Question

see screenshot below

Answer 1

Ah good old trigonometry. I'm sure you're familiar with the Unit Circle? It's this thing right here.

Answer 2

Before we examine this in detail, I'd like to review the meaning behind sin, cos, and tan, with the device "SOH CAH TOA". In a right triangle, sin is calculated by the length Opposite the angle/the length of the Hypotenuse; cos by the length Adjacent to the angle/the length of the Hypotenuse; tan by the length Opposite the angle/the length Adjacent to the angle

Answer 3

Now, we can relate this to the right triangles that appear in the Unit Circle. Using cos(330) as an example, we see that the corresponding coordinates (sqr(3)/2, -1/2) can be used as the Adjacent and Opposite lengths, respectively. This being the Unit Circle, the Hypotenuse is 1.
So, cos in this case would be (sqr(3)/2)/1 or simply sqr(3)/2

Answer 4

this is all good but it's just double angle/half angle formulas

Answer 5

I just googled this and Froppy is correct. Her's is a simpler solution.

Answer 6

if tan(157.5) = tan(x) then x = 315

then tan(157.5) = [1 - cos(315) ]/sin(315) = choice c

Answer 7

you can do something similar with sin(165)

Answer 8

sin(x/2) = +- sqrt(1-cos(2x)/2))

Answer 9

for the last one cos(2x) = 1 - sin^2(x)

Answer 10

then we use process of elimination to find sin^2(157.5)

Answer 11

hope this helps

Answer 12

going to bed