Question

In \( \triangle ABC \), if \( 3\sin A + 4\cos B = 6 \) and \( 4\sin B + 3\cos A = 1 \), then the angle \( C \) is:

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{6} \)

Hint:

Triangle Trigonometry Systems}
Use angle sum identity \( A + B + C = \pi \)
Convert to known sin/cos expressions and solve algebraically
Evaluate the remaining angle after finding \( A \) and \( B \)

Answer 1

The Correct Option is D

Use angle sum identity: \[ A + B + C = \pi \Rightarrow C = \pi - (A + B) \] From the given: \[ 3\sin A + 4\cos B = 6 \quad \text{(1)} \] \[ 4\sin B + 3\cos A = 1 \quad \text{(2)} \] Solving these equations using known trigonometric values (or via substitution/graphing), the values of \( A \) and \( B \) are obtained such that \( C = \frac{\pi}{6} \).