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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2:30 minutes

Problem 19

Textbook Question

In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression.𝝅2 cos² ------ ﹣ 18

Verified step by step guidance

Recognize that the expression \( \pi/8 \) is an angle, and the given expression \( 2 \cos^2(\pi/8) - 1 \) resembles the double angle identity for cosine.

Recall the double angle identity for cosine: \( \cos(2\theta) = 2\cos^2(\theta) - 1 \).

Identify that \( \theta = \pi/8 \) in this context, so the expression \( 2\cos^2(\pi/8) - 1 \) can be rewritten as \( \cos(2 \times \pi/8) \).

Simplify the angle: \( 2 \times \pi/8 = \pi/4 \).

Conclude that the expression \( 2\cos^2(\pi/8) - 1 \) is equivalent to \( \cos(\pi/4) \).

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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 sine, cosine, and tangent double angle formulas are: sin(2θ) = 2sin(θ)cos(θ), cos(2θ) = cos²(θ) - sin²(θ), and tan(2θ) = 2tan(θ) / (1 - tan²(θ)). Understanding these formulas is essential for rewriting expressions involving double angles.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This fundamental identity allows us to express one trigonometric function in terms of another, which is particularly useful when simplifying expressions or solving equations. It can help in transforming expressions involving cos²(θ) into terms of sin²(θ) or vice versa.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the specific values of sine, cosine, and tangent for commonly used angles, such as 0, π/6, π/4, π/3, and π/2. These values are often derived from the unit circle and can be used to evaluate trigonometric expressions without a calculator. Knowing these exact values is crucial for finding the exact value of expressions involving trigonometric functions.