If x sin theta = y sin left( theta + frac{2pi | vedclass

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

Question

(B) Let $x \sin 	heta = y \sin \left( 	heta + \frac{2\pi}{3} ight) = z \sin \left( 	heta + \frac{4\pi}{3} ight) = k$.Then $x = \frac{k}{\sin

Vedclass · Accepted Answer

$xy + yz + zx = 0$

Vedclass · Answer

(B) Let $x \sin 	heta = y \sin \left( 	heta + \frac{2\pi}{3} ight) = z \sin \left( 	heta + \frac{4\pi}{3} ight) = k$.Then $x = \frac{k}{\sin 	heta}$,$y = \frac{k}{\sin \left( 	heta + \frac{2\pi}{3} ight)}$,and $z = \frac{k}{\sin \left( 	heta + \frac{4\pi}{3} ight)}$.Consider the sum $xy + yz + zx = k^2 \left[ \frac{1}{\sin 	heta \sin \left( 	heta + \frac{2\pi}{3} ight)} + \frac{1}{\sin \left( 	heta + \frac{2\pi}{3} ight) \sin \left( 	heta + \frac{4\pi}{3} ight)} + \frac{1}{\sin \left( 	heta + \frac{4\pi}{3} ight) \sin 	heta} ight]$.Using the identity $\sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)]$,the denominator terms simplify.Alternatively,using the property that $\sin 	heta + \sin \left( 	heta + \frac{2\pi}{3} ight) + \sin \left( 	heta + \frac{4\pi}{3} ight) = 0$,we can show that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0$.Multiplying by $xyz$,we get $yz + zx + xy = 0$.

If $x \sin \theta = y \sin \left( \theta + \frac{2\pi}{3} \right) = z \sin \left( \theta + \frac{4\pi}{3} \right)$,then:

Similar Questions

If $\sin \beta$ is the geometric mean between $\sin \alpha$ and $\cos \alpha,$ then $\cos 2\beta$ is equal to

If $\sin \alpha = \frac{336}{625}$ and $450^\circ < \alpha < 540^\circ$,then $\sin \left( \frac{\alpha}{4} \right) = $

If $\sec \theta = x + \frac{1}{4x}$ where $0^{\circ} < \theta < 90^{\circ}$,then $\sec \theta + \tan \theta$ is equal to:

If $\theta$ is an acute angle and $\sin\theta = \frac{p - 6}{8 - p}$,then $p$ must satisfy:

The value of $x$ for the maximum value of $(\sqrt{3} \sin x + \cos x)$ is .....$^o$

Vedclass Test Series

Exam Paper Generator

Online Exam Module