If `sinA=(336)/(625)where450^(@)ltAlt540^(@), then sin""(A)/(4)=`

Options

A. `3//5`
B. `-3//5`
C. `4//5`
D. `-4//5`

Answer

To solve the problem, we need to find the value of \( \sin \frac{A}{4} \) given that \( \sin A = \frac{336}{625} \) and \( 450^\circ < A < 540^\circ \).

### Step-by-Step Solution:

1. **Identify the Quadrant for Angle A**:
Since \( A \) is between \( 450^\circ \) and \( 540^\circ \), it lies in the second quadrant where sine is positive and cosine is negative.

2. **Use the Pythagorean Identity**:
We know the identity:
\[ \sin^2 A + \cos^2 A = 1 \]
Given \( \sin A = \frac{336}{625} \), we can find \( \cos A \):
\[ \cos^2 A = 1 - \sin^2 A = 1 - \left(\frac{336}{625}\right)^2 \]

3. **Calculate \( \sin^2 A \)**:
\[ \sin^2 A = \left(\frac{336}{625}\right)^2 = \frac{112896}{390625} \]

4. **Calculate \( \cos^2 A \)**:
\[ \cos^2 A = 1 - \frac{112896}{390625} = \frac{390625 - 112896}{390625} = \frac{277729}{390625} \]

5. **Find \( \cos A \)**:
Since \( A \) is in the second quadrant, \( \cos A \) will be negative:
\[ \cos A = -\sqrt{\frac{277729}{390625}} = -\frac{527}{625} \]

6. **Use the Half-Angle Formula**:
We can use the half-angle formula for cosine:
\[ \cos A = 2 \cos^2 \frac{A}{2} - 1 \]
Setting \( \cos A = -\frac{527}{625} \):
\[ 2 \cos^2 \frac{A}{2} - 1 = -\frac{527}{625} \]

7. **Solve for \( \cos^2 \frac{A}{2} \)**:
Rearranging gives:
\[ 2 \cos^2 \frac{A}{2} = -\frac{527}{625} + 1 = \frac{98}{625} \]
Therefore,
\[ \cos^2 \frac{A}{2} = \frac{49}{625} \]

8. **Find \( \cos \frac{A}{2} \)**:
Since \( \frac{A}{2} \) is between \( 225^\circ \) and \( 270^\circ \) (third quadrant), \( \cos \frac{A}{2} \) will also be negative:
\[ \cos \frac{A}{2} = -\frac{7}{25} \]

9. **Use the Half-Angle Formula for Sine**:
We can now use the half-angle formula for sine:
\[ \cos \frac{A}{2} = \sqrt{1 - \sin^2 \frac{A}{2}} \]
Rearranging gives:
\[ \sin^2 \frac{A}{2} = 1 - \cos^2 \frac{A}{2} = 1 - \left(-\frac{7}{25}\right)^2 = 1 - \frac{49}{625} = \frac{576}{625} \]

10. **Find \( \sin \frac{A}{2} \)**:
Taking the square root gives:
\[ \sin \frac{A}{2} = \sqrt{\frac{576}{625}} = \frac{24}{25} \]

11. **Use the Half-Angle Formula for Sine Again**:
Now, we can find \( \sin \frac{A}{4} \):
\[ \sin \frac{A}{4} = \sqrt{\frac{1 - \cos \frac{A}{2}}{2}} = \sqrt{\frac{1 + \frac{7}{25}}{2}} = \sqrt{\frac{\frac{32}{25}}{2}} = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

### Final Answer:
\[ \sin \frac{A}{4} = \frac{4}{5} \]