$5 s i n x + 4 c o s x = 3 ⟹ 4 s i n x - 5 c o s x =$

4

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4 \sqrt{2}

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3 \sqrt{2}

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\sqrt{2}

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Solution

The correct option is D $4 \sqrt{2}$
$5 sin x + 4 cos x = 3$
Squaring both sides
$25 {sin}^{2} x + 16 {cos}^{2} x + 2 * 5 * 4 sin x cos x = 9 \Rightarrow 9 {sin}^{2} x + 7 + 40 sin x cos x = 0 \Rightarrow 9 {sin}^{2} x + 7 = - 40 sin x cos x$

Now, $4 sin x - 5 cos x = \sqrt{(4 sin x - 5 cos x)^{2}} = \sqrt{16 {sin}^{2} x + 25 {cos}^{2} x - 40 sin x cos x} = \sqrt{16 {sin}^{2} x + 25 {cos}^{2} x + 9 {sin}^{2} x + 7} = \sqrt{25 {sin}^{2} x + 25 {cos}^{2} x + 7} = \sqrt{25 + 7} = \sqrt{32} = 4 \sqrt{2}$

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