3D Geometry
Direction Cosines and Angles
Grade 12
Question:
<p>A line makes the same angle θ with X-axis and Z-axis. If the angle β, which it makes with Y-axis, is such that \(\sin 2β = 3 \sin 2θ\), then the value of \(\cos^2 θ\) is</p>
<p>(a) \(\frac{1}{5}\)</p>
<p>(b) \(\frac{2}{5}\)</p>
<p>(c) \(\frac{3}{5}\)</p>
<p>(d) \(\frac{2}{3}\)</p>
Step-by-Step Solution
Key Concept: Use the property that the sum of squares of direction cosines equals 1, and substitute the given relation between angles.
Step 1: Since \(l^2 + m^2 + n^2 = 1\), we have \(\cos^2 θ + \cos^2 β + \cos^2 θ = 1\) Step 2: Substituting \(\sin^2 β = 3 \sin^2 θ\): \(2\cos^2 θ + 1 - 3\sin^2 θ = 1\) Step 3: \(2\cos^2 θ - 3(1 - \cos^2 θ) = 0\) Step 4: \(5\cos^2 θ = 3\) \(∴ \cos^2 θ = \frac{3}{5}\) Answer is (c).
Correct Answer: C