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

Sum and Difference Identities

Trigonometry

6. Trigonometric Identities and More Equations

Sum and Difference Identities

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2: minutes

Problem 5

Textbook Question

In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. b. Write the expression as the cosine of an angle.cos 50° cos 20° + sin 50° sin 20°

Verified step by step guidance

Recognize the trigonometric identity for the cosine of a difference: \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \).

Identify the given expression: \( \cos 50^\circ \cos 20^\circ + \sin 50^\circ \sin 20^\circ \).

Compare the given expression with the identity to determine \( \alpha \) and \( \beta \).

Notice that \( \alpha = 50^\circ \) and \( \beta = 20^\circ \).

Write the expression as \( \cos(50^\circ - 20^\circ) \).

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Key Concepts

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

Cosine of Angle Difference Formula

The cosine of the difference of two angles, α and β, is given by the formula cos(α - β) = cos(α)cos(β) + sin(α)sin(β). This formula is fundamental in trigonometry as it allows for the simplification of expressions involving the cosine of angle differences into products of cosines and sines.

Trigonometric Values

Trigonometric values such as cos(50°) and sin(20°) represent the ratios of the sides of a right triangle relative to its angles. Understanding these values is essential for evaluating trigonometric expressions and applying them in various contexts, including solving problems involving angles and distances.

Angle Representation

In trigonometry, angles can be represented in various forms, including degrees and radians. Converting between these forms is crucial for accurate calculations. In this context, recognizing that the expression cos(50°)cos(20°) + sin(50°)sin(20°) can be rewritten as cos(50° - 20°) demonstrates the application of the cosine difference formula.