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Common solution of two trig equations

This type of question, asked 4 times: 11 May 2023 S1, 25 Apr 2025 S2, 20 Apr 2026 S1, 15 May 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    11 May 2023 S1

    135
    If angle θ\theta in [0,2π][0, 2\pi] satisfies both the equations cot⁡θ=3\cot\theta = \sqrt{3} and 3sec⁡θ+2=0\sqrt{3}\sec\theta + 2 = 0, then θ\theta is equal to

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  2. Question 2

    25 Apr 2025 S2

    107
    The common principal solution of the equations sin⁡θ=−12\sin\theta= -\frac{1}{2} and tan⁡θ=13\tan\theta=\frac{1}{\sqrt{3}} is

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  3. Question 3

    20 Apr 2026 S1

    57
    The number of common solutions of the pair of equations 2sin⁡2θ−2cos⁡2θ+1=02\sin^2\theta - 2\cos^2\theta + 1 = 0 and 2sin⁡2θ+3sin⁡θ−2=02\sin^2\theta + 3\sin\theta - 2 = 0 in the interval [0,2π][0, 2\pi], is

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  4. Question 4

    15 May 2026 S2

    52
    If α\alpha and β\beta are the solutions of 2cos⁡θ=12\cos\theta = 1 in the interval [0,2π)[0, 2\pi), then the solutions of 2cos⁡θ+cos⁡(α+β)=02\cos\theta + \cos(\alpha + \beta) = 0 in the interval [0,2π)[0, 2\pi) are...

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