Question

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

Answer 1

`(17pi)/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 (c) =O^(-1) (O (c)) = c`.

`cos((17pi)/5)= cos (3pi + ((2pi)/5))=-cos((2pi)/5)= - -cos 72^o=- 0.3090`.

You can see the basic difference between evaluating directly

`cos^(-1)(-0.3090) = (3pi)/5 or - (3pi)/15 or (7pi)/5 or -(7pi)5, (17pi)/5 or -(17pi)/5`..... and

`cos^(-1)cos ((17pi)/5) = cos^(-1)(cos((17pi)/5)) = (1 7pi)/5`.

For that matter, if the expression is

`cos^(-1)(cos((3pi)/5))`, the value is `(3pi)/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..