Question

if sin theta = 3/4 and cos theta is negative find csc theta and cot theta

Answer 1

Recall that

$\cos^2\theta+\sin^2\theta=1\implies\cos\theta=\pm\sqrt{1-\sin^2\theta}$

But we're specifically told that cosine is negative, so we take the negative root above, and then

$\cos\theta=-\sqrt{1-\left(\dfrac34\right)^2}=-\dfrac{\sqrt7}4$

Now,

$\csc\theta=\dfrac1{\sin\theta}=\dfrac43$

$\cot\theta=\dfrac{\cos\theta}{\sin\theta}=-\dfrac{\sqrt7}3$