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

Defining the Unit Circle

Trigonometry

3. Unit Circle

Defining the Unit Circle

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Problem 3.36

Textbook Question

Find the angular speed ω for each of the following.

a gear revolving 300 times per min

Verified step by step guidance

Start by understanding that angular speed \( \omega \) is the rate at which an object rotates or revolves relative to another point, expressed in radians per second.

Recognize that the gear is revolving 300 times per minute. This is the frequency of the gear's rotation.

Convert the frequency from revolutions per minute to revolutions per second by dividing by 60, since there are 60 seconds in a minute.

Use the formula for angular speed: \( \omega = 2\pi \times \text{frequency} \). This formula comes from the fact that one complete revolution is \( 2\pi \) radians.

Substitute the frequency in revolutions per second into the formula to find \( \omega \).

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Key Concepts

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

Angular Speed

Angular speed, denoted as ω, measures how quickly an object rotates around a central point, typically expressed in radians per second. It is calculated by dividing the angle of rotation (in radians) by the time taken for that rotation. In this context, understanding angular speed is crucial for converting revolutions per minute (RPM) into a standard unit of angular measurement.

Conversion of Units

To solve problems involving angular speed, it is often necessary to convert units. For instance, when given revolutions per minute, one must convert this to radians per second since angular speed is typically expressed in these units. Knowing that one revolution equals 2π radians and there are 60 seconds in a minute is essential for accurate conversion.

Revolutions to Radians

A revolution is a complete turn around a circle, equivalent to 2π radians. This relationship is fundamental in trigonometry and physics, as it allows for the conversion of rotational motion into a linear format. Understanding this conversion is key when calculating angular speed from a given number of revolutions, as it directly impacts the final result.