If `sin theta = (5)/(13), 0 lt theta lt (pi)/(2)` then find the value of following.
`cos 3 theta`

Answer

To find the value of \( \cos 3\theta \) given that \( \sin \theta = \frac{5}{13} \) and \( 0 < \theta < \frac{\pi}{2} \), we can follow these steps:

### Step 1: Find \( \cos \theta \)
Since we know \( \sin \theta \), we can use the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \sin \theta = \frac{5}{13} \):
\[ \left(\frac{5}{13}\right)^2 + \cos^2 \theta = 1 \]
Calculating \( \left(\frac{5}{13}\right)^2 \):
\[ \frac{25}{169} + \cos^2 \theta = 1 \]
Now, subtract \( \frac{25}{169} \) from both sides:
\[ \cos^2 \theta = 1 - \frac{25}{169} = \frac{169}{169} - \frac{25}{169} = \frac{144}{169} \]
Taking the square root (since \( \theta \) is in the first quadrant, \( \cos \theta \) is positive):
\[ \cos \theta = \frac{12}{13} \]

### Step 2: Use the formula for \( \cos 3\theta \)
The formula for \( \cos 3\theta \) is:
\[ \cos 3\theta = 4\cos^3 \theta - 3\cos \theta \]
Substituting \( \cos \theta = \frac{12}{13} \):
\[ \cos 3\theta = 4\left(\frac{12}{13}\right)^3 - 3\left(\frac{12}{13}\right) \]

### Step 3: Calculate \( \left(\frac{12}{13}\right)^3 \)
Calculating \( \left(\frac{12}{13}\right)^3 \):
\[ \left(\frac{12}{13}\right)^3 = \frac{12^3}{13^3} = \frac{1728}{2197} \]

### Step 4: Substitute back into the formula
Now substituting back into the formula:
\[ \cos 3\theta = 4 \cdot \frac{1728}{2197} - 3 \cdot \frac{12}{13} \]
Calculating \( 3 \cdot \frac{12}{13} \):
\[ 3 \cdot \frac{12}{13} = \frac{36}{13} \]

### Step 5: Find a common denominator
To combine the two fractions, we need a common denominator:
\[ \frac{36}{13} = \frac{36 \cdot 169}{13 \cdot 169} = \frac{6084}{2197} \]
Now substituting back:
\[ \cos 3\theta = \frac{6912}{2197} - \frac{6084}{2197} = \frac{828}{2197} \]

### Final Answer
Thus, the value of \( \cos 3\theta \) is:
\[ \cos 3\theta = \frac{828}{2197} \]