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,


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°


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