Question

How do you evaluate `cos A = 0.5878`?

Answer 1

If by `0.5878` you mean

`\sqrt{{5-\sqrt{5}}/8}=0.5877852523`

(to ten decimal places), then the answer is `A={3\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(\pi/5)={\sqrt{5}+1}/4`.

Then

`sin({3\pi}/{10})={\sqrt{5}+1}/4`

because `{3\pi}/{10}+\pi/5=\pi/2`.

Then

`cos^2({3\pi}/{10})=1-sin^2({3\pi}/{10})=1-({\sqrt{5}+1}/4)^2={5-\sqrt{5}}/8`.