In Exercises 13 to 18, classify each angle by quadrant, and...

In Exercises 13 to $18,$ classify each angle by quadrant, and state the measure of the positive angle with measure less than $360^{\circ}$ that is coterminal with the given angle.
$$\alpha=-872^{\circ}$$

Key Concepts

Quadrant Classification

This concept involves determining in which of the four quadrants of a coordinate system the terminal side of an angle lies when the angle is drawn in standard position. Standard position means that the angle's vertex is at the origin and its initial side is along the positive x-axis. For angles, especially negative ones, multiple rotations may be necessary to establish their quadrant, which is useful for understanding the sign and value of trigonometric functions in that quadrant.

Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position, even though they may be measured differently due to additional rotations. They are generally found by adding or subtracting full rotations (360° in degrees) to the given angle. This concept is essential for reducing angles to a standard range, typically between 0° and 360°, which simplifies the evaluation of trigonometric functions and comparisons between angles.

Modulo Arithmetic in Angle Reduction

Modulo arithmetic is applied to angle measurement to 'wrap around' values greater than or less than a full circle. By taking an angle modulo 360°, one finds the equivalent angle within the conventional range of 0° to 360°. This technique is crucial for finding the least positive coterminal angle and for solving problems involving periodicity in trigonometry.