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A straight line is equally inclined to the x-axis and the y-axis, making an angle α with each. If its angle θ with the z-axis satisfies sin²θ = 2 sin²α, determine α.

1. π/4
2. π/3
3. π/6
4. π/2

Correct answer: π/4

Solution

Direction cosines (cos a, cos a, cos th) give 2cos^2 a + cos^2 th = 1. Using sin^2 th = 2 sin^2 a gives cos^2 th = 2cos^2 a - 1. Equating: 2cos^2 a - 1 = 1 - 2cos^2 a => cos^2 a = 1/2 => alpha = pi/4.

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