If x cos = y cos [ - ( 2 3 ) ] = z cos [ + ( 2 3 ) ] , then x + y + z is equal to

Question

If $\frac{x}{\cos θ} = \frac{y}{\cos [θ - (\frac{2 π}{3})]} = \frac{z}{\cos [θ + (\frac{2 π}{3})]},$ then $x + y + z$ is equal to

Smart Achievers · Accepted Answer

(0)

We have $\frac{x}{\cos θ} = \frac{y}{\cos [θ - (\frac{2 π}{3})]} = \frac{z}{\cos [θ + (\frac{2 π}{3})]}$

Therefore, each ratio is equal to

$\frac{x + y + z}{\cos θ + \cos [θ - (\frac{2 π}{3})] + \cos [θ + (\frac{2 π}{3})]}$

$= \frac{x + y + z}{\cos θ + 2 \cos θ \cos (\frac{2 π}{3})}$

$= \frac{x + y + z}{0}$

$\Rightarrow x + y + z = 0$

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