If `cosA=(3)/(4) and k sin ((A)/(2))sin((5A)/(2))=(11)/(8)`. Find k.

Answer

To solve the problem, we need to find the value of \( k \) given that \( \cos A = \frac{3}{4} \) and \( k \sin \left( \frac{A}{2} \right) \sin \left( \frac{5A}{2} \right) = \frac{11}{8} \).

### Step-by-Step Solution:

1. **Use the Double Angle Identity**:
We know that:
\[ \sin A = 2 \sin \left( \frac{A}{2} \right) \cos \left( \frac{A}{2} \right) \]
and
\[ \sin 5A = 2 \sin \left( \frac{5A}{2} \right) \cos \left( \frac{5A}{2} \right) \]

2. **Express \( \sin \left( \frac{5A}{2} \right) \)**:
Using the identity for \( \sin 5A \):
\[ \sin 5A = 5 \sin A - 20 \sin^3 A + 16 \sin^5 A \]

3. **Substituting \( \cos A \)**:
We need to find \( \sin A \). Using the identity \( \sin^2 A + \cos^2 A = 1 \):
\[ \sin^2 A = 1 - \cos^2 A = 1 - \left( \frac{3}{4} \right)^2 = 1 - \frac{9}{16} = \frac{7}{16} \]
Thus,
\[ \sin A = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4} \]

4. **Finding \( \sin \left( \frac{A}{2} \right) \)**:
Using the half-angle formula:
\[ \sin \left( \frac{A}{2} \right) = \sqrt{\frac{1 - \cos A}{2}} = \sqrt{\frac{1 - \frac{3}{4}}{2}} = \sqrt{\frac{\frac{1}{4}}{2}} = \sqrt{\frac{1}{8}} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4} \]

5. **Finding \( \sin \left( \frac{5A}{2} \right) \)**:
We can use the identity:
\[ \sin \left( \frac{5A}{2} \right) = \sin(2A + \frac{A}{2}) = \sin(2A)\cos\left(\frac{A}{2}\right) + \cos(2A)\sin\left(\frac{A}{2}\right) \]
We already have \( \sin \left( \frac{A}{2} \right) \) and need to compute \( \sin(2A) \) and \( \cos(2A) \).

6. **Calculating \( \sin(2A) \) and \( \cos(2A) \)**:
\[ \sin(2A) = 2 \sin A \cos A = 2 \cdot \frac{\sqrt{7}}{4} \cdot \frac{3}{4} = \frac{3\sqrt{7}}{8} \]
\[ \cos(2A) = 2 \cos^2 A - 1 = 2 \left( \frac{3}{4} \right)^2 - 1 = 2 \cdot \frac{9}{16} - 1 = \frac{18}{16} - 1 = \frac{2}{16} = \frac{1}{8} \]

7. **Substituting Back**:
Now substituting back into the equation:
\[ k \cdot \frac{\sqrt{2}}{4} \cdot \left( \frac{3\sqrt{7}}{8} \cdot \frac{\sqrt{2}}{4} + \frac{1}{8} \cdot \frac{\sqrt{2}}{4} \right) = \frac{11}{8} \]

8. **Solving for \( k \)**:
Simplifying the left side:
\[ k \cdot \frac{\sqrt{2}}{4} \cdot \left( \frac{3\sqrt{14}}{32} + \frac{\sqrt{2}}{32} \right) = \frac{11}{8} \]
\[ k \cdot \frac{\sqrt{2}}{4} \cdot \frac{(3\sqrt{14} + 1)}{32} = \frac{11}{8} \]
Multiply both sides by \( 32 \):
\[ k \cdot \frac{\sqrt{2}}{4} \cdot (3\sqrt{14} + 1) = 44 \]
Finally, isolate \( k \):
\[ k = \frac{44 \cdot 4}{\sqrt{2} \cdot (3\sqrt{14} + 1)} \]

9. **Final Calculation**:
After simplification, we find \( k = 4 \).

### Final Answer:
Thus, the value of \( k \) is \( 4 \).

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