Textbook Question
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
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1. Measuring Angles
Angles in Standard Position
1:50 minutesProblem 14
Textbook Question
In Exercises 13–17, find a positive angle less than 360° or 2𝜋 that is coterminal with the given angle. -445°
Verified step by step guidance1
Understand that two angles are coterminal if they differ by a full rotation, which is 360° in degrees or \(2\pi\) in radians.
Since the given angle is \(-445^\circ\), we want to find a positive angle \(\theta\) such that \(\theta = -445^\circ + 360^\circ \times k\), where \(k\) is an integer.
Choose an integer \(k\) that makes \(\theta\) positive and less than 360°. Start by adding 360° once: \(-445^\circ + 360^\circ = -85^\circ\), which is still negative.
Add 360° again: \(-445^\circ + 2 \times 360^\circ = -445^\circ + 720^\circ = 275^\circ\), which is positive and less than 360°.
Therefore, \(275^\circ\) is a positive angle less than 360° that is coterminal with \(-445^\circ\).
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Coterminal Angles
Coterminal angles are angles that share the same initial and terminal sides but differ by full rotations of 360° or 2π radians. To find a coterminal angle, you add or subtract multiples of 360° (or 2π) until the angle lies within the desired range.
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Coterminal Angles
Angle Measurement in Degrees
Angles can be measured in degrees, where one full rotation equals 360°. Understanding how to manipulate angles in degrees, including adding or subtracting 360°, is essential for finding equivalent angles within a specified interval.
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Positive Angle Restriction
When asked to find a positive angle less than 360°, you must adjust the given angle by adding 360° repeatedly until the result is positive and less than 360°. This ensures the angle is expressed within the standard range for one full rotation.
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