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a sin t + b cos t = root(a^2 + b^2) gives tan t = a/b: evaluate

This type of question, asked 3 times: CDS 2 2022, CDS 1 2026, CDS 2 2026. All 3 questions below, oldest first.

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  1. Question 1

    CDS 2 2022

    52
    If 11sin⁡θ+60cos⁡θ=6111\sin\theta + 60\cos\theta = 61; 0<θ<90∘0 < \theta < 90^\circ, then what is the value of 660(tan⁡θ+cot⁡θ)\sqrt{660(\tan\theta + \cot\theta)} ?

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  2. Question 2

    CDS 1 2026

    43
    If 3sin⁡θ+4cos⁡θ=53\sin\theta + 4\cos\theta = 5, then what is a value of 4tan⁡θ+3cot⁡θ4\tan\theta + 3\cot\theta ?

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  3. Question 3

    CDS 2 2026

    46
    If 5sin⁡θ+12cos⁡θ=135\sin\theta + 12\cos\theta = 13, where 0<θ<π20 < \theta < \frac{\pi}{2}, then what is tan⁡θ+cot⁡θ\tan\theta + \cot\theta equal to ?

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