Question

Find tan theta if cos theta = 1/4 and sin theta <0

Answer 1

`\cos\theta=\frac14` is positive, and we want
`\sin\theta` to be negative, which together means
`\theta` is an angle that lies in the fourth quadrant.

Use the Pythagorean identity to solve for
`\sin\theta`:

`\sin^2\theta+\cos^2\theta=1\implies\sin\theta=-√(1-\cos^2\theta)=-\frac{√(15)}4`

Then by definition of tangent, we have

`\tan\theta=(\sin\theta)/(\cos\theta)=\frac{-\frac{√(15)}4}{\frac14}=-√(15)`