Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

6. Trigonometric Identities and More Equations

Double Angle Identities

Trigonometry

6. Trigonometric Identities and More Equations

Double Angle Identities

Previous problemNext problem

2:53 minutes

Problem 43

Textbook Question

In Exercises 43–44, express each product as a sum or difference.sin 6x sin 4x

Verified step by step guidance

Recognize that the expression $ \sin A \sin B $ can be rewritten using the product-to-sum identities.

Recall the product-to-sum identity: $ \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] $.

Identify $ A = 6x $ and $ B = 4x $ in the given expression $ \sin 6x \sin 4x $.

Substitute $ A = 6x $ and $ B = 4x $ into the identity: $ \sin 6x \sin 4x = \frac{1}{2} [\cos(6x - 4x) - \cos(6x + 4x)] $.

Simplify the expression: $ \sin 6x \sin 4x = \frac{1}{2} [\cos(2x) - \cos(10x)] $.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Product-to-Sum Formulas

Product-to-sum formulas are trigonometric identities that allow the conversion of products of sine and cosine functions into sums or differences. For example, the product of two sine functions can be expressed as a sum of cosine functions. This transformation simplifies the process of integration and solving trigonometric equations.

Sine Function Properties

The sine function is a periodic function defined for all real numbers, with a range of [-1, 1]. It is essential to understand its properties, such as its periodicity (period of 2π) and symmetry (odd function), to manipulate and simplify expressions involving sine. Recognizing these properties aids in applying the correct identities when transforming products into sums.

Angle Addition and Subtraction Identities

Angle addition and subtraction identities express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of those angles. These identities are crucial when working with products of sine functions, as they provide a systematic way to rewrite expressions, facilitating easier calculations and interpretations in trigonometric problems.