The problem is as follows:

Evaluate: $\frac{1}{\sqrt{2}} \sin 45^\circ + 4 \cos 60^\circ - \frac{\sqrt{3}}{2} \tan 30^\circ$

Let's solve each trigonometric term one by one using known values.

Step-by-Step Solution

1. Evaluate $\sin 45^\circ$: $\sin 45^\circ = \frac{\sqrt{2}}{2}$ Therefore, $\frac{1}{\sqrt{2}} \sin 45^\circ = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{2} = \frac{1}{2}$

2. Evaluate $\cos 60^\circ$: $\cos 60^\circ = \frac{1}{2}$ Then, $4 \cos 60^\circ = 4 \cdot \frac{1}{2} = 2$

3. Evaluate $\tan 30^\circ$: $\tan 30^\circ = \frac{\sqrt{3}}{3}$ Therefore, $\frac{\sqrt{3}}{2} \tan 30^\circ = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{3} = \frac{3}{6} = \frac{1}{2}$

4. Combine all terms: $\frac{1}{2} + 2 - \frac{1}{2} = 2$

Final Answer:

$\boxed{2}$

Would you like a further breakdown of any steps or additional details?

Here are some related questions to expand on this concept:

1. How do we evaluate trigonometric values for other standard angles (like $90^\circ$ and $120^\circ$)?
2. What are reference triangles, and how are they used in trigonometry?
3. How would the solution change if an angle were given in radians instead of degrees?
4. What is the purpose of using trigonometric identities in simplifying expressions?
5. How can we derive the exact values for trigonometric functions without using a calculator?

Tip: Memorizing the exact values for common angles in trigonometry, like $30^\circ$, $45^\circ$, and $60^\circ$, will make problems like this much easier to solve without a calculator.