In a triangle ABC, 3sinA + 4cosB = 6 and 4sinB + 3cosA = 1. Find the measure of angle C.

Options

A. `(pi)/(4)`
B. `(pi)/(6)`
C. `(pi)/(3)`
D. `(5pi)/(6)`

Answer

To solve the problem step by step, we will start with the given equations and manipulate them to find the measure of angle C in triangle ABC.

### Step 1: Write down the given equations
We have two equations:
1. \( 3 \sin A + 4 \cos B = 6 \) (Equation 1)
2. \( 4 \sin B + 3 \cos A = 1 \) (Equation 2)

### Step 2: Square both equations
Squaring Equation 1:
\[ (3 \sin A + 4 \cos B)^2 = 6^2 \]
Expanding this gives:
\[ 9 \sin^2 A + 24 \sin A \cos B + 16 \cos^2 B = 36 \quad \text{(Equation 3)} \]

Squaring Equation 2:
\[ (4 \sin B + 3 \cos A)^2 = 1^2 \]
Expanding this gives:
\[ 16 \sin^2 B + 24 \sin B \cos A + 9 \cos^2 A = 1 \quad \text{(Equation 4)} \]

### Step 3: Add both squared equations
Now, we will add Equation 3 and Equation 4:
\[ (9 \sin^2 A + 16 \cos^2 B) + (16 \sin^2 B + 9 \cos^2 A) + (24 \sin A \cos B + 24 \sin B \cos A) = 36 + 1 \]
This simplifies to:
\[ 9(\sin^2 A + \cos^2 A) + 16(\sin^2 B + \cos^2 B) + 24(\sin A \cos B + \sin B \cos A) = 37 \]
Using the identity \(\sin^2 x + \cos^2 x = 1\):
\[ 9(1) + 16(1) + 24(\sin A \cos B + \sin B \cos A) = 37 \]
Thus, we have:
\[ 25 + 24(\sin A \cos B + \sin B \cos A) = 37 \]

### Step 4: Isolate the sine terms
Now, isolate the sine terms:
\[ 24(\sin A \cos B + \sin B \cos A) = 37 - 25 \]
\[ 24(\sin A \cos B + \sin B \cos A) = 12 \]
Dividing both sides by 24:
\[ \sin A \cos B + \sin B \cos A = \frac{12}{24} = \frac{1}{2 \]

### Step 5: Use the sine addition formula
Using the sine addition formula:
\[ \sin A \cos B + \sin B \cos A = \sin(A + B) \]
Thus, we have:
\[ \sin(A + B) = \frac{1}{2} \]

### Step 6: Find angle C
Since \(A + B + C = \pi\), we can write:
\[ A + B = \pi - C \]
Thus:
\[ \sin(\pi - C) = \frac{1}{2} \]
Using the property of sine:
\[ \sin C = \frac{1}{2} \]
This implies:
\[ C = \frac{\pi}{6} \quad \text{(or 30 degrees)} \]

### Final Answer
The measure of angle C is:
\[ C = 30^\circ \quad \text{or} \quad C = \frac{\pi}{6} \text{ radians.} \]