Multiple Choice
Which of the following expressions finds the measure of an angle that is coterminal with a angle?
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1. Measuring Angles
Coterminal Angles
2:07 minutesProblem 1.1.66
Textbook Question
In Exercises 57β70, find a positive angle less than or that is coterminal with the given angle. 25π 6
Verified step by step guidance1
Understand that two angles are coterminal if they differ by an integer multiple of \(2\pi\). This means we can add or subtract \(2\pi\) to the given angle to find coterminal angles.
Given the angle \(\frac{25\pi}{6}\), we want to find a positive angle \(\theta\) such that \(0 \leq \theta < 2\pi\) and \(\theta\) is coterminal with \(\frac{25\pi}{6}\).
To do this, subtract multiples of \(2\pi\) from \(\frac{25\pi}{6}\) until the result lies between \$0$ and \(2\pi\). Since \(2\pi = \frac{12\pi}{6}\), subtract \(\frac{12\pi}{6}\) as many times as needed.
Calculate \(\frac{25\pi}{6} - 2\pi = \frac{25\pi}{6} - \frac{12\pi}{6} = \frac{13\pi}{6}\). Check if \(\frac{13\pi}{6}\) is less than \(2\pi\); if not, subtract \(2\pi\) again.
Repeat the subtraction: \(\frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6}\). Since \(\frac{\pi}{6}\) is between \$0$ and \(2\pi\), this is the positive coterminal angle less than or equal to \(2\pi\).
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2mKey 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 2Ο radians. To find a coterminal angle, you add or subtract multiples of 2Ο until the angle lies within the desired range, such as between 0 and 2Ο.
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Coterminal Angles
Angle Measurement in Radians
Angles can be measured in radians, where 2Ο radians equal one full rotation (360 degrees). Understanding how to convert and manipulate angles in radians is essential for solving problems involving coterminal angles and angle normalization.
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Finding Positive Angles Less Than or Equal to 2Ο
To find a positive angle less than or equal to 2Ο that is coterminal with a given angle, repeatedly subtract 2Ο from the angle until the result falls within the interval [0, 2Ο]. This process normalizes the angle to a standard position.
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