Problem 53 - Precalculus

Question

For the following exercises, find the angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal to the given angle. $$ 1400^{\circ} $$

Step-by-Step Solution

Answer

320° is coterminal with 1400°.

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same terminal side when the angles are in standard position. They differ by full rotations, meaning they differ by multiples of 360°. Our task is to find an angle between 0° and 360° that is coterminal with 1400°.

Step 2: Calculate Full Rotations

First, determine how many full rotations are within 1400° by dividing it by 360°. Calculate \( \frac{1400}{360} \approx 3.8889 \). This means there are 3 full rotations and a fraction of a rotation left over.

Step 3: Find Equivalent Angle

To find the angle within 0° and 360° that is coterminal with 1400°, subtract 3 full rotations from 1400°. Calculate \( 1400 - (3 \times 360) = 1400 - 1080 = 320 \).

Step 4: Ensure Angle is Within Range

The angle of 320° is already in the desired range between 0° and 360°. Thus, it is the coterminal angle.

Key Concepts

Angle Measurement

Angles can be measured in several ways, but the most common units are degrees. An angle measures the amount of rotation between two intersecting lines or rays.
For example, a full circle has a measurement of 360 degrees, representing a complete rotation around a point.
When we measure angles, we consider direction.

• A positive angle results from a counterclockwise rotation.
• A negative angle results from a clockwise rotation.

Understanding angles involves recognizing that different angles can share the same terminal side if they differ by full rotations of 360 degrees.
This concept is crucial when working with coterminal angles, as it allows us to find angles that "land" on the same position as others.

Standard Position

When we say an angle is in "Standard Position", it refers to how we describe the location of the angle on a coordinate grid. Here are the characteristics of angles in standard position:

• The vertex of the angle is located at the origin, which is the point (0,0) on the coordinate plane.
• The initial side of the angle lies along the positive x-axis.
• The terminal side of the angle can be anywhere on the coordinate plane, determined by the measurement of the angle.

Being in standard position means any identical angle shares the same initial side, allowing for comparison between various angles by acknowledging their terminal positions.
This is essential for recognizing coterminal angles since they share the same ending point on the circle after multiple rotations.

Full Rotations

Full rotations refer to a complete 360-degree turn around a circle.
This is equivalent to going all the way around a circle exactly once, landing back at the starting point.
In the context of angles, full rotations are important because:

• They help in identifying coterminal angles, which are angles that land at the same terminal side after possibly more than one rotation.
• By subtracting or adding multiples of 360 degrees, we can always relate an angle back to its coterminal counterpart within the first full rotation (0° to 360°).
• Understanding this concept allows us to "simplify" any large or negative angle into a more manageable angle while preserving its direction and magnitude.

Using full rotations, as shown in the problem, helps reduce larger angles like 1400° to a simpler equivalent angle like 320°, making it easier to work with angles in practical applications.