Question

What are the six trig function values of `5pi`?

Answer 1

`sin(5pi) = 0`
`cos(5pi) = -1`
`tan(5pi) = 0`
`sec(5pi) = -1`
`csc(5pi)` is `"undefined"`
`cot(5pi)` is `"undefined"`

Explanation:

First, let's note that sine and cosine are periodic with period `2pi`, meaning for any integer `k`,
`sin(x) = sin(x+k2pi)` and `cos(x) = cos(x+k2pi)`
With this, we get the first two desired values:

`sin(5pi) = sin(pi + 2(2pi)) = sin(pi) = 0`
and
`cos(5pi) = cos(pi+2(2pi)) = cos(pi) = -1`

The remaining trig functions are just functions of sine and cosine.

`tan(5pi) = sin(5pi)/cos(5pi) = 0/(-1) = 0`

`sec(5pi) = 1/cos(5pi) = 1/(-1) = -1`

`csc(5pi) = 1/sin(5pi) = 1/0` is `"undefined"`

`cot(5pi) = cos(5pi)/sin(5pi) = (-1)/0` is `"undefined"`