Problem 41 - Precalculus Mathematics for Calculus

Question

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$1110^{\circ}$$

Step-by-Step Solution

Answer

The coterminal angle is \(30^{\circ}\).

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position. To find a coterminal angle between \(0^{\circ}\) and \(360^{\circ}\), you need to add or subtract full rotations (\(360^{\circ}\)) from the given angle until it falls within the desired range.

Step 2: Calculate Rotations

To find how many full \(360^{\circ}\) rotations are in \(1110^{\circ}\), divide it by \(360^{\circ}\): \[\frac{1110}{360} \approx 3.0833.\] This means there are 3 full rotations in \(1110^{\circ}\).

Step 3: Find Coterminal Angle in Desired Range

Subtract the 3 full rotations from \(1110^{\circ}\): \[1110 - 3 \times 360 = 1110 - 1080 = 30^{\circ}.\]So, \(30^{\circ}\) is the angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(1110^{\circ}\).

Key Concepts

Angle Measure

Understanding angle measure is crucial when discussing angles and their properties. An angle is formed by two rays that share a common endpoint, often referred to as the vertex. The size of the angle is essentially the measure of rotation needed to align one of these rays with the other.When we talk about measuring angles, we use degrees as our unit. A full circle, or complete rotation, measures 360 degrees, denoted as \(360^{\circ}\). This degree measure applies whether we're discussing smaller angles, which form parts of a full circle, or angles larger than \(360^{\circ}\), which include full rotations and more.For instance, when dealing with the angle of \(1110^{\circ}\), it's helpful to know that this refers to an angle that encompasses more than three full rotations around the circle, plus an additional segment. This understanding aids in simplifying angles and identifying coterminal angles.

Standard Position

The concept of standard position is essential in understanding angles and their interactions. An angle is said to be in standard position when its vertex is at the origin of a coordinate plane, and its initial side, the starting side of the angle, lies along the positive x-axis.Placing angles in standard position facilitates easier comparison and computation concerning rotation and direction. Any angle drawn in standard position will have its terminal side, the side that represents the end of the angle's rotation, extending from that origin point.Knowing that angles with different measures, such as \(1110^{\circ}\) and \(30^{\circ}\), can have the same terminal side when placed in standard position is key. These angles are coterminal because despite differing in initial rotations, they end at the same point in the circle.

Full Rotation

A full rotation corresponds to a complete circle and is represented by \(360^{\circ}\). This measure is pivotal in understanding and working with angles that exceed the typical \(0^{\circ}\) to \(360^{\circ}\) range.In practical terms, if an angle measures more than \(360^{\circ}\), it signifies several complete rotations plus an extra segment. Determining the coterminal angle—the angle that shares the same endpoint within a single full rotation—involves reducing the number of full rotations embedded in the angle.To simplify an angle like \(1110^{\circ}\), for example, we calculate the number of complete \(360^{\circ}\) turns it contains and subtract these to find an equivalent angle between \(0^{\circ}\) and \(360^{\circ}\). As shown, we subtracted three full rotations from \(1110^{\circ}\) to end up with \(30^{\circ}\), demonstrating the utility of understanding full rotations in angle measurement.