If $x\cos{ heta = y\cos\left( heta + \frac{2\pi}{3} ight)} = z\cos\left( heta + \frac{4\pi}{3} ight)$, then the value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is equal to

Question

If $x\cos{	heta = y\cos\left( 	heta + \frac{2\pi}{3} ight)} = z\cos\left( 	heta + \frac{4\pi}{3} ight)$, then the value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is equal to

Dr. Abubacker · Accepted Answer

(c)
$x\cos{	heta = y\cos\left( 	heta + \frac{2\pi}{3} ight)}$
$= z\cos\left( 	heta + \frac{4\pi}{3} ight) = k$ (say)
$\Rightarrow \ \cos{	heta = \frac{k}{x}},\cos\left( 	heta + \frac{2\pi}{3} ight) = \frac{k}{y}$
And $\cos\left( 	heta + \frac{4\pi}{3} ight) = \frac{k}{z}$
Hence, $\frac{k}{x} + \frac{k}{y} + \frac{k}{z} = \cos{	heta + \cos{\left( 	heta + \frac{2\pi}{3} ight) + \cos\left( 	heta + \frac{4\pi}{3} ight)}}$
$= \cos{	heta - \cos\left( \frac{\pi}{3} - 	heta ight) - \cos\left( \frac{\pi}{3} + 	heta ight)}$
$= \cos{	heta - 2\cos{\frac{\pi}{3}\cos{	heta = 0}}}$

Q. If $x\cos{\theta = y\cos\left( \theta + \frac{2\pi}{3} \right)} = z\cos\left( \theta + \frac{4\pi}{3} \right)$, then the value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is equal to

Explanation

Reviewed by Dr. Abubacker

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