When Hyperbolic function is zero?

Question

If we have the determinant of matrix $\sinh x \cosh x=0$

Then $\sinh(x)=0$ or $\cosh(x)=0$

If $\sinh(x)=0$, then $x=0, \pi, 2\pi, 3\pi$

And $\cosh(x)=0$ then $x=\pi/2, 3\pi/2, 5\pi/2$

Is that correct or not? How can we find the value of $x$ ?

Answer 1

The hyperbolic functions are quite different from the circular ones. For one thing, they are not periodic.

For your equation, the double-"angle" formula can be used:

$\sinh x \cosh x = 0$

$\frac 12 \sinh 2x = 0$

$\sinh 2x = 0$

The only solution to that is $2x = 0 \implies x = 0$.

Alternatively, you can simply observe that $\cosh x$ is always non-zero, and the only solution comes from $\sinh x = 0$.

Updated: in the complex numbers, $2x = k\pi i \implies x = \frac 12 k\pi i, k \in \mathbb{Z}$. See my comment below.