Question

What is cos^-1(5/13)?

Answer 1

`cos^(-1)(5/13) = 1.176005 + npi, AAn in ZZ`

Explanation:

`cos^(-1)(5/13) = theta`
`rArr` `color(white)("XXXX")` `cos(theta) =5/13`

This ratio gives us a Pythagorean triangle
with hypotenuse `=13`
`color(white)("XX")`adjacent side `=5`
`color(white)("XX")`opposite side `=12`

Unfortunately this is not one of the standard triangles with a simple angle.
The only ways to solve it are by using a calculator, or trig tables, or an infinite series approximation.
This gives `theta = 1.176005` radians

Note that the same `5:13` triangular ratios also exist with a congruent reference triangle in Quadrant III.
Therefore adding `pi` (and any integer multiple of `pi`) will also give a valid value for `theta`.