We are tasked with solving:

\[
\frac{3}{4} \sin^2 \left(4 \left( \frac{\pi}{8} \right) \right)
\]

### Step 1: Simplify the angle inside the sine function
First, simplify the expression inside the sine function:

\[
4 \times \frac{\pi}{8} = \frac{4\pi}{8} = \frac{\pi}{2}
\]

Thus, the expression becomes:

\[
\frac{3}{4} \sin^2 \left(\frac{\pi}{2}\right)
\]

### Step 2: Evaluate \(\sin \left(\frac{\pi}{2}\right)\)
We know that:

\[
\sin \left(\frac{\pi}{2}\right) = 1
\]

### Step 3: Apply the result
Substituting \(\sin \left(\frac{\pi}{2}\right) = 1\) into the expression:

\[
\frac{3}{4} \times 1^2 = \frac{3}{4}
\]

Thus, the value of the expression is:

\[
\boxed{\frac{3}{4}}
\]

Would you like more details or have any questions?

Here are some related questions:
1. What is the general formula for \(\sin^2 x\)?
2. How do we evaluate trigonometric functions for other special angles?
3. What is the value of \(\sin(\frac{\pi}{4})\) and \(\cos(\frac{\pi}{4})\)?
4. How can you use the Pythagorean identity with \(\sin^2 x\) and \(\cos^2 x\)?
5. How does \(\sin\) function behave over different quadrants of the unit circle?

**Tip:** Memorizing key angles like \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{3}\) for sine and cosine can help with quick calculations.

Learn how to solve the trigonometric expression (3/4) sin^2(4(pi/8)) with step-by-step simplification. This solution covers key trigonometric concepts such as angle reduction, sine function evaluation, and applying identities. Suitable for high school students in Grades 9-12.

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