Problem 28 - Algebra 2

Question

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(225^{\circ}\)

Step-by-Step Solution

Answer

The positive coterminal angle is \(585^{\circ}\) and the negative coterminal angle is \(-135^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side. To find coterminal angles, we can add or subtract full rotations of the circle, which is usually in multiples of \(360^{\circ}\).

Step 2: Finding the Positive Coterminal Angle

To find a positive coterminal angle, we add \(360^{\circ}\) to the given angle. So for \(225^{\circ}\): \[225^{\circ} + 360^{\circ} = 585^{\circ}\]Therefore, \(585^{\circ}\) is a positive coterminal angle for \(225^{\circ}\).

Step 3: Finding the Negative Coterminal Angle

To find a negative coterminal angle, we subtract \(360^{\circ}\) from the given angle. So for \(225^{\circ}\): \[225^{\circ} - 360^{\circ} = -135^{\circ}\]Hence, \(-135^{\circ}\) is a negative coterminal angle for \(225^{\circ}\).

Key Concepts

Angle Measure

Angle measures are a way to describe how much a ray has rotated around its starting point, known as the vertex, from a fixed starting position, usually the positive x-axis. This measure is typically in degrees, with a full circle equating to \(360^{\circ}\).
An angle can be visualized as the amount of turn between two straight paths. Consider a standard clock with the hands pointing to 12 o'clock, representing \(0^{\circ}\). If you rotate the minute hand around the face of the clock once back to 12 o'clock, this corresponds to \(360^{\circ}\) because it made a full turn.
More specific measurements, like \(45^{\circ}\) or \(225^{\circ}\), indicate smaller sections of rotation corresponding to the clock's hands moving a smaller distance.

Positive Angle

Positive angles are generated when a rotation is made in a counterclockwise direction from the start position. This direction is standard for measuring angles and is defined as positive.
For example, if you have a starting angle of \(225^{\circ}\) and you want to find a positive coterminal angle, you simply add \(360^{\circ}\). This is because one complete counterclockwise rotation doesn't change the terminal side's overall position, i.e.

• \(225^{\circ} + 360^{\circ} = 585^{\circ}\)
  

So, \(585^{\circ}\) lands in the same position, just achieved through additional rotation, making it a positive coterminal angle.

Negative Angle

Negative angles come into play when the rotation around the circle is clockwise. This means that you're essentially going backwards compared to the usual direction of measuring angles.
To find a negative coterminal angle for a given angle, you subtract a full rotation or \(360^{\circ}\) from it. This tells us how far backward we've turned from the starting point.

• For \(225^{\circ}\), subtracting \(360^{\circ}\) gives you \(225^{\circ} - 360^{\circ} = -135^{\circ}\)

This means \(-135^{\circ}\) is simply the angle you get by rotating \(225^{\circ}\) in the opposite (clockwise) direction.

Full Rotation

A full rotation is the same as making one complete circle, and it equates to \(360^{\circ}\). It represents returning to the initial direction after traveling around the circle.
This concept is crucial for understanding coterminal angles because we add or subtract \(360^{\circ}\) to find another angle with the same terminal side position.
Think of it as walking around a circular track: starting from one point and trudging the entire circle back to where you began. Whether you take an extra lap by adding \(360^{\circ}\) or take a step back by subtracting \(360^{\circ}\), you're still at the same start or stop point position.
For angles, using full rotations helps to find other angles that appear to stop at the same place, like \(225^{\circ}\), \(585^{\circ}\), and \(-135^{\circ}\). These measures exhibit the same terminal position due to full rotation addition or subtraction.