Use the Pythagorean Theorem to find tan ɵ, given in Quadrant III.

Make a sketch.

x = ?

y = -77

r = 85

Use ol' Pythagoras to find x.

But since we're in Quadrant III, x = -36.

So . Knocked it outta the park.

If sin ɵ = -½ in Quadrant IV, what is tan ɵ?

Make a sketch.

x = ?

y = -1

r = 2

A little Pythagorean Theorizing gives us the value of our missing side, x.

So .

Now rationalize the denominator:

If  in Quadrant II, what is sin ɵ?

Make a sketch.

x = -60

y = ?

r = 109

Use Pythagoras to find y.

Since we're in Quadrant II, y stays positive. Now grab the opposite side of our triangle over the hypotenuse to find the sine.

Use the Pythagorean identity to find sin ɵ, given cos ɵ = -0.4.

First let's scrawl the Pythagorean identity down for safekeeping:

sin² ɵ + cos² ɵ = 1

Now all we've gotta do is plug in what we know and solve for what we don't know.

sin² ɵ + (-0.4)² = 1

sin² ɵ + 0.16 = 1

Subtract 0.16 from both sides.

sin² ɵ = 1 – 0.16

sin² ɵ = 0.84

That's the sine squared, so snag the square root of both sides.

sin ɵ ≈ 0.917

Use a sum or difference identity to find the exact value of sin(105°).

We don't have any special triangles with 105°, so let's split it up into a sum of two angles that are more familiar to us.

105° = 60° + 45°

Yeah, that'll work. Now grab the sine sum identity and get down to business.

sin(α + β ) = sin(α)cos(β) + cos(α)sin(β)

sin(105°) = sin(60° + 45°)= sin(60°)cos(45°) + cos(60°)sin(45°)

Use a sum or difference identity to find the exact value of cos(120°).

How's about we split that ugly 120° angle into something easier to work with?

120° = 90° + 30°

Perfect. Let's use the cosine sum identity.

cos(α + β) = cos(α)cos(β) – sin(α)sin(β)

Use a sum or difference identity to find the exact value of cos(15°).

We'll need to split this guy up into a difference of two easier angles.

15° = 45° – 30°

Awesomesauce. Bust out the cosine difference identity.

cos(α – β) = cos(α)cos(β) + sin(α)sin(β)

Use a sum or difference identity to find tan(165°).

First, rewrite 165° as a sum or difference.

165° = 120° + 45°

Which means:

We need to clean up our denominator, so multiply by the conjugate over the conjugate:

And our answer is: