If `5 sin theta + 12 cos theta =13`, then what is `5 cos theta -12 sin theta` equal to ?

Options

A. `-2`
B. `-1`
C. 0
D. 1

Answer

To solve the problem, we start with the equation given:

\[ 5 \sin \theta + 12 \cos \theta = 13 \]

We need to find the value of:

\[ 5 \cos \theta - 12 \sin \theta \]

### Step 1: Identify A, B, and C
From the equation \( A \sin \theta + B \cos \theta = C \), we can identify:
- \( A = 5 \)
- \( B = 12 \)
- \( C = 13 \)

### Step 2: Use the formula
We can use the formula for finding \( A \cos \theta - B \sin \theta \):
\[ A \cos \theta - B \sin \theta = \sqrt{A^2 + B^2 - C^2} \]

### Step 3: Calculate \( A^2 + B^2 - C^2 \)
Now, we calculate:
- \( A^2 = 5^2 = 25 \)
- \( B^2 = 12^2 = 144 \)
- \( C^2 = 13^2 = 169 \)

Now substitute these values into the formula:
\[ A^2 + B^2 - C^2 = 25 + 144 - 169 \]

### Step 4: Simplify the expression
Calculating the above:
\[ 25 + 144 = 169 \]
\[ 169 - 169 = 0 \]

### Step 5: Conclusion
Thus, we find that:
\[ 5 \cos \theta - 12 \sin \theta = \sqrt{0} = 0 \]

So, the final answer is:
\[ 5 \cos \theta - 12 \sin \theta = 0 \]

---