# What does cos(arctan((3pi)/4))  equal?

Mar 11, 2018

$\cos \left(\arctan \left(\frac{3 \pi}{4}\right)\right) = \frac{4}{\sqrt{9 {\pi}^{2} + 16}}$

#### Explanation:

This is a rather curious question, since $\frac{3 \pi}{4}$ looks like an angle and not the result of applying $\tan$ to an angle.

Nevertheless $\frac{3 \pi}{4}$ is a value taken by $\tan$ somewhere in Q1.

Consider a right angled triangle with angle $\theta$ and sides $\text{opposite} = 3 \pi$, $\text{adjacent} = 4$ and $\text{hypotenuse} = \sqrt{9 {\pi}^{2} + 16}$.

Then:

$\tan \theta = \text{opposite"/"adjacent} = \frac{3 \pi}{4}$

$\cos \theta = \text{adjacent"/"hypotenuse} = \frac{4}{\sqrt{9 {\pi}^{2} + 16}}$

Then:

$\cos \left(\arctan \left(\frac{3 \pi}{4}\right)\right) = \cos \theta = \frac{4}{\sqrt{9 {\pi}^{2} + 16}}$