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√(1-\sin^2\theta)`

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

`\cos\theta=-√(1-\left(\frac34\right)^2)=-\frac{\sqrt7}4`

Now,

`\csc\theta=\frac1{\sin\theta}=\frac43`

`\cot\theta=(\cos\theta)/(\sin\theta)=-\frac{\sqrt7}3`