Find where sec(1+|cos x|) changes sign.

Question

I'm stuck on a question about finding where $\sec{(1+|\cos{x}|)}$ changes sign on the interval from $(-\infty,\infty)$.

Here's what I came up with:

$1+|\cos{x}|=\frac{\pi}{2}+\pi k$, where $k$ is an integer.

$|\cos{x}|=\frac{\pi}{2}-1+\pi k$

$\cos{x}=\pm(\frac{\pi}{2}-1+\pi k$)

$x=\arccos{(\frac{\pi}{2}-1+\pi (0))}\approx0.963+2\pi k$

$x=\arccos{(-\frac{\pi}{2}+1-\pi (0))}\approx2.178+2\pi k$

But the actual answer seems to be $0.963+\pi k$ and $2.178+\pi k$.

I'm not sure how to come up with this, since I thought the rule was $\arccos{\theta}=\arccos{\theta}+2\pi k$.

Thanks!

Answer 1

I think I figured it out:

Since $\cos{\theta}=\cos{(-\theta)}$ the actual rule is $\arccos{\theta}=\pm\arccos{\theta}+2\pi k$.

Therefore, my answers become

$0.963+2\pi k$

$-0.963+2\pi k$

$2.178+2\pi k$

$-2.178+2\pi k$

which simplify to $0.963+\pi k$ and $2.178+\pi k$.