In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. - 38𝜋/9

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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 \(2\pi\) as many times as needed to find an equivalent angle within the desired range.

Given the angle \(\frac{38\pi}{9}\), we want to find a positive angle \(\theta\) such that \(0 \leq \theta < 2\pi\) and \(\theta\) is coterminal with \(\frac{38\pi}{9}\).

Express the problem as: find an integer \(k\) such that \(\theta = \frac{38\pi}{9} - 2\pi k\) and \(0 \leq \theta < 2\pi\).

Rewrite \(2\pi\) with a denominator of 9 to combine terms easily: \(2\pi = \frac{18\pi}{9}\). Then, \(\theta = \frac{38\pi}{9} - \frac{18\pi}{9}k = \frac{(38 - 18k)\pi}{9}\).

Choose the integer \(k\) so that \(\theta\) lies between \(0\) and \(2\pi\) (i.e., \(0 \leq \frac{(38 - 18k)\pi}{9} < 2\pi\)). Solve this inequality to find the appropriate \(k\), then substitute back to find \(\theta\).

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Key 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 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 work with radians, including converting between improper fractions and mixed numbers, is essential for manipulating and simplifying angle expressions.

Modulo Operation with Angles

Finding a positive angle less than or equal to 2π that is coterminal with a given angle involves using the modulo operation with 2π. This process effectively reduces the angle by subtracting multiples of 2π until it falls within the standard interval [0, 2π).

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