Problem 43 - Precalculus with Calculus Previews

Question

In Problems \(41-44\), find the angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 400^{\circ} $$

Step-by-Step Solution

Answer

The angle 400° is coterminal with 40°.

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position on the coordinate plane. Such angles differ from each other by integer multiples of full circle rotations, which are 360 degrees.

Step 2: Calculate the Difference

To find an angle coterminal with 400°, we need to find how much more than 360° this angle is. Subtract 360° from 400°: \[ 400^ ext{°} - 360^ ext{°} = 40^ ext{°}. \]

Step 3: Determine the Coterminal Angle

Since the angle measured clockwise from the positive x-axis within the 360° range is 40°, this means 400° is coterminal with 40°.

Key Concepts

Angle Measurement

Angles are a way to measure the rotation between two intersecting lines or rays emanating from a common point known as the vertex. This measurement can be represented in degrees (°), with a full circle measuring 360°.

It's important to understand how angles greater than 360° are interpreted. These angles indicate that a point has completed one or more full rotations and then moved beyond that. For example, an angle of 400° means a full rotation of 360° plus an additional 40° rotation.

This concept is crucial for finding coterminal angles, which are angles that land on the same terminal side. You can determine a coterminal angle by simply adding or subtracting multiples of 360° from the original angle.

Full Circle Rotation

A full circle rotation is synonymous with a complete turn around a circle, which measures 360°. This concept plays a vital role in understanding how additional rotations affect angle measurement.

When we mention a full circle rotation, it means that the traveling direction continues beyond the starting point after covering the whole circle. For angles greater than 360°, full circle rotation is the excess amount that continues after returning to the start point.

Any angle can be represented within the typical range of 0° to 360° by considering how the excess over 360° (or more full rotations) should be viewed. Take 400°; it includes one full circle and 40°, making it coterminal with just 40°, since 400°-360° gives us 40°.

Standard Position

In geometric and trigonometric contexts, angles are often measured in standard position. This means starting from the positive x-axis on the coordinate plane and moving counter-clockwise.

Understanding standard position aids in visualizing and comparing angles. If two angles are coterminal, they will have their sides and terminal points coinciding when placed in standard position.

By using the standard position, we can easily compute and recognize that angles like 400° and 40° end up in the same spot, confirming they are coterminal. They both measure from the positive x-axis, representing the same direction and terminal side.