To find the value of \( \sin(22.5^\circ) \), we can use a half-angle identity for sine, as \( 22.5^\circ = 45^\circ / 2 \).

The half-angle formula for sine is:
\[
\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}
\]
Since \( 22.5^\circ \) is in the first quadrant, where sine values are positive, we use the positive root:
\[
\sin(22.5^\circ) = \sqrt{\frac{1 - \cos(45^\circ)}{2}}
\]

Now, substitute \( \cos(45^\circ) = \frac{\sqrt{2}}{2} \):
\[
\sin(22.5^\circ) = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}
\]

Simplify inside the square root:
\[
= \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2}
\]

This is the exact form of \( \sin(22.5^\circ) \). In decimal form, this value is approximately:
\[
\sin(22.5^\circ) \approx 0.3827
\]

Would you like further details or have any questions?

Here are some related questions:
1. How can the cosine half-angle identity be applied for \( \cos(22.5^\circ) \)?
2. What are the half-angle identities for tangent?
3. How would you use half-angle identities to find \( \sin(15^\circ) \)?
4. Can you derive \( \cos(22.5^\circ) \) using the half-angle identity?
5. What is the approximate decimal value for \( \cos(22.5^\circ) \)?

**Tip:** Memorizing values for common angles can help solve trigonometric problems more quickly.

Learn how to find sin(22.5°) with a detailed solution using the half-angle identity for sine. This problem covers key trigonometric concepts, including using half-angle formulas for angle calculation, and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation