In Exercises 35–60, find the reference angle for each angle. - 11𝜋 / 4

Verified step by step guidance

1. Identify the given angle: \(\frac{11\pi}{4}\) radians.
2. Since the angle is greater than \(2\pi\), subtract multiples of \(2\pi\) to find a coterminal angle between \$0$ and \(2\pi\). Calculate \(\frac{11\pi}{4} - 2\pi\).
3. Simplify the subtraction: \(2\pi\) can be written as \(\frac{8\pi}{4}\), so subtract \(\frac{8\pi}{4}\) from \(\frac{11\pi}{4}\) to get the coterminal angle.
4. Determine the quadrant of the coterminal angle by comparing it to \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\).
5. Find the reference angle by calculating the acute angle between the coterminal angle and the nearest x-axis (either \$0$, \(\pi\), or \(2\pi\)), using the formula for reference angles 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 dealing with multiples of π, as it helps in locating the angle on the unit circle.

Coterminal Angles

Coterminal angles differ by full rotations of 2π radians but share the same terminal side. Finding coterminal angles helps reduce large or negative angles to an equivalent angle between 0 and 2π, which is useful for determining the reference angle.