If sin theta + sin 2theta + sin 3theta = sin | vedclass

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

Question

(A) Given equations are: $\sin 	heta + \sin 2	heta + \sin 3	heta = \sin \alpha$ ... $(i)$ $\cos 	heta + \cos 2	heta + \cos 3	heta = \cos \alpha$

Vedclass · Accepted Answer

$\alpha / 2$

Vedclass · Answer

(A) Given equations are: $\sin 	heta + \sin 2	heta + \sin 3	heta = \sin \alpha$ ... $(i)$ $\cos 	heta + \cos 2	heta + \cos 3	heta = \cos \alpha$ ... $(ii)$ Using the sum-to-product formula $\sin A + \sin B = 2 \sin(\frac{A+B}{2}) \cos(\frac{A-B}{2})$ in $(i)$: $(\sin 3	heta + \sin 	heta) + \sin 2	heta = \sin \alpha$ $2 \sin 2	heta \cos 	heta + \sin 2	heta = \sin \alpha$ $\sin 2	heta (2 \cos 	heta + 1) = \sin \alpha$ ... $(iii)$ Using the sum-to-product formula $\cos A + \cos B = 2 \cos(\frac{A+B}{2}) \cos(\frac{A-B}{2})$ in $(ii)$: $(\cos 3	heta + \cos 	heta) + \cos 2	heta = \cos \alpha$ $2 \cos 2	heta \cos 	heta + \cos 2	heta = \cos \alpha$ $\cos 2	heta (2 \cos 	heta + 1) = \cos \alpha$ ... $(iv)$ Dividing $(iii)$ by $(iv)$: $\frac{\sin 2	heta (2 \cos 	heta + 1)}{\cos 2	heta (2 \cos 	heta + 1)} = \frac{\sin \alpha}{\cos \alpha}$ $	an 2	heta = 	an \alpha$ $2	heta = \alpha$ $	heta = \alpha / 2$

If $\sin \theta + \sin 2\theta + \sin 3\theta = \sin \alpha$ and $\cos \theta + \cos 2\theta + \cos 3\theta = \cos \alpha$,then $\theta$ is equal to

Similar Questions

For $\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$,if $2 \cos \theta + \sin \theta = 1$ and $7 \cos \theta + 6 \sin \theta = k$,then the possible values of $k$ are:

Let $\alpha = \frac{1}{\sin 60^{\circ} \sin 61^{\circ}} + \frac{1}{\sin 62^{\circ} \sin 63^{\circ}} + \dots + \frac{1}{\sin 118^{\circ} \sin 119^{\circ}}$. Then the value of $\left(\frac{\operatorname{cosec} 1^{\circ}}{\alpha}\right)^2$ is $....$

If $\cos \theta - 4 \sin \theta = 1$,then $\sin \theta + 4 \cos \theta$ is equal to

If $p = \frac{2\sin \theta}{1 + \cos \theta + \sin \theta}$ and $q = \frac{\cos \theta}{1 + \sin \theta}$,then

If $\frac{3\pi}{4} < \alpha < \pi,$ then $\sqrt{\csc^2 \alpha + 2\cot \alpha}$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module