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Roots of a cos x + b sin x = c: use tan of the half-angle

This type of question, asked 2 times: 15 May 2023 S1, 14 May 2024 S2. All 2 questions below, oldest first.

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  1. Question 1

    15 May 2023 S1

    137
    Let a,b,ca, b, c be three non-zero real numbers such that the equation 3 acos⁡x+2bsin⁡x=c\sqrt{3}\,a\cos x + 2b\sin x = c, x∈[−π/2,π/2]x \in [-\pi/2, \pi/2], has two distinct real roots α\alpha and β\beta with α+β=π/3\alpha + \beta = \pi/3. Then the value of b/ab/a is

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  2. Question 2

    14 May 2024 S2

    147
    If acos⁡2θ+bsin⁡2θ=ca\cos 2\theta + b\sin 2\theta = c has α\alpha and β\beta as its roots, then tan⁡α+tan⁡β\tan\alpha + \tan\beta is

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