In Exercises 35–60, find the reference angle for each angle. 17𝜋 / 6

Verified step by step guidance

1. Identify the given angle: \(\frac{17\pi}{6}\) radians.
2. Since the angle is greater than \(2\pi\), subtract \(2\pi\) (which is \(\frac{12\pi}{6}\)) to find the equivalent angle between \(0\) and \(2\pi\): \(\frac{17\pi}{6} - 2\pi = \frac{17\pi}{6} - \frac{12\pi}{6} = \frac{5\pi}{6}\).
3. Determine the quadrant of the angle \(\frac{5\pi}{6}\). Since \(\frac{5\pi}{6}\) is between \(\frac{\pi}{2}\) and \(\pi\), it lies in the second quadrant.
4. For angles in the second quadrant, the reference angle \(\theta_r\) is calculated as \(\pi - \theta\). So, compute \(\theta_r = \pi - \frac{5\pi}{6} = \frac{\pi}{6}\).
5. Thus, the reference angle for \(\frac{17\pi}{6}\) is \(\frac{\pi}{6}\).

Key Concepts

Reference Angle

A reference angle is the acute angle formed between the terminal side of a given angle and the x-axis. It is always positive and less than or equal to 90°, used to simplify trigonometric calculations by relating any angle to a corresponding acute angle.

Angle Measurement in Radians

Angles can be measured in radians, where 2π radians equal 360 degrees. Understanding how to convert and interpret angles in radians is essential, especially for angles greater than 2π or negative angles, to find their equivalent positions on the unit circle.

Unit Circle and Coterminal Angles

The unit circle helps visualize angles and their positions. Coterminal angles differ by full rotations of 2π radians but share the same terminal side. Finding coterminal angles within one rotation simplifies determining the reference angle.