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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Problem 5.40

Textbook Question

Simplify each expression. See Example 4.
1 - 2 sin² 22 ½°

Verified step by step guidance

Recognize that the expression is in the form of a trigonometric identity. Specifically, it resembles the identity for the cosine of a double angle: \( \cos(2\theta) = 1 - 2\sin^2(\theta) \).

Identify \( \theta \) in the expression. Here, \( \theta = 22.5^\circ \).

Apply the double angle identity: \( 1 - 2\sin^2(22.5^\circ) = \cos(2 \times 22.5^\circ) \).

Calculate the angle: \( 2 \times 22.5^\circ = 45^\circ \).

Substitute back into the expression: \( \cos(45^\circ) \).

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Key Concepts

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

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This fundamental relationship allows us to express one trigonometric function in terms of another, which is essential for simplifying expressions involving sine and cosine.

Double Angle Formulas

Double angle formulas provide relationships for trigonometric functions of double angles, such as sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos²(θ) - sin²(θ). These formulas can be useful in simplifying expressions that involve angles that are multiples of a given angle.

Trigonometric Values of Special Angles

Certain angles, such as 0°, 30°, 45°, 60°, and 90°, have known sine and cosine values. For example, sin(30°) = 1/2 and cos(30°) = √3/2. Knowing these values can help in simplifying expressions involving trigonometric functions at these specific angles.