The value of `sin30^@ cos^2 60^@ + cos30^@ sin^2 60^@` is equal to

Answer

To solve the expression \( \sin 30^\circ \cos^2 60^\circ + \cos 30^\circ \sin^2 60^\circ \), we will follow these steps:

### Step 1: Identify the values of the trigonometric functions
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
- \( \cos 60^\circ = \frac{1}{2} \)

### Step 2: Substitute the values into the expression
Substituting the values we identified into the expression gives us:
\[ \sin 30^\circ \cos^2 60^\circ + \cos 30^\circ \sin^2 60^\circ = \left(\frac{1}{2}\right) \left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{3}}{2}\right)^2 \]

### Step 3: Calculate \( \cos^2 60^\circ \) and \( \sin^2 60^\circ \)
Calculating the squares:
- \( \cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)
- \( \sin^2 60^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \)

### Step 4: Substitute the squared values back into the expression
Now substituting these squared values back into the expression:
\[ \sin 30^\circ \cos^2 60^\circ + \cos 30^\circ \sin^2 60^\circ = \left(\frac{1}{2}\right) \left(\frac{1}{4}\right) + \left(\frac{\sqrt{3}}{2}\right) \left(\frac{3}{4}\right) \]

### Step 5: Simplify the expression
Calculating each term:
- First term: \( \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8} \)
- Second term: \( \frac{\sqrt{3}}{2} \cdot \frac{3}{4} = \frac{3\sqrt{3}}{8} \)

Now, combine the two terms:
\[ \frac{1}{8} + \frac{3\sqrt{3}}{8} = \frac{1 + 3\sqrt{3}}{8} \]

### Final Result
Thus, the value of the expression is:
\[ \frac{1 + 3\sqrt{3}}{8} \]

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