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

Explanation

Let 5 \sin \theta - 3 \cos \theta = x. Squaring both equations and adding them yields (3^2+5^2)(\sin^2 \theta + \cos^2 \theta) = 25 + x^2. This gives 34 = 25 + x^2, so x^2 = 9. Thus x = \pm 3. Given the options, -3 is the valid answer.

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