# How do you evaluate arc cos (-.3090)?

Jul 3, 2016

If by $- 0.3090$ we mean $\frac{1 - \setminus \sqrt{5}}{4}$, then the result is $\frac{3 \setminus \pi}{5}$.

#### Explanation:

The number $- 0.3090$ matches $\frac{1 - \setminus \sqrt{5}}{4}$ to all four reported significant digits, and it is assumed here that $\frac{1 - \setminus \sqrt{5}}{4}$ is the intended argument.

$\cos \left(\setminus \frac{\pi}{5}\right) = \frac{\setminus \sqrt{5} + 1}{4}$
$\cos \left(\frac{2 \setminus \pi}{5}\right) = \cos \left(\setminus \frac{\pi}{5}\right) - \left(\frac{1}{2}\right) = \textcolor{b l u e}{\frac{\setminus \sqrt{5} - 1}{4}}$

The blue figure is the negative of the given argument. So:

$\arccos \left(\frac{1 - \setminus \sqrt{5}}{4}\right)$
$= \setminus \pi - \arccos \left(\frac{\setminus \sqrt{5} - 1}{4}\right)$
$= \setminus \pi - \frac{2 \setminus \pi}{5} = \frac{3 \setminus \pi}{5}$.