Solveeit Logo
Login

Question

Mathematics Question on Trigonometric Equations

The possible values of θ∈(0,π)\theta \in \left(0, \pi\right) such that sin(θ\theta ) + sin(4 θsin(4\,\theta ) + sin(7 θ)=0sin(7\,\theta ) = 0 are :

A

π4,5π12,π2,2π3,3π4,8π9\frac{\pi}{4}, \frac{5\pi}{12}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{3\pi}{4}, \frac{8\pi}{9}

B

2π9,π4,π2,2π3,3π4,35π36\frac{2\pi}{9}, \frac{\pi}{4}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{3\pi}{4},\frac{ 35\pi}{36}

C

2π9,π4,π2,2π3,3π4,8π9\frac{2\pi}{9}, \frac{\pi}{4}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{3\pi}{4},\frac{ 8\pi}{9}

D

2π9,π4,4π9,π2,3π4,8π9\frac{2\pi}{9}, \frac{\pi}{4}, \frac{4\pi}{9}, \frac{\pi}{2}, \frac{3\pi}{4},\frac{ 8\pi}{9}

Answer

2π9,π4,4π9,π2,3π4,8π9\frac{2\pi}{9}, \frac{\pi}{4}, \frac{4\pi}{9}, \frac{\pi}{2}, \frac{3\pi}{4},\frac{ 8\pi}{9}

Explanation

Solution

sin4θ+2sin 4 θ cos 3θ=0∵θ, ∈(0,π)sin 4\theta+ 2sin\, 4 \,\theta \,cos \,3\theta = 0\quad\because\quad\theta,\,\in\left(0, \pi\right) sin 4θ(1+2 cos 3 θ)=0sin\, 4\theta\left(1 + 2 \,cos \,3\, \theta\right) = 0 sin 4θ=0sin \,4\theta= 0 \quad orcos 3θ=−12\quad cos \,3\theta= - \frac{1}{2} 4θ=nπ;n∈I4\theta = n\pi ; n\in I \quad or3θ=2nπ±2π3,n∈I\quad3\theta = 2n\pi \pm \frac{2\pi}{3}, n \in I θ=π4,π2,3π4\theta = \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4} \quad or θ=2π9,8π9,4π9\quad\theta = \frac{2\pi}{9}, \frac{8\pi}{9}, \frac{4\pi}{9}