CUET UG 2025 — Mathematics Arithmetic

A line in 3D space makes angles $\alpha$α with the positive x-axis, $\beta$β with the positive y-axis, and $\gamma$γ with the positive z-axis.

When a line makes angles $\alpha$α, $\beta$β, $\gamma$γ with the coordinate axes, the values $\cos \alpha$cosα, $\cos \beta$cosβ, $\cos \gamma$cosγ are called the direction cosines of the line.

The direction cosines satisfy the fundamental property:

$$\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$$cos2α+cos2β+cos2γ=1

This holds because the direction cosines represent components of a unit vector.

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Using the trigonometric identity $\sin^2\theta + \cos^2\theta = 1$sin2θ+cos2θ=1, we can write:

$$\sin^2\alpha = 1 - \cos^2\alpha$$sin2α=1−cos2α

$$\sin^2\beta = 1 - \cos^2\beta$$sin2β=1−cos2β

$$\sin^2\gamma = 1 - \cos^2\gamma$$sin2γ=1−cos2γ

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Adding all three equations:

$$\sin^2\alpha + \sin^2\beta + \sin^2\gamma = (1 - \cos^2\alpha) + (1 - \cos^2\beta) + (1 - \cos^2\gamma)$$sin2α+sin2β+sin2γ=(1−cos2α)+(1−cos2β)+(1−cos2γ)

$$= 3 - (\cos^2\alpha + \cos^2\beta + \cos^2\gamma)$$=3−(cos2α+cos2β+cos2γ)

$$= 3 - 1$$=3−1

$$= 2$$=2

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Therefore, $\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 2$sin2α+sin2β+sin2γ=2.