NAME:_________________________________
MATH 127 FINAL EXAM SAMPLE
All problems are worth 5 points unless otherwise noted. PLEASE SHOW ALL WORK!
1.) Find the exact value:
π 
π 
1 + tan 2   − csc 2  
6
4
 10π 
2.) Find the exact value of csc
 using reference angles.
 3 
Indicate the ref. angle and draw in standard position.
1.______________________
2.______________________
Ref. angle:_______________
π
1
3.) (10 points) Use the following equation to answer the questions: y = −2 cos x − 
2
2
a.) Find the period.
3a._____________________
b.) Find the amplitude.
3b._____________________
c.) Find the phase shift.
3c._____________________
d.) Graph the function over one period.
4.) (10 points) Find the following given:
cot θ = 5 and 180  < θ < 270  .
cos θ :________ sec θ :________
sin θ :________ csc θ :________
tan θ :________
5.) Find the distance from A to B in the figure below:
5.______________________
6.) At 10:00 AM on April 26, 2005, a building 300 feet high
casts a shadow 50 feet long. What is the angle of elevation
of the Sun?
6.______________________


3  
 .
7.) Find the exact value of cos tan −1  −


7



Please rationalize your answer.
7.______________________



x
  as an
8.) Use a right triangle to write sec sin −1 


2
 x + 4 

algebraic expression. Assume that x is positive and that
the given inverse trigonometric function is defined for the
expression in x.
9.) (10 pts) Prove the identity:
8.______________________
1 + sin θ 1 − sin θ
= 4 tan θ sec θ
−
1 − sin θ 1 + sin θ
10.) Find the EXACT value of: sin
5π
5π
π
π
sin
cos − cos
4
12
12
4
using a sum or difference formula.
11.) (10 pts) Prove the identity: sec(2θ ) =
sec 2 θ
2 − sec 2 θ
10._____________________
12.) (10 pts) Solve for θ on the interval [0, 2π ] :
(2 cosθ − 3 )(sin
2
θ − 1) = 0
12._____________________
13.) ( 10 points) To measure the distance through a mountain for a proposed tunnel, a point C is
chosen that can be reached from each end of the tunnel (see figure). Given: AC = 3800 meters,
BC = 2900 meters, and angle C is 110 degrees. Round all answers to two decimal places.
a.) Find the length of the tunnel.
13a.____________________
b.) Find the bearing from B to C.
(Assume line AB is horizontal)
13b.____________________
14.) Convert the rectangular equation x 2 y = 4 to a polar
equation that expresses r in terms of θ .
r = _____________________