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

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3:03 minutes

Problem 44

Textbook Question

In Exercises 43–44, express each product as a sum or difference.sin 7x cos 3x

Verified step by step guidance

Use the product-to-sum identities for trigonometric functions, which state that $ \sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)] $.

Identify $ A = 7x $ and $ B = 3x $ in the given expression $ \sin 7x \cos 3x $.

Substitute $ A $ and $ B $ into the identity: $ \sin 7x \cos 3x = \frac{1}{2} [\sin(7x + 3x) + \sin(7x - 3x)] $.

Simplify the expressions inside the sine functions: $ \sin(7x + 3x) = \sin 10x $ and $ \sin(7x - 3x) = \sin 4x $.

Express the product as a sum: $ \sin 7x \cos 3x = \frac{1}{2} [\sin 10x + \sin 4x] $.

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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 formula sin(A)cos(B) can be expressed as (1/2)[sin(A+B) + sin(A-B)]. These identities simplify calculations and are essential for integrating or solving trigonometric equations.

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that relate the angles of a right triangle to the ratios of its sides. The sine function represents the ratio of the opposite side to the hypotenuse, while the cosine function represents the ratio of the adjacent side to the hypotenuse. Understanding these functions is crucial for manipulating and transforming trigonometric expressions.

Angle Addition and Subtraction

Angle addition and subtraction formulas are used to express trigonometric functions of sums or differences of angles in terms of the functions of the individual angles. For instance, sin(A ± B) and cos(A ± B) provide relationships that are useful in simplifying expressions and solving equations. Mastery of these formulas is vital for effectively applying trigonometric identities.