The number of possible solutions of sin theta | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given equation: $\sin 7 	heta + \sin 	heta + \sin 4 	heta = 0$ Using the sum-to-product formula $\sin C + \sin D = 2 \sin \frac{C+D}{2} \cos \f

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(C) Given equation: $\sin 7 	heta + \sin 	heta + \sin 4 	heta = 0$ Using the sum-to-product formula $\sin C + \sin D = 2 \sin \frac{C+D}{2} \cos \frac{C-D}{2}$: $2 \sin 4 	heta \cos 3 	heta + \sin 4 	heta = 0$ $\sin 4 	heta (2 \cos 3 	heta + 1) = 0$ Case $1$: $\sin 4 	heta = 0 \implies 4 	heta = n \pi \implies 	heta = \frac{n \pi}{4}$. For $	heta \in (0, \pi)$,$n = 1, 2, 3$,so $	heta = \frac{\pi}{4}, \frac{\pi}{2}, \frac{3 \pi}{4}$. Case $2$: $\cos 3 	heta = -\frac{1}{2} = \cos \frac{2 \pi}{3} \implies 3 	heta = 2 n \pi \pm \frac{2 \pi}{3} \implies 	heta = \frac{2 n \pi}{3} \pm \frac{2 \pi}{9}$. For $	heta \in (0, \pi)$: If $n=0$,$	heta = \frac{2 \pi}{9}$ (since $-\frac{2 \pi}{9}$ is outside range). If $n=1$,$	heta = \frac{2 \pi}{3} \pm \frac{2 \pi}{9} = \frac{8 \pi}{9}, \frac{4 \pi}{9}$. If $n=2$,$	heta = \frac{4 \pi}{3} \pm \frac{2 \pi}{9}$ (both outside range). The solutions are $	heta \in \{\frac{\pi}{4}, \frac{\pi}{2}, \frac{3 \pi}{4}, \frac{2 \pi}{9}, \frac{4 \pi}{9}, \frac{8 \pi}{9}\}$. Total number of solutions $= 6$.

The number of possible solutions of $\sin \theta + \sin 4 \theta + \sin 7 \theta = 0$ for $\theta \in (0, \pi)$ is:

Similar Questions

Number of values of $x \in [0, 2\pi]$ satisfying the equation $\cot x - \cos x = 1 - \cot x \cos x$.

The number of solutions of $|\cos x| = \sin x$ in the interval $-4 \pi \leq x \leq 4 \pi$ is:

The number of distinct solutions of $\sec \theta + \tan \theta = \sqrt{3}$ for $0 \leqslant \theta \leqslant 2\pi$ is:

The number of elements in the set $S = \{\theta \in [0, 2\pi] : 3 \cos^4 \theta - 5 \cos^2 \theta - 2 \sin^2 \theta + 2 = 0\}$ is $...........$.

The general solution of $1+\sin ^{2} x=3 \sin x \cdot \cos x$,where $\tan x \neq \frac{1}{2}$,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module