If `2sin^(2)A=sin^(2)60^(@)+sin^(2)45^(@)`, then find the value of sin A.

Answer

To solve the equation \( 2\sin^2 A = \sin^2 60^\circ + \sin^2 45^\circ \), we will follow these steps:

### Step 1: Find the values of \(\sin 60^\circ\) and \(\sin 45^\circ\)

We know:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \]

### Step 2: Calculate \(\sin^2 60^\circ\) and \(\sin^2 45^\circ\)

Now we will square these values:
\[ \sin^2 60^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]
\[ \sin^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 3: Substitute these values into the equation

Now we substitute these squared values back into the original equation:
\[ 2\sin^2 A = \sin^2 60^\circ + \sin^2 45^\circ \]
\[ 2\sin^2 A = \frac{3}{4} + \frac{1}{2} \]

### Step 4: Find a common denominator and add the fractions

To add \(\frac{3}{4}\) and \(\frac{1}{2}\), we convert \(\frac{1}{2}\) to a fraction with a denominator of 4:
\[ \frac{1}{2} = \frac{2}{4} \]
Now we can add:
\[ 2\sin^2 A = \frac{3}{4} + \frac{2}{4} = \frac{5}{4} \]

### Step 5: Solve for \(\sin^2 A\)

Now we divide both sides by 2:
\[ \sin^2 A = \frac{5}{4} \div 2 = \frac{5}{8} \]

### Step 6: Find \(\sin A\)

Taking the square root of both sides gives us:
\[ \sin A = \sqrt{\frac{5}{8}} = \frac{\sqrt{5}}{\sqrt{8}} = \frac{\sqrt{5}}{2\sqrt{2}} \]

### Final Answer

Thus, the value of \(\sin A\) is:
\[ \sin A = \frac{\sqrt{5}}{2\sqrt{2}} \]

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