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

Introduction to Trigonometric Identities

Trigonometry

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

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Problem 5.34

Textbook Question

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.
cot α sin α

Verified step by step guidance

Recall the definition of cotangent: \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \).

Substitute the definition of \( \cot \alpha \) into the expression: \( \cot \alpha \sin \alpha = \frac{\cos \alpha}{\sin \alpha} \cdot \sin \alpha \).

Observe that the \( \sin \alpha \) in the numerator and denominator cancel each other out.

Simplify the expression to \( \cos \alpha \).

Conclude that the expression simplifies to a single trigonometric function: \( \cos \alpha \).

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Key Concepts

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

Cotangent Function

The cotangent function, denoted as cot(α), is the reciprocal of the tangent function. It can be expressed as cot(α) = cos(α) / sin(α). Understanding cotangent is essential for simplifying expressions involving trigonometric functions, particularly when combined with sine or cosine.

Sine Function

The sine function, sin(α), is a fundamental trigonometric function that relates the angle α to the ratio of the length of the opposite side to the hypotenuse in a right triangle. It is crucial for simplifying expressions that involve angles and is often used in conjunction with other trigonometric identities.

Fundamental Trigonometric Identities

Fundamental trigonometric identities are equations that relate the various trigonometric functions to one another. These include identities such as sin²(α) + cos²(α) = 1 and the reciprocal identities. Utilizing these identities is key to simplifying trigonometric expressions effectively.