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 5.4

Textbook Question

Match each expression in Column I with its value in Column II.
cos² (π/6) - sin² (π/6)

Verified step by step guidance

Recognize that the expression \( \cos^2(\pi/6) - \sin^2(\pi/6) \) is in the form of a trigonometric identity known as the cosine of a double angle: \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \).

Identify \( \theta \) as \( \pi/6 \) in this context, which means the expression can be rewritten using the double angle identity as \( \cos(2 \times \pi/6) \).

Simplify \( 2 \times \pi/6 \) to \( \pi/3 \).

Substitute back into the expression to get \( \cos(\pi/3) \).

Recall the value of \( \cos(\pi/3) \) from the unit circle or trigonometric tables.

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Key Concepts

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

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the lengths of its sides. For example, cos(θ) gives the ratio of the adjacent side to the hypotenuse, while sin(θ) gives the ratio of the opposite side to the hypotenuse. Understanding these functions is essential for evaluating expressions involving angles.

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, facilitating the simplification of expressions. In the given expression, this identity can be used to relate sin²(π/6) and cos²(π/6).

Angle Values

Specific angle values, such as π/6, correspond to known sine and cosine values. For instance, cos(π/6) = √3/2 and sin(π/6) = 1/2. Recognizing these standard values is crucial for evaluating trigonometric expressions quickly and accurately, as they provide the necessary numerical values to compute the expression.