Question

How do you find `cos(theta)` if `sin(theta)= 3/5` and `90 < theta < 180`?

Answer 1

`costheta=-4/5`

Explanation:

`90<theta<180` implies that we are in the second quadrant, where the cosine is negative and the sine is positive. So, keep in mind, `costheta` will be negative.

Recall the identity

`sin^2theta+cos^2theta=1`

This tells us that

`cos^2theta=1-sin^2theta -> costheta=+-sqrt(1-sin^2theta)`

In our situation, as we are in the second quadrant, `costheta=-sqrt(1-sin^2theta)`

We have `sintheta=3/5, sin^2theta=(3/5)^2=9/25`

So,

`costheta=-sqrt(1-9/25)`
`costheta=-sqrt(25/25-9/25)`

`costheta=-sqrt(16/25)=-4/5`