Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 89

Chapter 1, Problem 89

Find the absolute value of the radian measure of the angle that the second hand of a clock moves through in the given time. 3 minutes and 40 seconds

Verified step by step guidance

Understand that the second hand of a clock completes one full revolution (360 degrees or \(2\pi\) radians) in 60 seconds.

Convert the given time of 3 minutes and 40 seconds entirely into seconds: \(3 \times 60 + 40\) seconds.

Calculate the fraction of the full revolution the second hand moves through by dividing the total seconds by 60 seconds.

Multiply this fraction by the full angle in radians (\(2\pi\)) to find the radian measure of the angle moved.

Express the final answer as the absolute value of the radian measure, which will be a positive number.

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Key Concepts

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

Radian Measure of Angles

Radians are a way to measure angles based on the radius of a circle. One radian is the angle subtended by an arc equal in length to the radius. Since a full circle is 2π radians, this unit links linear and angular measurements, making it essential for problems involving circular motion.

Angular Velocity of the Second Hand

The second hand completes one full rotation (2π radians) every 60 seconds, so its angular velocity is 2π/60 radians per second. Understanding this constant rate allows calculation of the angle moved over any given time interval by multiplying angular velocity by time.

Time Conversion and Calculation

To find the angle moved, the given time (3 minutes and 40 seconds) must be converted entirely into seconds (220 seconds). Accurate time conversion is crucial for applying the angular velocity formula correctly and obtaining the precise radian measure of the angle.

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