If sin A = `3/5`, where `450^@ lt A lt 540^@`, then cos `A/2` is equal to:

Options

A. `(1)/(sqrt(10))`
B. `-sqrt((3)/(10))`
C. `(sqrt3)/(sqrt(10))`
D. None of these

Answer

To find the value of \( \cos \frac{A}{2} \) given that \( \sin A = \frac{3}{5} \) and \( 450^\circ < A < 540^\circ \), we can follow these steps:

### Step 1: Determine \( \cos A \)
We know the Pythagorean identity:
\[ \sin^2 A + \cos^2 A = 1 \]
Given \( \sin A = \frac{3}{5} \), we can find \( \cos A \):
\[ \sin^2 A = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \]
Substituting into the identity:
\[ \frac{9}{25} + \cos^2 A = 1 \]
\[ \cos^2 A = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]
Thus,
\[ \cos A = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5} \]

### Step 2: Determine the sign of \( \cos A \)
Since \( A \) is in the range \( 450^\circ < A < 540^\circ \) (which corresponds to the third quadrant), \( \cos A \) is negative. Therefore,
\[ \cos A = -\frac{4}{5} \]

### Step 3: Use the half-angle formula for cosine
The half-angle formula for cosine is:
\[ \cos \frac{A}{2} = \sqrt{\frac{1 + \cos A}{2}} \]
Substituting \( \cos A = -\frac{4}{5} \):
\[ \cos \frac{A}{2} = \sqrt{\frac{1 - \frac{4}{5}}{2}} = \sqrt{\frac{\frac{1}{5}}{2}} = \sqrt{\frac{1}{10}} = \frac{1}{\sqrt{10}} \]

### Step 4: Determine the sign of \( \cos \frac{A}{2} \)
Since \( A \) is between \( 450^\circ \) and \( 540^\circ \), \( \frac{A}{2} \) will be between \( 225^\circ \) and \( 270^\circ \). In this range, \( \cos \frac{A}{2} \) is negative. Therefore,
\[ \cos \frac{A}{2} = -\frac{1}{\sqrt{10}} \]

### Final Answer
Thus, the value of \( \cos \frac{A}{2} \) is:
\[ \cos \frac{A}{2} = -\frac{1}{\sqrt{10}} \]

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