Problem 63 - Algebra 2

Question

Which angle, in standard position, is NOT coterminal with the others? $$\begin{array}{llll}{\text { A. } 190^{\circ}} & {\text { B. }-170^{\circ}} & {\text { C. } 190^{\circ}} & {\text { D. } 550^{\circ}}\end{array}$$

Step-by-Step Solution

Answer

There doesn't exist a non-coterminal angle in the given options.

Step 1: Analyze Given Angles

The angles given are A. 190°, B. -170°, C. 190°, and D. 550°. The angles A and C are the same so they are coterminal by definition.

Step 2: Find Coterminal Angles for B and D

For B, we add 360° to -170° to find a positive angle that is coterminal with it. \(-170° + 360° = 190°\). For D, we subtract 360° from 550° twice to find a positive angle that is less than 360°. \(550° - 360° = 190°\) and then \(190° - 360° = -170°\).

Step 3: Determine the Non-Coterminal Angle

After processing the above steps, we realize that all angles are coterminal with each other, hence there does not exist a non-coterminal angle in the given options. This might be an error in the question.

Key Concepts

Standard Position

When studying angles, especially in a coordinate system, it is essential to understand what "standard position" means. An angle is said to be in standard position if its vertex is located at the origin of the coordinate plane and its initial side lies along the positive x-axis. From this position, the angle can rotate either clockwise or counterclockwise.

• Counterclockwise rotation results in a positive angle.
• Clockwise rotation results in a negative angle.

Understanding standard position helps us easily visualize and categorize angles as they rotate about the origin. For example, if you were to consider an angle of 90°, this angle in standard position will be pointing directly upwards from the x-axis.

Positive Angle

A positive angle is one that is measured by a counterclockwise rotation from the initial side along the positive x-axis. Positive angles are commonly used because they represent the standard or typical direction of angle measurement in mathematics.

• Angles like 45°, 90°, and even 190° fall into this category when visualized in standard position.
• To convert a negative angle into a positive one, add 360° to it, ensuring the angle lies within the first rotation.

For example, if an angle is -170°, as in the exercise, adding 360° to it results in a positive angle of 190°, making it easier to work with, especially in identifying coterminal angles.

Degrees

Degrees are a unit of measurement for angles based on dividing a circle into 360 equal parts. This is an ancient system that still widely forms the basis for angle measurements in various fields.

• One complete rotation around a circle is equal to 360°.
• Half a circle or a straight angle is 180°.
• A right angle is 90°.

This measurement system allows us to easily understand and calculate angle measures. In the exercise, each angle is given in degrees, highlighting the simplicity and familiarity of this unit. Whether positive or negative, angles are typically measured in degrees, providing a standardized way to describe and compute rotational positions.

Angle Subtraction

Angle subtraction is a technique used to calculate equivalent measures within the standard 0° to 360° range. This is particularly helpful when dealing with angles that exceed a full rotation or are given as negative.

• For angles greater than 360°, subtract 360° until the angle falls within the 0° to 360° range.
• For the angle 550° in our exercise, subtract 360° to find 190°, an equivalent positive angle within the standard range.

Similarly, if a given angle is negative, adding 360° allows finding a positive equivalent. Angle subtraction ensures that even when angles appear complex, they can be converted into a more familiar context for easier interpretation and understanding of coterminality.