# How do you evaluate cos A = 0.5878?

Oct 2, 2016

If by $0.5878$ you mean

$\setminus \sqrt{\frac{5 - \setminus \sqrt{5}}{8}} = 0.5877852523$

(to ten decimal places), then the answer is $A = \frac{3 \setminus \pi}{10}$ radians or 54°.

#### Explanation:

In this answer https://socratic.org/questions/how-do-i-evaluate-cos-pi-5-without-using-a-calculator#225722 it is proved that

$\cos \left(\setminus \frac{\pi}{5}\right) = \frac{\setminus \sqrt{5} + 1}{4}$.

Then

$\sin \left(\frac{3 \setminus \pi}{10}\right) = \frac{\setminus \sqrt{5} + 1}{4}$

because $\frac{3 \setminus \pi}{10} + \setminus \frac{\pi}{5} = \setminus \frac{\pi}{2}$.

Then

${\cos}^{2} \left(\frac{3 \setminus \pi}{10}\right) = 1 - {\sin}^{2} \left(\frac{3 \setminus \pi}{10}\right) = 1 - {\left(\frac{\setminus \sqrt{5} + 1}{4}\right)}^{2} = \frac{5 - \setminus \sqrt{5}}{8}$.