In Exercises 35–60, find the reference angle for each angle. -13𝜋/3

Verified step by step guidance

1. Identify the given angle: \(\frac{13\pi}{3}\) radians.
2. Since the angle is greater than \(2\pi\), find its coterminal angle by subtracting multiples of \(2\pi\) until the angle lies between \(0\) and \(2\pi\). Use the formula: \(\theta_{coterminal} = \theta - 2\pi \times k\), where \(k\) is an integer.
3. Calculate the coterminal angle for \(\frac{13\pi}{3}\) by finding the appropriate \(k\) such that \(0 \leq \theta_{coterminal} < 2\pi\).
4. Determine the quadrant in which the coterminal angle lies by comparing it to \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\).
5. Find the reference angle by measuring the acute angle between the coterminal angle and the nearest x-axis (either \(0\), \(\pi\), or \(2\pi\)), using the formula depending on the quadrant.

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 an angle in the first quadrant.

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 when working with multiples of π, as it helps in locating the angle on the unit circle.

Unit Circle and Quadrants

The unit circle divides the coordinate plane into four quadrants, each affecting the sign and value of trigonometric functions. Knowing which quadrant an angle lies in helps determine the reference angle by measuring the smallest angle to the x-axis.