Question

$5 \sin x + 4 \cos x = 3 \Rightarrow 4 \sin x - 5 \cos x =$

• A. 4
• B. $4 \sqrt{2}$
• C. $3 \sqrt{2}$
• D. $ \sqrt{2}$

Answer 1

Answer: (B)

$5 \sin x+4 \cos x=3$
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, $4 \sin x-5 \cos x=\sqrt{(4 \sin x-5 \cos x)^{2}}$
$=\sqrt{16 \sin ^{2} x+25 \cos ^{2} x-40 \sin x \cos x}$
$=\sqrt{16 \sin ^{2} x+25 \cos ^{2} x+9 \sin ^{2} x+7}$
$=\sqrt{25 \sin ^{2} x+25 \cos ^{2} x+7}$
$=\sqrt{25+7}=\sqrt{32}=$
$4 \sqrt{2}$