Math Analysis Problem Set Chapter 5
Name_________________________________
Due:
Directions: Show all work for the problems that follow. Circle your answers. No credit will be awarded for answers
that are missing the appropriate work. Calculators are allowed, but you don’t need it for all the problems. Both
accuracy of answers and good process are important components of all exercises in Math Analysis. You may
work with another student or with a group, but you may NOT work with a tutor or another teacher.
Combine the fractions and simplify to a multiple of a power of a basic trigonometric function.
1.
sin 
1  cos

1  cos
sin 
Write each expression in factored form as an algebraic expression of a single
trigonometric function.
2. sin 2  
2
1
c s c
Find all solutions to the equation in the intervals 0,2  . You do not need a
calculator.
3. sin x tan2 x  sin x
Prove the identity.
4. sec   cos  
6.
tan 
csc 
5. 1  tan 2  
sec   1
sin 

ta n 
1  cos
 


8. cos   x   y   sin x  y 

 2

7.
9.
1
1  sin 2 
cos
cos

 2 sec 
1  sin  1  si n 
sin 4x  sin 2x  2sin 3x cos x
10. tan u  v  
tanu  tan v
1  tanu tan v
11. sin 4x  (4 sin x cos x)(2 cos2 x  1)
(hint: start with sin4x=sin2(2x)
12. cos 3 x  sin 3 x  cos x  sin x 1  cos x sin x 
Identify a simple function that has the same graph. Then confirm your choice with a
proof.
13.
sin  cos 

csc  sec 
Write the expression as the sine, cosine, or tangent of an angle.
14.
tan

5
1  tan
 tan

5

tan
3

3
Prove the double-angle identity.
2 tan u
15. tan 2u  
1  tan 2 u
16. cot 2u 
cot 2 u  1
2 cot u
Write the expression as one involving only sinx or cosx
17. sin 2x  cos 2x
18. A rectangular tunnel is cut through a mountain to make a road. The upper
vertices of the rectangle are on the circle x 2  y2  400 .
a) Show that the cross-sectional area of the end of the tunnel is 400sin2 .
b) Find the dimensions of the rectangular end of the tunnel that maximizes
its cross-sectional area.