If `a=3,b=4, c=5`, then the value of `sin 2B` is

Options

A. `24/25`
B. `25/24`
C. `1/25`
D. none

Answer

To find the value of \( \sin 2B \) given \( a = 3 \), \( b = 4 \), and \( c = 5 \), we can follow these steps:

### Step 1: Use the double angle formula for sine
The double angle formula for sine states:
\[ \sin 2B = 2 \sin B \cos B \]
We need to find \( \sin B \) and \( \cos B \) first.

### Step 2: Use the cosine rule to find \( \cos B \)
The cosine rule states:
\[ \cos B = \frac{a^2 + c^2 - b^2}{2ac} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ \cos B = \frac{3^2 + 5^2 - 4^2}{2 \cdot 3 \cdot 5} \]
Calculating the squares:
\[ \cos B = \frac{9 + 25 - 16}{30} = \frac{18}{30} = \frac{3}{5} \]

### Step 3: Use the Pythagorean identity to find \( \sin B \)
Using the identity \( \sin^2 B + \cos^2 B = 1 \):
\[ \sin^2 B = 1 - \cos^2 B \]
Substituting the value of \( \cos B \):
\[ \sin^2 B = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \]
Taking the square root:
\[ \sin B = \frac{4}{5} \]

### Step 4: Substitute \( \sin B \) and \( \cos B \) into the double angle formula
Now we can substitute \( \sin B \) and \( \cos B \) into the double angle formula:
\[ \sin 2B = 2 \sin B \cos B = 2 \cdot \frac{4}{5} \cdot \frac{3}{5} \]
Calculating this:
\[ \sin 2B = 2 \cdot \frac{12}{25} = \frac{24}{25} \]

### Final Answer
Thus, the value of \( \sin 2B \) is:
\[ \sin 2B = \frac{24}{25} \]

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