Question

Find the value of each expression using the given information.

Answer 1

Recall that $\cos^2\theta+\sin^2\theta=1$. So

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

Given that both $\cos\theta$ and $\tan\theta$, and knowing that $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$, it follows that we should expect $\sin\theta>0$, so we take the positive root above.

Now

$\sin\theta=\sqrt{1-\left(-\dfrac15\right)^2}=\dfrac{2\sqrt6}5$

Then

$\cot\theta=\dfrac{\cos\theta}{\sin\theta}=\dfrac{-\frac15}{\frac{2\sqrt6}5}=-\dfrac1{2\sqrt6}$