# How do you calculate cos(tan^-1 (-12 / 5))?

May 15, 2015

If you have a graphing calculator, you can simply punch in the equation.
I am using the TI-84 Plus with it in 'degree mode'

When I put in
The equation listed above I got the answer of .3846153846
Which would be rounded to .385 or .38 degrees.

May 15, 2015

A triangle with sides $5$, $12$ and $13$ is a right angled triangle, since ${5}^{2} + {12}^{2} = 25 + 144 = 169 = {13}^{2}$.

Let us use $\theta$ to denote the (second largest) angle between the side of length $5$ and the hypotenuse - which has length $13$.

Then $\tan \theta = \frac{12}{5}$ and $\cos \theta = \frac{12}{13}$

$\tan \left(- \theta\right) = - \tan \theta = - \frac{12}{5}$

So ${\tan}^{- 1} \left(- \frac{12}{5}\right) = - \theta$

and $\cos \left({\tan}^{- 1} \left(- \frac{12}{5}\right)\right) = \cos \left(- \theta\right) = \cos \left(\theta\right) = \frac{12}{13}$