Problem 317 - Precalculus

Question

For the following exercises, find the angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 420^{\circ} $$

Step-by-Step Solution

Answer

The angle coterminal with \(420^{\circ}\) is \(60^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. To find an angle coterminal with a given angle, we can add or subtract full rotations (360°) until the angle falls within the desired range, which is between 0° and 360° for this exercise.

Step 2: Subtract Full Rotations

The given angle is 420°. To find an angle between 0° and 360° that is coterminal with 420°, we subtract 360°: \[ 420° - 360° = 60° \]

Step 3: Verify the Solution

Check if 60° is between 0° and 360°. Since it is, we do not need any further adjustments. This confirms that 60° is the coterminal angle.

Key Concepts

Angle Measurement

Angle measurement is a way to determine how much an angle opens or rotates. Angles are typically measured in degrees or radians, but for most everyday uses, degrees are the common currency of angle measurement.

• 1 full circle, or rotation, is 360 degrees (°).
• A degree (\(^\circ\)) is a unit of measurement of a plane angle.
• Angles can start at 0° and increase clockwise or counterclockwise.

Understanding angle measurement is essential in everything from basic geometry to advanced trigonometry. We use it to describe the orientation of figures and objects in space.
Whether it's the degree of a fold, the tilt of a ramp, or the direction on a compass, angle measurement is a critical component of describing and understanding the spatial environment around us.

Degrees

Degrees are a unit of measurement used to express the size of an angle. The notation for degrees is the degree symbol (\(^\circ\)), placed after a number. When we say 180°, we mean an angle that stretches half of a full rotation.

• There are 360 degrees in a full rotation.
• A right angle has 90 degrees.
• Acute angles are less than 90 degrees, while obtuse angles are above 90 but less than 180 degrees.

Once you grasp the concept of measuring angles in degrees, you’ll find applying it is straightforward. Just imagine a cake cut into 360 thin slices, each slice represents a degree. When working with angles, understanding how many of those slices it takes to complete the angle you see can help in visualizing and calculating other related problems, like finding coterminal angles.

Full Rotations

A full rotation refers to a complete turn or circle, which is equivalent to 360 degrees. The notion of adding or subtracting full rotations is central to understanding coterminal angles.
When you have an angle greater than 360°, it means the angle has completed one or more full rotations. For instance:

• 420° is 360° (a full rotation) plus an extra 60°.
• By subtracting 360° from 420°, you find the simpler equivalent angle of 60° within a single rotation.
• This shows how multiple rotations can end at the same terminal side as those from the initial angle.

Utilizing the concept of full rotations makes solving angle problems like finding coterminal angles easier. It allows you to translate an angle back into the first full rotation (0° to 360°), simplifying how the angle is perceived and used in various applications.