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

3. Unit Circle

Trigonometric Functions on the Unit Circle

Trigonometry

3. Unit Circle

Trigonometric Functions on the Unit Circle

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2:31 minutes

Problem 28

Textbook Question

In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator.sin² 𝜋 + cos² 𝜋 10 10

Verified step by step guidance

Recognize that the expression \( \sin^2 \theta + \cos^2 \theta = 1 \) is a fundamental Pythagorean identity in trigonometry.

Identify that the given expression \( \sin^2 \frac{\pi}{10} + \cos^2 \frac{\pi}{10} \) matches the form of the Pythagorean identity.

Apply the identity directly to the expression: \( \sin^2 \frac{\pi}{10} + \cos^2 \frac{\pi}{10} = 1 \).

Understand that this identity holds true for any angle \( \theta \), including \( \frac{\pi}{10} \).

Conclude that the value of the expression is determined by the identity, without needing further calculation.

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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, the relationship sin²(θ) + cos²(θ) = 1 holds true. This fundamental identity is derived from the Pythagorean theorem and is essential in trigonometry for simplifying expressions involving sine and cosine functions.

Values of Trigonometric Functions at Specific Angles

Trigonometric functions have specific values at key angles, such as 0, π/2, π, and 3π/2. For example, sin(π) = 0 and cos(π) = -1. Knowing these values allows for quick calculations and simplifications in trigonometric expressions.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using identities and known values to reduce complex expressions to simpler forms. This process often includes substituting known values for sine and cosine at specific angles, which can lead to straightforward numerical results.