if pi/2 is less than or equal to theta is less than or equal to pi and sin of theta equals 5/9 find cos theta and cot theta

Answer 1

Sketching this would result in this:

Notice that the angle I've marked as θ (theta) is actually the reference angle but that is the most useful when you're dealing with this kind of problem (from my perspective).

We can make this sketch because we know it is quadrant 2 (greater then π/2 and less than π) and because sin(θ) = opposite/hypotenuse making the opposite side 5 and the hypotenuse 9.

You can solve for "x" using the Pythagorean Theorem. 5² + x² = 9² or x = √(81 - 25) = 2√14 but this is in Q2 so it is actually -2√14

Then, cos(θ) = adjacent/hypotenuse = -2√14/9

And cot(θ) = adjacent/opposite = -2√14/5

Answer 2

cos² θ + sin²θ = 1

tan θ = sin θ / cos θ