Question

How to find the value of x if `sec x = -5`?

Answer 1

`secx = -5`

We rewrite the secant function and do some algebric manipulation.

`1/cosx = -5`
`1 = -5cosx`
`-1/5 = cosx`
`cosx = -1/5`

We take the arccosine of both sides

`arccos(cosx) = arccos(-1/5)`

And evaluate that on a calculator, it comes out to approximately `101.5º`

The arccosine has a proper range so it outputs every value uniquely (We know it only outputs values on the range of `0 <= theta <= 180º`

But since we know that `cos(x+360) = cosx` we can further rewrite the answer to `S ~= 101.5º + 360nº`

Where `n` is an integer.

And that isn't all done, since there is another angle, this time on the third quadrant with the same cosine, we know that `cos(360-x) = cos(x)` and that `cos(-x) = cos(x)`

Therefore, `cos(360-101.5) = cos(101.5)`, so `cos(258.5) ~= -1/5` too.

Which leaves us
`S ~= 101.5º + 360nº and 258.5º + 360nº` where `n` is an integer.