Question

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

Answer 1

If by `-0.3090` we mean `(1-\sqrt{5})/4`, then the result is `(3\pi)/5`.

Explanation:

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

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

`cos(\pi/5)=(\sqrt{5}+1)/4`
`cos({2\pi}/5)=cos(\pi/5)-(1/2)=color(blue)((\sqrt{5}-1)/4)`

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

`arccos((1-\sqrt{5})/4)`
`=\pi-arccos((\sqrt{5}-1)/4)`
`=\pi-{2\pi}/5={3\pi}/5`.