If a line makes angles $\frac{2\pi}{3}$ and $\frac{\pi}{3}$ with positive direction of x-axis any y-axis respectively, then the acute angle made by the line with positive z-axis is :

$\frac{\pi}{4}$

The correct answer is Option (2) → $\frac{\pi}{4}$

let angles made with $x, y, z$ be $α,β,γ$

$\cos^α+\cos^2β+\cos^2γ=1$

so $\cos^2\frac{2π}{3}+\cos^2\frac{π}{3}+\cos^2γ=1$

$\frac{1}{4}+\frac{1}{4}+\cos^2γ=1$

so $\cos^2γ=\frac{1}{2}⇒\cos γ=\frac{1}{\sqrt{2}}$

$γ$ is acute

so $γ=\frac{\pi}{4}$