The number of solutions of sin x + 3x + 5x = 0 in the interval [...

\left(\text{sin x} + \text{\sin 5x}\right) + \text{\sin 3x} = 0$<br />$\text{2 \sin 3x \cos 2x} + \text{\sin 3x} = 0$<br />$\text{\sin 3x} \left(\text{2 \cos 2x} + 1\right) = 0$<br />$\text{\sin 3x} = 0$and$\text{2 \cos 2x} + 1 = 0$<br />$\sin 3x = 0$<br />Graph f(x) = \sin 3x<br><img alt="" src="/api/proxy-image?url=https$x = \dfrac{2 \pi}{3}$or$x = \pi$or$x = \dfrac{4 \pi}{3}\text{,}$<br>$∴ \text\sin 3 x = 0$have 3 solutions<br />Graph f(x) = \cos 2x<br><img alt="" src="/api/proxy-image?url=https$\text\cos 2 x = - \dfrac{1}{2}$<br>$∴ x = \dfrac{2\pi}{3}and\dfrac{4\pi}{3}$<br>So the solution of given equation in$[ \dfrac{\pi}{2}, \dfrac{3\pi}{2} ]$are$\dfrac{2\pi}{3}and\dfrac{4\pi}{3}