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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2:33 minutes

Problem 39

Textbook Question

In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression.cos² 15° - sin² 15°

Verified step by step guidance

Recognize that the expression \( \cos^2 15^\circ - \sin^2 15^\circ \) can be rewritten using the double-angle identity for cosine: \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \).

Identify that \( \theta = 15^\circ \) in this context, so the expression becomes \( \cos(2 \times 15^\circ) \).

Calculate \( 2 \times 15^\circ = 30^\circ \).

Use the known value of \( \cos 30^\circ \), which is \( \frac{\sqrt{3}}{2} \).

Conclude that \( \cos^2 15^\circ - \sin^2 15^\circ = \cos 30^\circ = \frac{\sqrt{3}}{2} \).

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Key Concepts

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

Double-Angle Formulas

Double-angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For example, the cosine double-angle formula states that cos(2θ) = cos²(θ) - sin²(θ). This formula is essential for simplifying expressions involving angles that are multiples of a given angle, such as 30°, 45°, or 15°.

Half-Angle Formulas

Half-angle formulas allow us to express trigonometric functions of half angles in terms of the functions of the original angle. For instance, sin(θ/2) and cos(θ/2) can be derived from the sine and cosine of θ. These formulas are particularly useful when dealing with angles that are not standard, such as 15°, as they help in calculating exact values.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This fundamental relationship between sine and cosine is crucial for deriving other trigonometric identities and simplifying expressions. In the context of the given expression, it can be used to relate sin²(15°) to cos²(15°) and vice versa, aiding in the calculation of the exact value.