If `5costheta+12sintheta=13,0^(@)ltthetalt90^(@)` , then the value of sin `theta` is

Options

A. a)`(5)/(13)`
B. b)`-(12)/(13)`
C. c)`(6)/(13)`
D. d)`(12)/(13)`

Answer

To solve the equation \( 5 \cos \theta + 12 \sin \theta = 13 \) for \( \sin \theta \) where \( 0^\circ < \theta < 90^\circ \), we can follow these steps:

### Step 1: Rewrite the equation
Given:
\[ 5 \cos \theta + 12 \sin \theta = 13 \]

### Step 2: Divide the entire equation by 13
To simplify the equation, we divide every term by 13:
\[ \frac{5}{13} \cos \theta + \frac{12}{13} \sin \theta = 1 \]

### Step 3: Recognize the coefficients
Let:
\[ a = \frac{5}{13} \quad \text{and} \quad b = \frac{12}{13} \]
We can see that \( a^2 + b^2 = \left(\frac{5}{13}\right)^2 + \left(\frac{12}{13}\right)^2 \).

### Step 4: Calculate \( a^2 + b^2 \)
Calculating:
\[ a^2 = \left(\frac{5}{13}\right)^2 = \frac{25}{169} \]
\[ b^2 = \left(\frac{12}{13}\right)^2 = \frac{144}{169} \]
Adding these:
\[ a^2 + b^2 = \frac{25}{169} + \frac{144}{169} = \frac{169}{169} = 1 \]

### Step 5: Use the sine and cosine identity
Since \( a^2 + b^2 = 1 \), we can interpret this in terms of sine and cosine:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
This means we can express \( \sin \theta \) and \( \cos \theta \) as:
\[ \sin \theta = \frac{12}{13} \quad \text{and} \quad \cos \theta = \frac{5}{13} \]

### Step 6: Conclusion
Thus, the value of \( \sin \theta \) is:
\[ \sin \theta = \frac{12}{13} \]