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 3.21

Textbook Question

Use the formula v = r ω to find the value of the missing variable.

r = 12 m , ω = 2π/3 radians per sec

Verified step by step guidance

Identify the given values: radius \( r = 12 \) meters and angular velocity \( \omega = \frac{2\pi}{3} \) radians per second.

Recall the formula for linear velocity: \( v = r \cdot \omega \).

Substitute the given values into the formula: \( v = 12 \cdot \frac{2\pi}{3} \).

Simplify the expression by multiplying the numbers: \( v = 12 \times \frac{2\pi}{3} \).

Complete the multiplication to find the linear velocity \( v \).

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Key Concepts

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

Angular Velocity (ω)

Angular velocity (ω) measures how quickly an object rotates around a central point, expressed in radians per second. It indicates the angle covered per unit of time, allowing us to understand the rotational speed of an object. In this question, ω is given as 2π/3 radians per second, which is essential for calculating linear velocity.

Radius (r)

The radius (r) is the distance from the center of a circle to any point on its circumference. In the context of circular motion, the radius plays a crucial role in determining the linear velocity of an object moving along a circular path. Here, the radius is provided as 12 meters, which is necessary for applying the formula v = rω.

Linear Velocity (v)

Linear velocity (v) represents the speed of an object moving along a circular path and is calculated using the formula v = rω. This formula shows that linear velocity is directly proportional to both the radius and the angular velocity. By substituting the given values of r and ω into this formula, we can find the missing variable, which is the linear velocity.