If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.[CBSE 2015]

Key point: This problem is based on the application of direction cosines.

We know that if α, β, γ are the angles that a line makes with x , y and z axis respectively then cos α , cos β and cos γ are called its direction cosines and,

cos² α + cos² β + cos² γ = 1

Here,

α = 90° , β = 60° and γ = θ

∴ cos² 90° + cos² 60° + cos² θ = 1

⇒ 0 + (1/2)² + cos² θ = 1

∴ cos² θ = 1 – 1/4 = 3/4

⇒ cos θ = ±√(3/4) = ±(√3)/2

∴ θ = 30° or 150°

But θ is acute. So, θ = 30°