Question

If 3 sin x + 4 cos x = 5, then 4 sin x − 3 cos x =

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Answer 1

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