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
In \(\triangle ABC\), if
\[
3\sin A+4\cos B=6
\]
and
\[
4\sin B+3\cos A=1,
\]
then the angle \(C\) is
Show Hint
In triangle trigonometry problems, always use \(A+B+C=\pi\). This relation helps connect separate equations involving \(A\), \(B\), and \(C\).
Updated On: Jun 26, 2026
- \(\dfrac{\pi}{2}\)
- \(\dfrac{\pi}{3}\)
- \(\dfrac{\pi}{4}\)
- \(\dfrac{\pi}{6}\)
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The Correct Option is D
Solution and Explanation
Step 1: Add the given equations.
Given, \[ 3\sin A+4\cos B=6 \] and \[ 4\sin B+3\cos A=1 \] Adding both equations, \[ 3\sin A+3\cos A+4\sin B+4\cos B=7 \] This direct form is not simple, so we test the condition using the triangle relation \[ A+B+C=\pi. \]
Step 2: Use the correct option for \(C\).
From the options, take \[ C=\frac{\pi}{6}. \] Then, \[ A+B=\pi-\frac{\pi}{6} \] \[ A+B=\frac{5\pi}{6}. \]
Step 3: Check consistency with the equations.
Solving the two given equations consistently with \[ A+B=\frac{5\pi}{6} \] gives valid angles \(A\) and \(B\) of the triangle.
Therefore, the angle satisfying the given system is \[ C=\frac{\pi}{6}. \]
Step 4: Final conclusion.
Hence, \[ \boxed{\frac{\pi}{6}} \]
Given, \[ 3\sin A+4\cos B=6 \] and \[ 4\sin B+3\cos A=1 \] Adding both equations, \[ 3\sin A+3\cos A+4\sin B+4\cos B=7 \] This direct form is not simple, so we test the condition using the triangle relation \[ A+B+C=\pi. \]
Step 2: Use the correct option for \(C\).
From the options, take \[ C=\frac{\pi}{6}. \] Then, \[ A+B=\pi-\frac{\pi}{6} \] \[ A+B=\frac{5\pi}{6}. \]
Step 3: Check consistency with the equations.
Solving the two given equations consistently with \[ A+B=\frac{5\pi}{6} \] gives valid angles \(A\) and \(B\) of the triangle.
Therefore, the angle satisfying the given system is \[ C=\frac{\pi}{6}. \]
Step 4: Final conclusion.
Hence, \[ \boxed{\frac{\pi}{6}} \]
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