Question

Find the value of each expression using the given information.

Cotθ and sinθ; cosθ= -1/5, tanθ<0

Answer 1

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

`\sin\theta=\pm√(1-\cos^2\theta)`

Given that both
`\cos\theta<0` and
`\tan\theta<0`, and knowing that
`\tan\theta=(\sin\theta)/(\cos\theta)`, it follows that we should expect
`\sin\theta>0`, so we take the positive root above.

Now

`\sin\theta=√(1-\left(-\frac15\right)^2)=\frac{2\sqrt6}5`

Then

`\cot\theta=(\cos\theta)/(\sin\theta)=\frac{-\frac15}{\frac{2\sqrt6}5}=-\frac1{2\sqrt6}`