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

Verified step by step guidance

1. <Step 1: Understand the concept of a reference angle. A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. It is always positive and less than or equal to 90 degrees (or \( \frac{\pi}{2} \) radians).>
2. <Step 2: Identify the given angle. The angle provided is \( \frac{11\pi}{4} \).>
3. <Step 3: Convert the angle to a standard position if necessary. Since \( \frac{11\pi}{4} \) is greater than \( 2\pi \), subtract \( 2\pi \) (or \( \frac{8\pi}{4} \)) from \( \frac{11\pi}{4} \) to find an equivalent angle within the first full rotation.>
4. <Step 4: Determine the quadrant in which the angle lies. After finding the equivalent angle, determine which quadrant it is in by comparing it to \( \pi \) and \( \frac{3\pi}{2} \).>
5. <Step 5: Calculate the reference angle. Depending on the quadrant, subtract the equivalent angle from \( \pi \) or \( 2\pi \) to find the reference angle.>

Key Concepts

Reference Angle

The reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. It is always measured as a positive angle and is typically between 0 and π/2 radians (0° and 90°). For angles greater than 2π or negative angles, the reference angle helps simplify trigonometric calculations by relating them to angles within the first quadrant.

Angle Measurement

Angles can be measured in degrees or radians, with radians being the standard unit in trigonometry. One full rotation (360°) is equivalent to 2π radians. Understanding how to convert between these two units is essential for finding reference angles, especially when dealing with angles expressed in radians, such as 11π/4.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each corresponding to specific ranges of angle measures. The first quadrant contains angles from 0 to π/2, the second from π/2 to π, the third from π to 3π/2, and the fourth from 3π/2 to 2π. Knowing which quadrant an angle lies in helps determine its reference angle and the sign of its trigonometric functions.