<p>A line makes the same angle \(\theta\) with each of the \(x\) and \(z\) axis. If the angle \(\beta\), which it makes with \(y\)-axis, is such that \(\sin^2\beta = 3\sin^2\theta\), then \(\cos^2\theta\) equals</p>
Step-by-Step Solution
Key Concept: Use the fundamental constraint that for direction cosines: cos²α + cos²β + cos²γ = 1, where α, β, γ are angles with x, y, z axes respectively. Since the line makes angle θ with both x and z axes, we have cos²θ + cos²β + cos²θ = 1.
Step 1: Let the line make angles α = θ, β = β, and γ = θ with the x, y, and z axes respectively. Step 2: Using the fundamental property of direction cosines: cos^2α + cos^2β + cos^2γ = 1 ⟹ cos^2θ + cos^2β + cos^2θ = 1 ⟹ 2cos^2θ + cos^2β = 1 ... (i) Step 3: Given: sin^2β = 3sin^2θ Therefore: cos^2β = 1 - sin^2β = 1 - 3sin^2θ Also: sin^2θ = 1 - cos^2θ ⟹ cos^2β = 1 - 3(1 - cos^2θ) = 1 - 3 + 3cos^2θ = 3cos^2θ - 2 ... (ii) Step 4: Substitute equation (ii) into equation (i): 2cos^2θ + (3cos^2θ - 2) = 1 5cos^2θ - 2 = 1 5cos^2θ = 3 cos^2θ = 3/5 ∴ Answer: D (cos^2θ = 3/5)
Correct Answer: D