Quiz 3
Math 1060–5
Friday, September 14, 2012
Key
Directions: Show all work for full credit. Clearly indicate all answers. Simplify all mathematical expressions completely. No calculators are allowed.
1. Consider the angle θ =
11π
.
3
(6 points each)
(a) Find the reference angle, θ0 .
The reference angle is the angle between θ and the horizontal axis, which is θ0 =
4π − 11π
= π3 .
3
(b) Using your answer to (1a), find sin θ and cos θ.
√
11π
π
3
sin
= − sin = −
3
3
2
11π
π
1
cos
= cos =
3
3
2
We know that sine is negative and cosine is positive because θ is in the fourth
quadrant.
3
2. Given an angle θ in Quadrant II such that sin θ = , find the values of the six trigono5
metric functions of θ. (14 points)
Using that sin2 θ + cos2 θ = 1, we can find cosine
cos2 θ = 1 − sin2 θ
2
16
3
9
2
=
cos θ = 1 −
=1−
5
25
25
r
16
4
cos θ = ±
=−
(We need negative because cosine is negative in Quadrant II)
25
5
Using reciprocals and that tan θ = sin θ/ cos θ, we get:
sin θ =
3
5
4
5
3
tan θ = −
4
cos θ = −
csc θ =
5
3
5
4
4
cot θ = −
3
sec θ = −
3. Find the amplitude and period of, and graph each of the following trigonometric functions. Include at least two periods for each graph. (12 points each)
x
+2
(a) y = −3 sin
4
Amplitude = | − 3| = 3
2π
Period =
= 8π
1/4
5
6
1
2π
-
−1
(b) y =
1
cos x (#37 from 4.5)
3
Amplitude =
Period =
1
3
2π
= 2π
1
6
1
−2π −π
1
3
− 13
−1
π
2π-