If 3 +4 =5, then 4 -3 = - Vidyadip

MCQ

If $3\sin\text{x}+4\cos\text{x}=5,$ then $4\sin\text{x}-3\cos\text{x}=$

✓
$0$
B
$5$
C
$1$
D
None of these

✓

Answer

Correct option: A.

$0$

$3\sin\text{x}+4\cos\text{x}=5$
$\frac{3}{5}\sin\text{x}+\frac{4}{5}\cos\text{x}=1$
Let $\cos\alpha=\frac35$ and $\sin\alpha=\frac45.$
$\therefore\cos\alpha\sin\text{x}+\sin\alpha\cos\text{x}=1$
$\Rightarrow\sin(\alpha+\text{x})=\sin\frac\pi2$
$\Rightarrow\alpha+\text{x}=\frac\pi2$
$\Rightarrow\text{x}=\frac{\pi}{2}-\alpha\cdots(1)$
We have to find the value of $4\sin\text{x}-3\cos\text{x}.$
$4\sin\Big(\frac\pi2-\alpha\Big)-3\cos\Big(\frac\pi3-\alpha\Big) ...\{$From eq $(1)\}$
$=4\cos\alpha-3\sin\alpha$
$=4\times\frac35-3\times\frac45$ $\Big(\because\cos\alpha=\frac35$ and $\sin\alpha=\frac{4}{5}\Big)$
$=0$

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