Question

How do you find the exact value of `cos[arc tan(-2/3)]`?

Answer 1

The exact value is apparently `3/sqrt(13)`.

However, you'll have to use a calculator to find `arctan(-2/3)`. It's an irrational radian value that you'll get to be `-33.69006752598...etc^o`. Wolfram Alpha simply gives it as:

`-180/pi * arctan(-2/3)`
which doesn't really give a satisfactory exact answer (such as `2/sqrt2`). It's just the conversion of `arctan(-2/3)` to degrees...

If you just take this exact result and take the `cos` of it, you'll get `3/sqrt(13)`, but it probably won't be obvious at all that it's equal to that from just looking at it.

`sqrt13 = 3.605551275...etc`.

`3/sqrt13 ~~ 0.83205...etc`

Answer 2

Find `cos (arctan (-2/3))`

Explanation:

`tan x = -2/3`

Calculator gives; `x = -33.69` and `x = -33.69 + 180 = 146.31`

Therefor, `cos x = +- 0.83`