Find

Where is cosine positive?

Cosine is positive in Quadrants I and IV, but inverse cosine is bounded by [0°, 180°]. We must be in Quadrant I. Remember your reference triangles. What angle gives us the cosine value we want?

ɵ = 45°

Find .

We're hunting down the angle that'll give us this value when we take the sine. So our first question is, where is sine negative?

Sine is negative in Quadrants III and IV, but inverse sine is bounded by . We must be chillin' in Quadrant IV.

Reference triangle time.

Here, ρ = 45°. That's just our reference angle, though.

In Quadrant IV, ɵ = 360° – 45°.

That means our actual angle is ɵ = 315°, or -45°.

Find

Where is our buddy tangent positive?

Tangent is positive in Quadrants I and III, but inverse tangent is bounded by . So the angle we want is hiding out in Quadrant I somewhere.

Remember your reference triangles.

Boom:

ɵ = 60°

Find sin(tan^-1(-1)).

Don't freak out. This problem isn't as gnarly as it looks. What we're really asking is, "What's the sine of the angle whose tangent is -1?" All we've gotta do is solve this from the inside out: first find the angle whose tangent is -1, then grab the sine of that.

Where is tangent negative?

Tangent is negative in Quadrants II and IV, but inverse tangent is bounded by . So we're in Quadrant IV.

You know the drill. Bust out a reference triangle in the fourth quadrant.

Here, ρ = 45°, so our angle is -45° or 360° – 45° = 315°. However you wanna slice it. We'll go with ɵ = -45°.

Now we want the sine of that angle.

By ASTC, we know that sine is negative in Quadrant IV, so