If 3\sin \theta+5\cos \theta=5, then what is the value of 5\sin \theta-3\cos \theta equal to?

Explanation

Let x = 5\sin\theta - 3\cos\theta. Squaring and adding both equations: (3\sin\theta + 5\cos\theta)^2 + (5\sin\theta - 3\cos\theta)^2 = 5^2 + x^2. This simplifies to (3^2+5^2)(\sin^2\theta+\cos^2\theta) = 25 + x^2 \implies 34(1) = 25 + x^2 \implies x^2 = 9. Thus, x = \pm 3. Note that if \cos\theta=1 and \sin\theta=0, the given equation 3(0)+5(1)=5 is satisfied. Substituting into the expression: 5(0)-3(1) = -3.

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