If cos A = `3//4` . then what is the value of sin`((A)/(2))sin((3A)/(2))` ?

Options

A. `5//8`
B. `5//16`
C. `5//24`
D. `7//32`

Answer

To solve the problem, we need to find the value of \( \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) \) given that \( \cos A = \frac{3}{4} \).

### Step-by-Step Solution:

1. **Use the Product-to-Sum Formula:**
We can use the identity:
\[ \sin c \sin d = \frac{1}{2} \left( \cos(c - d) - \cos(c + d) \right) \]
Here, let \( c = \frac{A}{2} \) and \( d = \frac{3A}{2} \).

2. **Calculate \( c - d \) and \( c + d \):**
- \( c - d = \frac{A}{2} - \frac{3A}{2} = -A \)
- \( c + d = \frac{A}{2} + \frac{3A}{2} = 2A \)

3. **Substituting into the Identity:**
Now substituting these values into the identity:
\[ \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) = \frac{1}{2} \left( \cos(-A) - \cos(2A) \right) \]
Since \( \cos(-A) = \cos A \), we have:
\[ \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) = \frac{1}{2} \left( \cos A - \cos(2A) \right) \]

4. **Finding \( \cos(2A) \):**
We can use the double angle formula:
\[ \cos(2A) = 2\cos^2 A - 1 \]
Given \( \cos A = \frac{3}{4} \):
\[ \cos(2A) = 2\left(\frac{3}{4}\right)^2 - 1 = 2 \cdot \frac{9}{16} - 1 = \frac{18}{16} - 1 = \frac{18}{16} - \frac{16}{16} = \frac{2}{16} = \frac{1}{8} \]

5. **Substituting Back:**
Now substitute \( \cos A \) and \( \cos(2A) \) back into our equation:
\[ \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) = \frac{1}{2} \left( \frac{3}{4} - \frac{1}{8} \right) \]

6. **Finding a Common Denominator:**
To subtract \( \frac{3}{4} \) and \( \frac{1}{8} \), we need a common denominator:
\[ \frac{3}{4} = \frac{6}{8} \]
So,
\[ \frac{3}{4} - \frac{1}{8} = \frac{6}{8} - \frac{1}{8} = \frac{5}{8} \]

7. **Final Calculation:**
Now substitute this back into our equation:
\[ \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) = \frac{1}{2} \cdot \frac{5}{8} = \frac{5}{16} \]

### Final Answer:
Thus, the value of \( \sin\left(\frac{A}{2}\right) \sin\left(\frac{3A}{2}\right) \) is \( \frac{5}{16} \).