Problem 37 - Precalculus Mathematics for Calculus

Question

The measures of two angles in standard position are given. Determine whether the angles are coterminal. $$155^{\circ}, \quad 875^{\circ}$$

Step-by-Step Solution

Answer

Angles 155° and 875° are coterminal.

Step 1: Understanding Coterminal Angles

Two angles are coterminal if they share the same initial and terminal sides when drawn in standard position, meaning they have the same position but may differ by a multiple of full rotations (360°). Thus, to check if two angles are coterminal, subtract the smaller angle from the larger and check if the result is a multiple of 360°.

Step 2: Compute the Difference Between Angles

Calculate the difference between the two given angles:\[ 875^{\circ} - 155^{\circ} = 720^{\circ} \]

Step 3: Check the Difference for Multiplicity of 360°

Determine if 720° is a multiple of 360° by dividing:\[ \frac{720}{360} = 2 \]Since the quotient is an integer, 720° is a multiple of 360°.

Step 4: Conclusion on Coterminality

The difference between the two angles, 720°, is a multiple of 360°, which means the angles 155° and 875° are coterminal because repeating a full circle (360°) twice brings you back to the original position.

Key Concepts

Understanding Angle Measures

Angle measures are typically expressed in degrees, which is a unit used to quantify the amount of rotation from one ray to another. Imagine standing with your arms out. If you turn from facing north to east, you'd have made a 90° turn. This helps in visualizing the direction or angle of rotation.
Angles can be positive or negative:

• A positive angle is measured in a counterclockwise rotation from the ray's initial side.
• A negative angle is measured in a clockwise rotation.

Additionally, when working with angles, we often need to identify angles that seem different in value but occupy the same geometrical position. These are called coterminal angles.

Explaining Standard Position of Angles

When we discuss angles, it's crucial to know what standard position means. An angle in standard position has its vertex at the origin of the coordinate plane.

• The initial side of the angle lies along the positive x-axis.
• The terminal side is what 'swings' out from the initial side to form the angle.

Understanding standard position is key to analyzing an angle's direction and its equivalent values, such as coterminal angles. By knowing the standard position, you can determine whether two angles that look different actually coincide.

Exploring Multiples of 360 Degrees

A complete turnaround in rotation is 360°, similar to making a full circle. So, when an angle exceeds this value, it indicates that you've circled back to the same point one or more times.
Here's why multiples of 360° matter:

• If the difference between two angles is a multiple of 360°, they are said to be coterminal.
• Mathematically, if an angle is expressed as \( n \times 360° + b \) (where b is a base angle and n is an integer), it shares its terminal side with b.

For instance, angles of 870° and 510° both reduce to a 150° base angle after removing full rotations, making them coterminal. Understanding these multiples helps in determining when angles share the same terminal side even if their measures initially differ.