In Exercises 44–48, find the reference angle for each angle.
- 11𝜋/3

Verified step by step guidance

1. First, understand that the reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. It is always between 0 and \( \frac{\pi}{2} \).
2. Since the given angle is \( \frac{11\pi}{3} \), which is greater than \( 2\pi \), we need to find its equivalent angle between 0 and \( 2\pi \) by subtracting multiples of \( 2\pi \). Use the formula: \( \theta_{equiv} = \theta - 2\pi \times k \), where \( k \) is an integer chosen so that \( \theta_{equiv} \) lies in \( [0, 2\pi) \).
3. Calculate \( k \) such that \( \frac{11\pi}{3} - 2\pi k \) is between 0 and \( 2\pi \). Since \( 2\pi = \frac{6\pi}{3} \), subtract \( 2\pi \) multiples accordingly.
4. Once you find the equivalent angle \( \theta_{equiv} \), determine which quadrant it lies in by comparing it to \( \frac{\pi}{2} \), \( \pi \), and \( \frac{3\pi}{2} \).
5. Finally, find the reference angle based on the quadrant: - Quadrant I: reference angle = \( \theta_{equiv} \) - Quadrant II: reference angle = \( \pi - \theta_{equiv} \) - Quadrant III: reference angle = \( \theta_{equiv} - \pi \) - Quadrant IV: reference angle = \( 2\pi - \theta_{equiv} \)

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 acute angle in the first quadrant.

Angle Reduction Using Coterminal Angles

Coterminal angles differ by full rotations of 2π radians (360°). To find a reference angle for large angles, first reduce the angle by subtracting multiples of 2π until it lies between 0 and 2π, making it easier to analyze its position in the unit circle.

Quadrants and Sign of Angles

The position of an angle in the coordinate plane (quadrants I-IV) determines how to calculate its reference angle. Knowing the quadrant helps identify whether to subtract the angle from π, 2π, or use the angle directly to find the acute reference angle.