# How do you evaluate cos^-1(cos((17 pi)/5))?

May 16, 2016

$\frac{17 \pi}{5}$

#### Explanation:

An inverse function might be single-valued or many-valued. But applying an inverse function and the function in succession, over an operand, returns the operand.

${O}^{- 1} O \left(c\right) = {O}^{- 1} \left(O \left(c\right)\right) = c$.

$\cos \left(\frac{17 \pi}{5}\right) = \cos \left(3 \pi + \left(\frac{2 \pi}{5}\right)\right) = - \cos \left(\frac{2 \pi}{5}\right) = - - \cos {72}^{o} = - 0.3090$.

You can see the basic difference between evaluating directly

${\cos}^{- 1} \left(- 0.3090\right) = \frac{3 \pi}{5} \mathmr{and} - \frac{3 \pi}{15} \mathmr{and} \frac{7 \pi}{5} \mathmr{and} - \left(7 \pi\right) 5 , \frac{17 \pi}{5} \mathmr{and} - \frac{17 \pi}{5}$..... and

${\cos}^{- 1} \cos \left(\frac{17 \pi}{5}\right) = {\cos}^{- 1} \left(\cos \left(\frac{17 \pi}{5}\right)\right) = \frac{1 7 \pi}{5}$.

For that matter, if the expression is

${\cos}^{- 1} \left(\cos \left(\frac{3 \pi}{5}\right)\right)$, the value is $\frac{3 \pi}{5}$.

Note that cos (+-(17/5)pi) = cos (+-(7/5)pi) = cos (+-(3/5)pi))= -0.3090

Cosine is negative, in 2nd and 3rd quadrants..