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 5.36

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 t tan t

Verified step by step guidance

Recognize that \( \cot t \) is the cotangent of \( t \), which is defined as \( \frac{1}{\tan t} \).

Substitute \( \cot t \) with \( \frac{1}{\tan t} \) in the expression \( \cot t \tan t \).

The expression becomes \( \frac{1}{\tan t} \times \tan t \).

Apply the property of multiplication that states \( a \times \frac{1}{a} = 1 \) for any non-zero \( a \).

Conclude that \( \frac{1}{\tan t} \times \tan t = 1 \), simplifying the expression to a constant.

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Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Fundamental identities, such as the Pythagorean identities, reciprocal identities, and quotient identities, serve as the foundation for simplifying trigonometric expressions. Understanding these identities is crucial for manipulating and simplifying expressions like 'cot t tan t'.

Cotangent and Tangent Functions

The cotangent (cot) and tangent (tan) functions are fundamental trigonometric functions defined as the ratios of the sides of a right triangle. Specifically, cotangent is the reciprocal of tangent, expressed as cot t = 1/tan t. Recognizing the relationship between these functions is essential for simplifying expressions involving them, such as 'cot t tan t', which simplifies to 1.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves using identities and algebraic manipulation to reduce complex expressions to simpler forms. This process often includes factoring, combining like terms, and substituting equivalent expressions. In the case of 'cot t tan t', applying the identity that cot t is the reciprocal of tan t leads to a straightforward simplification to 1.