Answer: C
Given: \( 3\cos\theta - 4\sin\theta = 5 \) Then, \((3\cos\theta - 4\sin\theta)^2 = 5^2\) \[ = 9\cos^2\theta - 2 \cdot 3\cos\theta \cdot 4\sin\theta + 16\sin^2\theta = 25 \] \[ = 9(1 - \sin^2\theta) - 24\cos\theta\sin\theta + 16(1 - \cos^2\theta) = 25 \] \[ = 9 - 9\sin^2\theta - 24\cos\theta\sin\theta + 16 - 16\cos^2\theta = 25 \] \[ = 25 - (9\sin^2\theta + 2 \cdot 3\sin\theta \cdot 4\cos\theta + 16\cos^2\theta) = 25 \] \[ \Rightarrow -(3\sin\theta + 4\cos\theta)^2 = 25 - 25 \Rightarrow (3\sin\theta + 4\cos\theta)^2 = 0 \Rightarrow 3\sin\theta + 4\cos\theta = 0 \]
Given: \( 3\cos\theta - 4\sin\theta = 5 \) Then, \((3\cos\theta - 4\sin\theta)^2 = 5^2\) \[ = 9\cos^2\theta - 2 \cdot 3\cos\theta \cdot 4\sin\theta + 16\sin^2\theta = 25 \] \[ = 9(1 - \sin^2\theta) - 24\cos\theta\sin\theta + 16(1 - \cos^2\theta) = 25 \] \[ = 9 - 9\sin^2\theta - 24\cos\theta\sin\theta + 16 - 16\cos^2\theta = 25 \] \[ = 25 - (9\sin^2\theta + 2 \cdot 3\sin\theta \cdot 4\cos\theta + 16\cos^2\theta) = 25 \] \[ \Rightarrow -(3\sin\theta + 4\cos\theta)^2 = 25 - 25 \Rightarrow (3\sin\theta + 4\cos\theta)^2 = 0 \Rightarrow 3\sin\theta + 4\cos\theta = 0 \]