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 3.15

Textbook Question

Use the formula ω = θ/t to find the value of the missing variable.

θ = 3π/4 radians, t = 8 sec

Verified step by step guidance

Identify the given values: \( \theta = \frac{3\pi}{4} \) radians and \( t = 8 \) seconds.

Recall the formula for angular velocity: \( \omega = \frac{\theta}{t} \).

Substitute the given values into the formula: \( \omega = \frac{\frac{3\pi}{4}}{8} \).

Simplify the expression by dividing the numerator by the denominator: \( \omega = \frac{3\pi}{4 \times 8} \).

Further simplify the expression: \( \omega = \frac{3\pi}{32} \).

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Key Concepts

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

Angular Displacement (θ)

Angular displacement, represented by θ, measures the angle through which an object has rotated about a fixed point. It is typically expressed in radians, where 2π radians correspond to a full rotation (360 degrees). In this question, θ is given as 3π/4 radians, indicating that the object has rotated three-quarters of the way around a circle.

Time (t)

Time (t) in this context refers to the duration over which the angular displacement occurs. It is a crucial variable in calculating angular velocity, as it provides the timeframe for the rotation. In the question, t is specified as 8 seconds, which will be used to determine how quickly the object is rotating.

Angular Velocity (ω)

Angular velocity (ω) is a measure of how quickly an object rotates around a point, defined as the rate of change of angular displacement over time. It is calculated using the formula ω = θ/t, where θ is the angular displacement and t is the time taken. In this case, substituting the given values will yield the angular velocity in radians per second.