IIT JEE , AIEEE Forum - Mathematics , Trignometry - Prove that 5 sin theta-3cos theta=+-3

matsci

If 3sin theta+5cos theta=5

Then prove that 5 sin theta-3cos theta=+-3

rudra.panda

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rudra.panda

$( 3 \sin \theta + 5 \cos \theta)^2=25\\\Rightarrow9(1-\cos^2\theta)+25(1-\sin^2\theta)+30\sin \theta \cos \theta=25\\\Rightarrow9-9\cos^2\theta +25-25\sin^2\theta+30 \sin \theta \cos \theta=25\\\Rightarrow9\cos^2\theta+25\sin^2\theta-30\sin\theta.\cos\theta=9\\\Rightarrow(3\cos\theta-5\sin\theta)^2=9\\\Rightarrow3\cos\theta-5\sin\theta=\pm3\\\Rightarrow5\sin\theta-3\cos\theta=\pm3(proved)$

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vnkt.swaroop

3sin theta+5cos theta=5

let 5sin theta -3cos theta=k

square both of them and solve them simultaneously.

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