Question

How do you find the exact value of `tan(theta) = -4`?

Answer 1

`theta = {(-1.32586 +-npi " radians"),("or"),(-75.9638^@ +- n*180^@):}`
`AAn in ZZ`

Explanation:

`tan(theta)=-4` is not descriptive of any of the standard triangles so the best we can do is use a calculator:
`theta = arctan(tan(theta)) = arctan(-4) = -1.32582` (radians)

Since `tan(x) = tan(x+pi)`
adding any integer multiple of `pi` to `theta` will give the same value for `tan(theta)` and are therefore also valid solutions.

`-1.32582 " radians" = -75.9638^@`

Note that these values are not "exact" (as requested in the question) but they are quite accurate and the best I can see how to determine.