Find a positive angle less than 2𝜋 that is coterminal with 16𝜋 3

Verified step by step guidance

1. Understand that two angles are coterminal if they differ by an integer multiple of \(2\pi\). This means we can add or subtract \(2\pi\) any number of times to find coterminal angles.
2. Given the angle \(\frac{16\pi}{3}\), we want to find an angle \(\theta\) such that \(0 < \theta < 2\pi\) and \(\theta = \frac{16\pi}{3} - 2\pi k\) for some integer \(k\).
3. Express \(2\pi\) with a denominator of 3 to combine terms easily: \(2\pi = \frac{6\pi}{3}\). So, \(\theta = \frac{16\pi}{3} - k \times \frac{6\pi}{3} = \frac{16\pi - 6\pi k}{3}\).
4. Find the integer \(k\) such that \(\theta\) lies between 0 and \(2\pi\) (i.e., \(0 < \theta < \frac{6\pi}{3}\)). This involves solving inequalities for \(k\).
5. Once you find the appropriate \(k\), substitute back to get \(\theta = \frac{16\pi - 6\pi k}{3}\), which will be the positive coterminal angle less than \(2\pi\).

Key Concepts

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 interval.

Angle Measurement in Radians

Angles can be measured in radians, where 2π radians equal one full rotation (360 degrees). Understanding radian measure is essential for working with trigonometric functions and converting between angles.

Modulo Operation with Angles

Finding an angle coterminal within a specific range involves using the modulo operation with 2π. This process reduces any angle to an equivalent angle between 0 and 2π by subtracting multiples of 2π.