sin 2   cos 2   1

sin  cos 
11

sin  cos 
0
tan   cot   sec  csc  
sec   cos 
sin 2 
b – 5pts)

sec   cos  1  cos 2 
1
 cos 
sec   cos  cos 

1
sec   cos 
 cos 
cos 
1  cos 2 
 cos 2
1  cos 
cos 
1  cos 2 

1  cos 2 
sin 2 

1  cos 2 
2
3
3
8) Suppose that sin   , tan   0 and cos    ,    
.
3
5
2
By drawing triangles in the appropriate quadrants (as in Problem 3) we find that
cos α = -√5 / 3 and that sin β = -4/5
a-5pts) Find sin(   ) .
sin(α+β) = sin α cos β + sin β cos α = (2/3)(-3/5) + (-4/5)( -√5/3) = (4 √5 – 6) / 15
b-5pts) Find cos(2 ) .
cos (2 α) = cos2 α – sin2 α = (-√5/3)2 – (2/3)2 = 1/9
9 – 10pts) Solve the following trigonometric equation for all θ in the range
0 ≤ θ ≤ 2π.
3
cos   
2
Consult the Unit Circle for the appropriate triangles!
θ = 5π/6, 7π/6
10 – 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.
Using the law of sines, we find that sin γ = 3/2 sin(40o). This gives two possible
angles:
γ = 74.62o or 105.38o
CASE 1:
γ = 74.62o
α = 65.38o
a = 2 sin(65.38o)/sin(40o) = 2.83
CASE 2:
γ = 105.38o
α = 34.62o
a = 2 sin(34.62o)/sin(40o) = 1.77
BONUS – 10pts
Verify the following identity:
sin(   )sin(   )  cos 2   cos 2 
sin(    ) sin(    )  sin  cos   sin  cos  sin  cos   sin  cos  
 sin 2  cos 2   sin 2  cos 2 

 
   cos  sin
 sin 2  cos 2   cos 2  cos 2   sin 2  cos 2   cos 2  cos 2 

 cos  sin   cos
2
2
 cos 2   cos 2 
2
2
2
  cos  
2
