1. Tardigrade
  2. Question
  3. Mathematics
  4. The equation sin 3 θ=4 sin θ ⋅ sin 2 θ . sin 4 θ in 0 ≤ θ ≤ π has:

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The equation $\sin 3 \theta=4 \sin \theta \cdot \sin 2 \theta . \sin 4 \theta$ in $0 \leq \theta \leq \pi$ has:

Trigonometric Functions

Solution:

$\sin 3 \theta=4 \sin \theta \sin 2 \theta \sin 4 \theta$
$\Rightarrow \sin 3 \theta=(2 \sin \theta)(2 \sin 2 \theta \sin 4 \theta)$
$\Rightarrow 3 \sin \theta-4 \sin ^3 \theta=2 \sin \theta(\cos 2 \theta-\cos 6 \theta)$
$\Rightarrow 3-4 \sin ^2 \theta=2(\cos 2 \theta-\cos 6 \theta)$ or $\sin \theta=0$
$\Rightarrow 3-2(1-\cos 2 \theta)=2 \cos 2 \theta-2 \cos 6 \theta$ or $\sin \theta=0$
$\Rightarrow 1=-2 \cos 6 \theta \Rightarrow \cos 6 \theta=\frac{-1}{2}$ or $\sin \theta=0$
$\therefore \sin \theta=0$ or $\cos 6 \theta=\frac{-1}{2}$
$\Rightarrow \theta= n \pi \text { or } \theta=\frac{2 n \pi \pm\left(\frac{2 \pi}{3}\right)}{6}=\frac{ n \pi}{3} \mp \frac{\pi}{9} $
$\Rightarrow \theta=0, \pi, \frac{\pi}{9}, \frac{\pi}{3} \mp \frac{\pi}{9}, \frac{2 \pi}{3} \mp \frac{\pi}{9}, \pi-\frac{\pi}{9}$
So eight solutions.