Question

Find value of sintheta+ sin3theta+sin5theta

Answer 1

Correct Question :)

Find the general solution of $\sf Sin( \theta) + Sin( 3\theta) + Sin( 5\theta) = 0$

Solution :)

$\sf \hookrightarrow \{Sin( 5\theta) + Sin( \theta) \} + Sin( 3\theta) = 0 \\ \\ \sf \hookrightarrow 2 Sin( \frac{ \theta + 5 \theta}{2})Cos( \frac{5 \theta - \theta}{2} ) + Sin(3 \theta) = 0\\ \\ \sf \hookrightarrow 2Sin(3 \theta)Cos(2 \theta) +Sin(3 \theta) = 0 \\ \\ \sf \hookrightarrow Sin(3 \theta) \bigg\{2Cos(2 \theta) + 1) \bigg \} = 0$

$\sf \bold{ \underline{For \: Sin(3 \theta) = 0 }}$

$\sf \hookrightarrow 3x = \frac{n}{\pi} \\ \\\sf \hookrightarrow x = \frac{n\pi}{3}$

$\sf \bold{ \underline{For \: 2Cos(2 \theta) + 1) = 0}}$

$\sf \hookrightarrow Cos(2 \theta) = - \frac{1}{2} \\ \\\sf \hookrightarrow Cos(2 \theta) = Cos(\pi - \frac{\pi}{3} ) \\ \\ \sf \hookrightarrow Cos(2 \theta) = Cos( \frac{2\pi}{3} ) \\ \\ \sf \hookrightarrow 2x = 2n\pi± \frac{2\pi}{3} \\ \\ \sf \hookrightarrow x = n\pi± \frac{\pi}{3}$

Formula used :)

$\star \: \: \sf Sin(x) + Sin(y) = 2Sin (\frac{x + y}{2}) Cos( \frac{x - y}{2} ) \\ \\ \star \: \: \sf Sin(x) =0 \: , \: x = n \pi \\ \\ \star \: \: \sf Cos(x) = Cos(y) \: , \: x = 2n \pi ± y$