If `theta` be positive acute angle and 5 `costheta+12sintheta=13` , then the value of `costheta` is

Options

A. `(12)/(13)`
B. `(5)/(13)`
C. `(5)/(12)`
D. `(1)/(5)`

Answer

To solve the equation \( 5 \cos \theta + 12 \sin \theta = 13 \) for \( \cos \theta \), follow these steps:

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

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

### Step 3: Introduce new variables
Let:
\[ x = \cos \theta \quad \text{and} \quad y = \sin \theta \]
Thus, the equation becomes:
\[ \frac{5x}{13} + \frac{12y}{13} = 1 \]
or
\[ 5x + 12y = 13 \]

### Step 4: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ x^2 + y^2 = 1 \]
Substituting \( y = \sqrt{1 - x^2} \) into the equation \( 5x + 12y = 13 \):
\[ 5x + 12\sqrt{1 - x^2} = 13 \]

### Step 5: Isolate the square root
Rearranging gives:
\[ 12\sqrt{1 - x^2} = 13 - 5x \]

### Step 6: Square both sides
Squaring both sides to eliminate the square root:
\[ 144(1 - x^2) = (13 - 5x)^2 \]
Expanding both sides:
\[ 144 - 144x^2 = 169 - 130x + 25x^2 \]

### Step 7: Combine like terms
Rearranging gives:
\[ 144 - 169 + 130x - 144x^2 - 25x^2 = 0 \]
This simplifies to:
\[ -169 + 130x - 169x^2 = 0 \]
or:
\[ 169x^2 - 130x + 25 = 0 \]

### Step 8: Use the quadratic formula
Applying the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 169, b = -130, c = 25 \):
\[ x = \frac{130 \pm \sqrt{(-130)^2 - 4 \cdot 169 \cdot 25}}{2 \cdot 169} \]
Calculating the discriminant:
\[ (-130)^2 - 4 \cdot 169 \cdot 25 = 16900 - 16900 = 0 \]
Thus:
\[ x = \frac{130}{2 \cdot 169} = \frac{130}{338} = \frac{65}{169} \]

### Step 9: Conclusion
Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = \frac{5}{13} \]