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SELECTED SOLUTIONS TO HOMEWORK
pg. 56, #48: Use the half-angle formula:
cos 2 π 12
= cos 2
( (^) π 6 2
1 + cos π 6 2
2
√ 3 2 2
√ 3 2 2
Another approach would use the difference formula:
cos 2 π 12
= cos 2
π
4
π
6
cos
π
4
cos
π
6
π
4
sin
π
6
Notice we get the same answer either way.
pg. 56 #8: We know tan x = 2 = 2
- We can view the^ opposite^ side as being 2, and the^ adjacent side as being 1. To find the sine and cosine, we need to know the radius. Applying the
Pythagorean theorem,
radius 2 = opposite 2
r 2 = 2 2
r =
Thus sin x = 2
5 and cos x = 1/
pg. 56 #32: Applying the sum formula, we have
cos
x +
π
2
= cos x cos
π
2
− sin x sin
π
2 = cos x · 0 − sin x · 1
= − sin x.
pg. 56 #16: We want to graph y = cos (πx/2). Since B = π/ 2 (B is the the coefficient of x!)
we know the period is
2 π π 2
= 2π ·
π
The graph of the cosine is the same as the graph of the sine, only shifted to the left π/ 2 ,
like so:
x
−
−1.57 1.57 3.14 4.71 6.
1
This problem’s function has no shifts, so its graph will look exactly the same as the graph
of the cosine, but for the fact that its period is different:
1
0
−1 1 2 3
x
