The half-angle formula for \(\sin^4 \theta\) can be derived using the half-angle identity for sine. Let's break it down step by step:

### Step 1: Start with the basic half-angle identity for sine:
\[
\sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{2}
\]

### Step 2: Express \(\sin^4\left(\frac{\theta}{2}\right)\):
Since we are looking for \(\sin^4\left(\frac{\theta}{2}\right)\), we square the half-angle identity:
\[
\sin^4\left(\frac{\theta}{2}\right) = \left(\frac{1 - \cos\theta}{2}\right)^2
\]

### Step 3: Simplify the expression:
\[
\sin^4\left(\frac{\theta}{2}\right) = \frac{(1 - \cos\theta)^2}{4}
\]

Thus, the formula for \(\sin^4\left(\frac{\theta}{2}\right)\) is:
\[
\sin^4\left(\frac{\theta}{2}\right) = \frac{1 - 2\cos\theta + \cos^2\theta}{4}
\]

This is the half-angle expression for \(\sin^4\left(\frac{\theta}{2}\right)\).

Would you like further details or clarifications?

Here are 5 questions to expand on the concept:
1. What is the half-angle formula for \(\cos^2\theta\)?
2. How do we derive the double-angle identity for sine?
3. How can the expression for \(\sin^4 \theta\) be integrated over an interval?
4. What are some real-world applications of the half-angle identities?
5. How does \(\sin^4\theta\) behave as \(\theta\) approaches 0 or \(\pi\)?

**Tip:** Half-angle formulas are especially useful in simplifying trigonometric integrals and solving equations where direct application of standard identities may be complex.

Explore the half-angle formula for sin^4(θ/2) with a detailed derivation, breaking down trigonometric identities step by step. Learn key concepts such as half-angle identities and how to simplify complex trigonometric expressions. 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