Name: Trigonometric Equation to Convert Cosine or Sine
Uploaded: 2019-07-19
Duration: 4 min 11 s
Description: Watch the video to learn how to solve trigonometric equations involving both sine and cosine simultaneously to find the general solution of the given equation.

Question

Find the general solution of the equation:

\sin x + \sin 3 x + \sin 5 x = 0

.

Accepted Answer

Consider the given equation $\sin x + \sin 3 x + \sin 5 x = 0$

$\Rightarrow 2 \sin (\frac{5 x + x}{2}) \cos (\frac{5 x - x}{2}) + \sin 3 x = 0$ $(∵ \sin C + \sin D = 2 \sin (\frac{C + D}{2}) \cos (\frac{C - D}{2}))$

$\Rightarrow 2 \sin 3 x \cos 2 x + \sin 3 x = 0$

$\Rightarrow \sin 3 x [2 \cos 2 x + 1] = 0$

$\Rightarrow \sin 3 x = 0$ or $2 \cos 2 x + 1 = 0$

$\Rightarrow \sin 3 x = \sin 0$ or $\cos 2 x = - \frac{1}{2}$

$\Rightarrow \sin 3 x = \sin 0$ or $\cos 2 x = - \cos \frac{π}{3}$

$\Rightarrow \sin 3 x = \sin 0$ or $\cos 2 x = \cos (π - \frac{π}{3}) = \cos \frac{2 π}{3}$

$\Rightarrow 3 x = n π \pm 0$ or $2 x = 2 m π \pm \frac{2 π}{3}$ where $m, n \in Z$

$\Rightarrow x = \frac{n π}{3}$ or $x = m π \pm \frac{π}{3}$

Thus, the general solution for the given trigonometric equation is $x = \frac{n π}{3}$ or $x = m π \pm \frac{π}{3}$ where $m, n \in Z$ .

Find the general solution of the equation: $\sin x + \sin 3 x + \sin 5 x = 0$ .

Important Questions on Trigonometric Equations

Learn and Prepare for any exam you want !

Find the general solution of the equation: sin⁡x+sin⁡3x+sin⁡5x=0.

Important Questions on Trigonometric Equations

Learn and Prepare for any exam you want !

Find the general solution of the equation: $\sin x + \sin 3 x + \sin 5 x = 0$ .