Hello, Mike!

ticbol did an excellent job of explaining.

Here are my versions . . .

I'll assume the angle are between and .

Find the exact value of:

We want the angle whose sine is

You are expected to know that:

Sine is negative in quadrants 3 and 4.

So we have a reference angle of in quadrants 3 and 4.

Hence, the angles are: .

Inside, we have: .

Then we have: . . . . the angle whose cosine is

We know that: .

Cosine is negative in quadrants 2 and 3.

Therefore the angles are: .

Inside, we have: . . . . the angle whose sine is

Consider:

We don't know the exact value of this angle,

. . but we know it comes from this triangle: Code:

*
* |
* |
5 * | 4
* |
* θ |
* - - - - - *
3

Since sine is negative in quadrants 3 and 4,

. . the two possible angles look like this: Code:

| |
| |
-3 | | 3
- + - - - + - - - - - - - - - + - - - + -
: θ / | | \ θ :
-4: /5 | | 5\ :-4
: / | | \ :
* | | *

Since

. . the answers are: .