The minimum value of (3sinx−4cosx−10)(3sinx+4cosx−10) is

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The minimum value of $\sqrt{(3 \sin x - 4 \cos x - 10) (3 \sin x + 4 \cos x - 10)}$ is

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7.00

answer is D.

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Detailed Solution

$f (x) = 9 \sin^{2} x - 16 \cos^{2} x$

$- 10 (3 \sin x - 4 \cos x) - 10 (3 \sin x + 4 \cos x) + 100$

$= 25 \sin^{2} x - 60 \sin x + 84 = {(5 \sin x - 6)}^{2} + 48$

$∴$ the minimum value of $f (x)$ occurs when $\sin x = 1 ∴$ minimum value of $\sqrt{f (x)}$ is 7.

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