Problem 37 - Algebra and Trigonometry

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

The angles 155° and 875° are coterminal.

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but may have different measurements. For two angles to be coterminal, the difference between them must be a multiple of 360°. This means if we have angles \(a\) and \(b\), then \(a - b = 360n\) where \(n\) is an integer.

Step 2: Calculate the Difference Between Angles

Calculate the difference between the two given angles; \(b = 875^{\circ}\) and \(a = 155^{\circ}\). Use the formula: \(\text{Difference} = b - a = 875^{\circ} - 155^{\circ} = 720^{\circ}\).

Step 3: Check for Multiplicity of 360°

Determine if the calculated difference is a multiple of 360°. Divide the difference by 360°: \(\frac{720^{\circ}}{360^{\circ}} = 2\). Since 720° is exactly 2 times 360°, the angles differ by a multiple of 360°.

Step 4: Conclusion on Coterminality

Since the difference between the two angles calculated in Step 2 is a multiple of 360°, it means the angles are coterminal.

Key Concepts

Angle Measurement

When discussing angles, it's important to understand how they are measured. Angles can be measured in degrees, which is a unit of measurement for describing the rotation from one ray around a common point to another ray. There are 360 degrees in a full circle. This is because historically, the circle has been divided into 360 equal parts, each part being called a degree. To put it simply, a degree measures a part of the circumference of a circle. Angles can also be measured in radians, another unit where the circle is divided into 2\(\pi\). In everyday geometry, however, degrees are most commonly used, especially when looking at angles in a standard position. A standard position angle has its vertex at the origin of a coordinate plane and its initial side on the positive x-axis. Whether the angle is small or large, understanding angle measurement helps you compare and analyze different angles effectively.

Angle Difference

Understanding how to find the difference between two angles is crucial in determining if they are coterminal. The difference provides a way to see how far apart two angles are on a full 360-degree circle. For example, if you have two angles, 155° and 875°, you need to calculate the difference between them to see if they end up with the same terminal side. This is calculated by subtracting one angle measurement from the other:

• Difference = 875° - 155° = 720°

This subtraction tells us how many degrees one angle has to "travel" to be aligned with the other angle in a standard position. Recognizing this difference is the first step in assessing whether two angles could potentially be coterminal.

Multiplicity of 360 Degrees

The key to determining if two angles are coterminal is to check the multiplicity of 360 degrees in their difference. A significant aspect of angle calculation involves understanding how the 360-degree rotation reflects in mathematical problems. This concept is rooted in the fact that after a complete rotation in a circle, you end up where you started, so adding or subtracting 360 degrees to an angle gives another angle, but that angle is coterminal with the original. To see if the angles 155° and 875° are coterminal, you check if the difference between them is a multiple of 360°. After calculating the difference as 720°, you divide this by 360°:

• 720° / 360° = 2

Since 720° is 360° multiplied by 2, it confirms that 720° is indeed a multiple of 360°. This multiplicity indicates that after performing a complete circle, plus one additional complete circle, the angles end at the same position, affirming their coterminality. Understanding this concept is essential for identifying relationships between different angles in terms of their rotation.