Question

How do you find the exact value of `cos[arctan(-5/12)]`?

Answer 1

Notice that 5 and 12 are two sides of a right angled triangle whose hypotenuse is 13, since `5^2 + 12^2 = 25 + 144 = 169 = 13^2`

So if `theta` is the smallest angle in the `5`,`12`,`13` triangle then

`sin theta = 5/13`, `cos theta = 12/13` and `tan theta = 5/12`.

Then `tan (-theta) = -tan(theta) = -5/12`

So we are looking for `cos (-theta) = cos(theta) = 12/13`