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Use table for trigonometric function values of some common angles and simplify the resulting expression.

Let's simplify the given trigonometric expression:

\[
\sin 45^\circ \cos 60^\circ + \cos 30^\circ \sin 30^\circ
\]

We will use the known values of trigonometric functions for these angles from the standard table:

\[
\sin 45^\circ = \frac{1}{\sqrt{2}}, \quad \cos 60^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2}
\]

Now, substitute these values into the expression:

\[
\sin 45^\circ \cos 60^\circ + \cos 30^\circ \sin 30^\circ = \left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{2}\right) + \left(\frac{\sqrt{3}}{2}\right) \left(\frac{1}{2}\right)
\]

Simplify each term:

\[
\frac{1}{\sqrt{2}} \times \frac{1}{2} = \frac{1}{2\sqrt{2}}, \quad \frac{\sqrt{3}}{2} \times \frac{1}{2} = \frac{\sqrt{3}}{4}
\]

Now, add the two terms:

\[
\frac{1}{2\sqrt{2}} + \frac{\sqrt{3}}{4}
\]

To add these, we need a common denominator. The least common denominator between \(2\sqrt{2}\) and 4 is \(4\sqrt{2}\). So, let's rewrite the fractions:

\[
\frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4\sqrt{2}}, \quad \frac{\sqrt{3}}{4} = \frac{\sqrt{3}\sqrt{2}}{4\sqrt{2}}
\]

Now, adding them gives:

\[
\frac{\sqrt{2}}{4\sqrt{2}} + \frac{\sqrt{3}\sqrt{2}}{4\sqrt{2}} = \frac{\sqrt{2}(1 + \sqrt{3})}{4\sqrt{2}}
\]

Finally, the \(\sqrt{2}\) cancels out, resulting in:

\[
\frac{1 + \sqrt{3}}{4}
\]

Thus, the simplified result of the given expression is:

\[
\boxed{\frac{1 + \sqrt{3}}{4}}
\]

Would you like any further details or have any questions?

Here are some related questions:
1. How are trigonometric values derived for special angles?
2. What is the process for finding the least common denominator in fractional addition?
3. How do you simplify expressions involving square roots?
4. Can you explain the use of common trigonometric identities in simplifying expressions?
5. How does knowing trigonometric values help in geometry?

**Tip:** Always check if terms can be simplified by factoring or rationalizing when working with trigonometric expressions.

Use table for trigonometric function values of some common angles and simplify the resulting expression: sin(45°)cos(60°) + cos(30°)sin(30°).

Simplify the trigonometric expression sin(45°)cos(60°) + cos(30°)sin(30°) using standard trigonometric values of special angles. Detailed step-by-step solution suitable for Grades 10-12.

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