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

Sep 17, 2015

$\sec x = - 5$

We rewrite the secant function and do some algebric manipulation.

$\frac{1}{\cos} x = - 5$
$1 = - 5 \cos x$
$- \frac{1}{5} = \cos x$
$\cos x = - \frac{1}{5}$

We take the arccosine of both sides

$\arccos \left(\cos x\right) = \arccos \left(- \frac{1}{5}\right)$

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 $\cos \left(x + 360\right) = \cos x$ 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 $\cos \left(360 - x\right) = \cos \left(x\right)$ and that $\cos \left(- x\right) = \cos \left(x\right)$

Therefore, $\cos \left(360 - 101.5\right) = \cos \left(101.5\right)$, so $\cos \left(258.5\right) \cong - \frac{1}{5}$ too.

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