In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. 25𝜋 6

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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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주요 개념

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Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but differ by full rotations of 2π radians. To find a coterminal angle, you add or subtract multiples of 2π until the angle lies within the desired range, such as between 0 and 2π.

Angle Measurement in Radians

Angles can be measured in radians, where 2π 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.

Finding Positive Angles Less Than or Equal to 2π

To find a positive angle less than or equal to 2π that is coterminal with a given angle, repeatedly subtract 2π from the angle until the result falls within the interval [0, 2π]. This process normalizes the angle to a standard position.