What is the total length of any interval for which $-1 \leq \tan(x) \leq1$ and $\sin(2x)\geq0.5$?

Question

What is the total length of any interval for which $-1 \leq \tan(x) \leq 1$ and $\sin(2x)\geq0.5?$

We know that $0\leq x\leq \pi$, $\sin(2x)\geq 0.5$ can be rewritten as $\sin(x) \cos(x)\geq \frac{1}{4}$.

Answer 1

Since $x \in [0,\pi]$ the condition $-1\leq \tan \, x \leq 1$ is same as $0 \leq x \leq \frac {\pi} 4$ or $\frac {3\pi} 4 \leq x \leq \pi$. For $0 \leq x \leq \frac {\pi} 4$ the inequality $\sin (2x) \geq 0.5$ is equivalent to $2x \geq \frac {\pi} 6$. Can you handle the case $\frac {3\pi} 4 \leq x \leq \pi$?