Question

How do you evaluate `cos(tan^-1(3/7))` without a calculator?

Answer 1

`costheta=+-7/sqrt58` depending on
whether `theta` is in `I` quadrant or `III` quadrant.

Explanation:

Let `tan^(-1)(3/7)=theta`, then `tantheta=3/7`

Note that we are seeking `cos(tan^(-1)(3/7))` ie. `costheta`

As `tantheta=3/7`,

`sec^2theta=1+(3/7)^2=1+9/49=58/49`

Hence `sectheta=+-sqrt58/7`

and `costheta=+-7/sqrt58` depending on whether `theta` is in `I` quadrant or `III` quadrant.