A line in 3D space makes angles α with the positive x-axis, β with the positive y-axis, and γ with the positive z-axis.
When a line makes angles α, β, γ with the coordinate axes, the values cosα, cosβ, cosγ are called the direction cosines of the line.
The direction cosines satisfy the fundamental property:
cos2α+cos2β+cos2γ=1
This holds because the direction cosines represent components of a unit vector.
Using the trigonometric identity sin2θ+cos2θ=1, we can write:
sin2α=1−cos2α
sin2β=1−cos2β
sin2γ=1−cos2γ
Adding all three equations:
sin2α+sin2β+sin2γ=(1−cos2α)+(1−cos2β)+(1−cos2γ)
=3−(cos2α+cos2β+cos2γ)
=3−1
=2
Therefore, sin2α+sin2β+sin2γ=2.