If x cos θ = y cos(θ + 2π/3) = z cos(θ + 4π/3), Find 1/x + 1/y + 1/z
Question:
\[
x\cos\theta
=
y\cos\left(\theta+\frac{2\pi}{3}\right)
=
z\cos\left(\theta+\frac{4\pi}{3}\right)
\]
Find
\[
\frac1x+\frac1y+\frac1z
\]
Solution
\[ x\cos\theta = y\cos\left(\theta+\frac{2\pi}{3}\right) = z\cos\left(\theta+\frac{4\pi}{3}\right) =k \]
\[ x=\frac{k}{\cos\theta} \] \[ y=\frac{k}{\cos\left(\theta+\frac{2\pi}{3}\right)} \] \[ z=\frac{k}{\cos\left(\theta+\frac{4\pi}{3}\right)} \]
\[ \frac1x+\frac1y+\frac1z = \frac1k \left[ \cos\theta + \cos\left(\theta+\frac{2\pi}{3}\right) + \cos\left(\theta+\frac{4\pi}{3}\right) \right] \]
Using \[ \cos A+\cos\left(A+\frac{2\pi}{3}\right)+\cos\left(A+\frac{4\pi}{3}\right)=0 \]
\[ \therefore \frac1x+\frac1y+\frac1z=0 \]
\[ \boxed{0} \]