# How do you find the exact value of cos[arc tan(-2/3)]?

Jun 12, 2015

The exact value is apparently $\frac{3}{\sqrt{13}}$.

However, you'll have to use a calculator to find $\arctan \left(- \frac{2}{3}\right)$. It's an irrational radian value that you'll get to be $- 33.69006752598 \ldots e t {c}^{o}$. Wolfram Alpha simply gives it as:

$- \frac{180}{\pi} \cdot \arctan \left(- \frac{2}{3}\right)$
which doesn't really give a satisfactory exact answer (such as $\frac{2}{\sqrt{2}}$). It's just the conversion of $\arctan \left(- \frac{2}{3}\right)$ to degrees...

If you just take this exact result and take the $\cos$ of it, you'll get $\frac{3}{\sqrt{13}}$, but it probably won't be obvious at all that it's equal to that from just looking at it.

$\sqrt{13} = 3.605551275 \ldots e t c$.

$\frac{3}{\sqrt{13}} \approx 0.83205 \ldots e t c$

Jun 14, 2015

Find $\cos \left(\arctan \left(- \frac{2}{3}\right)\right)$

#### Explanation:

$\tan x = - \frac{2}{3}$

Calculator gives; $x = - 33.69$ and $x = - 33.69 + 180 = 146.31$

Therefor, $\cos x = \pm 0.83$