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

Introduction to Trigonometric Identities

Trigonometry

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

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Problem 5.40

Textbook Question

Simplify each expression.
±√[(1 + cos 20α)/2]

Verified step by step guidance

Recognize that the expression \( \pm \sqrt{\frac{1 + \cos 20\alpha}{2}} \) is related to the half-angle identity for cosine.

Recall the half-angle identity: \( \cos \theta = \pm \sqrt{\frac{1 + \cos 2\theta}{2}} \).

Identify that in the given expression, \( 20\alpha \) is playing the role of \( 2\theta \) in the identity.

Set \( 2\theta = 20\alpha \), which implies \( \theta = 10\alpha \).

Conclude that the expression simplifies to \( \pm \cos 10\alpha \) using the half-angle identity.

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Key Concepts

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

Cosine Function

The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is defined for all real numbers and is periodic with a period of 2π. Understanding the properties of the cosine function is essential for simplifying expressions involving cosine, such as the one in the question.

Half-Angle Identity

The half-angle identities are formulas that express trigonometric functions of half an angle in terms of the functions of the original angle. For cosine, the half-angle identity states that cos(θ/2) = ±√[(1 + cos θ)/2]. This identity is particularly useful for simplifying expressions like ±√[(1 + cos 20α)/2] by recognizing that it represents cos(10α).

Square Root and Absolute Value

The square root function returns the principal (non-negative) root of a number, while the ± symbol indicates that both the positive and negative roots are considered. In trigonometric simplifications, understanding how to handle square roots and the implications of the ± sign is crucial for accurately expressing the results, especially when dealing with angles and their trigonometric values.