Math 142 - Exam 3b
Lippman/Rasmussen – W11
NAME: _________________________
SHOW ALL WORK: Answers without adequate justification may not receive full credit.
Give exact answers wherever possible. Given angle answers in radians unless
otherwise specified.
1. Solve each of the following equations
a. (7pts) Solve for all possible solutions. Use the integer k to show multiple answers.
4 cos 2 ( x ) − 3 = 0
b. (10pts) Solve the following equation for θ , where 0 ≤ θ < 2π
−3sin (θ ) +3 = 2 cos 2 (θ )
WSBCTC
1
c. (7pts) Solve the following equation for θ , where 0 ≤ θ < 2π
sin ( 2θ ) + cos (θ ) = 0
2. (8pts) Find the exact values for the following (using rules from this chapter & unit circle
values)
a. sin(255o )
b. cos(15°) + cos(105°)
WSBCTC
2
3. (7pts) Rewrite
−5sin(15 x) − 4 cos(15 x)
4. (8pts) Given the sin θ =
in the form A sin( Bx + C )
3
Find the following 0o < θ < 90o
5
where 0 ≤ C < 2π
Show work (leave
answers exact)
sin ( 2θ ) = ____________
cos ( 2θ ) =____________
θ 
sin   =____________
2
WSBCTC
3
5. (8pts) Solve the following equation for α where 0 ≤ α < 2π
cos(2α ) = cos(α ) − 1
6. (7pts) Prove the following identity (i.e., establish the identity)
cos (α − β )
= cot β + tan α
cos α sin β
WSBCTC
4
7. (10pts) Simplify the following expression into one non-fractional trig expression
( cos ( t ) + sin ( t ) ) cos ( 2t )
( cos ( t ) − sin ( t ) ) sec ( t )
2
a.
2
2
2
sec2 (θ ) sin ( 2θ )
b.
2 tan (θ )
8.
(10pts) Given F (t ) = 4e
−0.5t
cos(2π t ) + (5 − t ) and
G (t ) = (3t + 1) cos(π t ) + 2
a. The Amplitude of F(t) is..(circle all that apply)
Increasing, Decreasing, Constant, Linear, Exponential
b. The Midline of F(t) is ?..(circle all that apply)
Increasing, Decreasing, Constant, Linear, Exponential
c. The Amplitude of G(t) is ..(circle all that apply)
Increasing, Decreasing, Constant, Linear, Exponential
d. The Midline of G(t) is ..(circle all that apply)
Increasing, Decreasing, Constant, Linear, Exponential
WSBCTC
5
e. Which equation takes longer to complete 1 cycle? (circle one)
F(t) or G(t)
9. (8pts) A string on a bass is plucked, and oscillates up and down 50 times per
second. The amplitude starts out at 0.5cm, and decreases by 15% per second.
Write an equation for the position, P, of the string as a function of time, t. Assume
the string starts at its lowest position.
WSBCTC
6