To find the value of \( \sin \frac{\pi}{8} \), we can use the half-angle formula for sine:

\[ \sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}} \]

We need to use this formula with \( x = \frac{\pi}{4} \), since \( \frac{\pi}{8} = \frac{\pi/4}{2} \). Therefore, we have:

\[ \sin \frac{\pi}{8} = \sin \left( \frac{\pi/4}{2} \right) = \sqrt{\frac{1 - \cos \frac{\pi}{4}}{2}} \]

We know that \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), so substituting this in:

\[ \sin \frac{\pi}{8} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \]

Now, simplify the expression inside the square root:

\[ \sin \frac{\pi}{8} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}} = \frac{\sqrt{2 - \sqrt{2}}}{2} \]

Therefore, the value of \( \sin \frac{\pi}{8} \) is:

\[ \sin \frac{\pi}{8} = \frac{\sqrt{2 - \sqrt{2}}}{2} \]

Would you like further details or have any questions?

Here are 5 questions you might ask:
1. How can we use the half-angle formula for other trigonometric functions?
2. What are the other ways to derive the value of \( \sin \frac{\pi}{8} \)?
3. Can you explain the derivation of the half-angle formula?
4. How does the half-angle formula work for cosine and tangent?
5. What are some common applications of the half-angle formula in calculus?

**Tip:** When dealing with trigonometric identities and formulas, always double-check your known values, such as \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), to ensure accuracy in your calculations.

Learn how to calculate \( \sin \frac{\pi}{8} \) with a step-by-step explanation of the half-angle formula. This detailed solution covers the half-angle formula for sine and provides a clear derivation of the result. Ideal for high school students studying trigonometry.

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