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 5.39

Textbook Question

Simplify each expression.
±√[(1 + cos 18x)/2]

Verified step by step guidance

Recognize that the expression \( \pm \sqrt{\frac{1 + \cos 18x}{2}} \) is related to the half-angle identity for cosine.

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

Identify that in the given expression, \( \theta = 18x \), so the expression represents \( \cos \frac{18x}{2} \).

Simplify \( \cos \frac{18x}{2} \) to \( \cos 9x \).

Conclude that the expression \( \pm \sqrt{\frac{1 + \cos 18x}{2}} \) simplifies to \( \pm \cos 9x \).

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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 periodic and oscillates between -1 and 1. 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 trigonometric identities that express trigonometric functions of half an angle in terms of the functions of the original angle. For cosine, the half-angle identity is cos(θ/2) = ±√[(1 + cos θ)/2]. This identity is crucial for simplifying expressions like ±√[(1 + cos 18x)/2] by recognizing it as a half-angle formula.

Square Root Properties

Square root properties involve the rules governing the manipulation of square roots, including the principle that √(a/b) = √a/√b and √(a) * √(a) = a. These properties are important for simplifying expressions under the square root, allowing for the extraction of factors and simplification of trigonometric expressions effectively.