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 5.2

Textbook Question

For each expression in Column I, choose the expression from Column II that completes an identity.
2. csc x = ____

Verified step by step guidance

Recognize that the cosecant function, \( \csc x \), is the reciprocal of the sine function.

Recall the identity: \( \csc x = \frac{1}{\sin x} \).

Identify the expression in Column II that matches \( \frac{1}{\sin x} \).

Verify that the chosen expression correctly represents the reciprocal relationship with \( \sin x \).

Conclude that the correct expression from Column II completes the identity for \( \csc x \).

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Key Concepts

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

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function. It is defined as csc(x) = 1/sin(x). Understanding this relationship is crucial for solving trigonometric identities, as it allows for the transformation of expressions involving sine into those involving cosecant.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and co-function identities. Recognizing these identities is essential for simplifying expressions and solving equations in trigonometry.

Reciprocal Identities

Reciprocal identities are a specific set of trigonometric identities that express the relationship between sine, cosine, tangent, and their respective cosecant, secant, and cotangent functions. For example, csc(x) = 1/sin(x) and sec(x) = 1/cos(x). These identities are fundamental for completing expressions and solving trigonometric equations.

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