The product of \(\cos(ax)\) and \(\sin(ax)\) can be rewritten using trigonometric identities. One of the common forms involves using the **double-angle identity** for sine. Here's the derivation:

\[
\cos(ax) \cdot \sin(ax) = \frac{1}{2} \sin(2ax)
\]

### Explanation:
- The identity \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\) is applied in reverse. In this case, \(\theta = ax\), so the product can be simplified as:

\[
2\cos(ax)\sin(ax) = \sin(2ax)
\]

Dividing both sides by 2:

\[
\cos(ax)\sin(ax) = \frac{1}{2}\sin(2ax)
\]

This is the simplified form of \(\cos(ax) \cdot \sin(ax)\).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How do you derive the double-angle identity for sine?
2. What are other double-angle identities in trigonometry?
3. Can you explain the product-to-sum identities in trigonometry?
4. How can this identity be used in integration problems involving trigonometric functions?
5. What is the general form of \(\cos(nx) \cdot \sin(nx)\) for any integer \(n\)?

**Tip:** Knowing trigonometric identities like double-angle or sum-to-product identities helps simplify complex expressions, especially in calculus and physics.

Understand how to simplify the expression cos(ax)sin(ax) using the double-angle identity. This solution includes a detailed explanation of trigonometric identities and their applications, 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