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 5.10

Textbook Question

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
csc θ - sin θ

Verified step by step guidance

Recall that the cosecant function is the reciprocal of the sine function: $ \csc \theta = \frac{1}{\sin \theta} $.

Substitute $ \csc \theta $ with $ \frac{1}{\sin \theta} $ in the expression: $ \frac{1}{\sin \theta} - \sin \theta $.

To combine the terms, find a common denominator, which is $ \sin \theta $.

Rewrite $ \sin \theta $ as $ \frac{\sin^2 \theta}{\sin \theta} $ to have the same denominator.

Combine the fractions: $ \frac{1 - \sin^2 \theta}{\sin \theta} $.

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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. Key identities include the Pythagorean identities, reciprocal identities, and co-function identities. Understanding these identities is essential for rewriting trigonometric expressions in terms of sine and cosine, which simplifies the problem.

Reciprocal Functions

Reciprocal functions are pairs of trigonometric functions that are defined as the reciprocal of each other. For example, the cosecant function (csc θ) is the reciprocal of the sine function (sin θ), expressed as csc θ = 1/sin θ. Recognizing these relationships allows for the conversion of expressions involving csc θ into terms of sin θ, facilitating simplification.

Simplification of Trigonometric Expressions

Simplification of trigonometric expressions involves rewriting them in a more manageable form, often eliminating fractions and combining like terms. This process typically requires the application of identities and algebraic manipulation. The goal is to express the function solely in terms of sine and cosine, which can make further analysis or calculations easier.