Find the number of solutions of the equations : $3^{\cos(x)}=\lvert \sin(x) \rvert$ and $x \in (-2\pi,2\pi)$.
I've tried matching the extreme limits on both sides of the equation, but didn't find anything promising.
$$
3^{\cos(x)} \in\left[\frac{1}{3},3\right]
\\
\lvert \sin(x) \rvert \in [0,1]
$$
How do I solve such equations ?
Best Answer
A graph indicates eight solutions. Four are $\pm \frac {\pi}2$ and $\pm \frac {3\pi}2$ where the value of both expressions is $1$. You can then observe that at $-2\pi,0,2\pi, 3^{\cos (x)} = 3 \gt |\sin (x)|$, and at $-\pi, \pi,\ 3^{\cos (x)} \gt |\sin (x)|=0$