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Find a direction angle using cos^2 a + cos^2 b + cos^2 g = 1

This type of question, asked 4 times: May 2022 S1, 15 May 2023 S2, 19 Apr 2025 S1, 13 May 2026 S1. All 4 questions below, oldest first.

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  1. Question 1

    May 2022 S1

    114
    A line makes the same angle α\alpha with each of the x and y axes. If the angle θ\theta, which it makes with the z-axis, is such that sin⁡2θ=2sin⁡2α\sin^2\theta = 2\sin^2\alpha, then the angle α\alpha is

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  2. Question 2

    15 May 2023 S2

    126
    A line makes 45∘45^\circ angle with positive X-axis and makes equal angles with positive Y-axis and Z-axis respectively, then the sum of the three angles which the line makes with positive X-axis, Y-axis and Z-axis is

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  3. Question 3

    19 Apr 2025 S1

    137
    If the directed line makes an angle 45∘45^{\circ} and 60∘60^{\circ} with the X and Y -axes respectively, then the obtuse angle θ\theta made by the line with the Z -axis is

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  4. Question 4

    13 May 2026 S1

    94
    A line makes the same angle α\alpha with each of the XX and ZZ axes. If the angle β\beta which it makes with the YY-axis is such that sin⁡2β=2sin⁡2α\sin^2\beta = 2\sin^2\alpha, then the value of cos⁡2α\cos^2\alpha is ...

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