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

1. Measuring Angles

Complementary and Supplementary Angles

Trigonometry

1. Measuring Angles

Complementary and Supplementary Angles

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Problem 3.92

Textbook Question

Through how many radians does the minute hand on a clock rotate in (a) 12 hr and (b) 3 hr?

Verified step by step guidance

Understand that the minute hand completes a full circle, which is 2π radians, in 60 minutes (1 hour).

For part (a), calculate the number of full circles the minute hand makes in 12 hours. Since 1 hour corresponds to 2π radians, multiply 12 by 2π.

For part (b), calculate the number of full circles the minute hand makes in 3 hours. Since 1 hour corresponds to 2π radians, multiply 3 by 2π.

Express the results for both (a) and (b) in terms of π to keep the answer in radians.

Review the concept that the radian measure is a way of expressing angles based on the radius of a circle, where 2π radians is equivalent to 360 degrees.

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Key Concepts

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

Radians and Degrees

Radians are a unit of angular measure used in mathematics, where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. There are 2π radians in a full circle (360 degrees), making radians a natural choice for calculations involving circular motion, such as the rotation of clock hands.

Angular Velocity

Angular velocity refers to the rate of change of angular position of an object, typically measured in radians per unit of time. For a clock, the minute hand completes one full rotation (2π radians) every 60 minutes, which allows us to calculate how many radians it rotates in any given time period by using the formula: radians = (time in minutes) × (angular velocity).

Time Conversion

To solve problems involving the rotation of clock hands, it is essential to convert time into a consistent unit. For instance, converting hours into minutes is necessary since the minute hand's movement is typically measured in minutes. This conversion allows for accurate calculations of the total radians rotated over specified time intervals.