Q. 34.8( 5 Votes )

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

Answer :

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,

cos^{2} α + cos^{2} β + cos^{2} γ = 1

Here,

α = 90° , β = 60° and γ = θ

∴ cos^{2} 90° + cos^{2} 60° + cos^{2} θ = 1

⇒ 0 + (1/2)^{2} + cos^{2} θ = 1

∴ cos^{2} θ = 1 – 1/4 = 3/4

⇒ cos θ = ±√(3/4) = ±(√3)/2

∴ θ = 30° or 150°

But θ is acute. So, θ = 30°

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