Problem 30 - Algebra 2

Question

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(-15^{\circ}\)

Step-by-Step Solution

Answer

Positive: \(345^{\circ}\); Negative: \(-375^{\circ}\).

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same terminal side in standard position. They can be found by adding or subtracting full rotations of \(360^{\circ}\). This means that if you have an angle \(\theta\), angles coterminal with \(\theta\) can be expressed as \(\theta + 360^{\circ}k\) where \(k\) is an integer.

Step 2: Finding a Positive Coterminal Angle

To find a positive coterminal angle with \(-15^{\circ}\), we add \(360^{\circ}\):\[-15^{\circ} + 360^{\circ} = 345^{\circ}\].So, \(345^{\circ}\) is a positive coterminal angle.

Step 3: Finding a Negative Coterminal Angle

To find a negative coterminal angle with \(-15^{\circ}\), we subtract \(360^{\circ}\):\[-15^{\circ} - 360^{\circ} = -375^{\circ}\].Thus, \(-375^{\circ}\) is a negative coterminal angle.

Key Concepts

Angle Measurement

When dealing with angles, especially in the context of problems involving coterminal angles, it's crucial to understand the concept of angle measurement. Angles can be measured in degrees or radians, but for this discussion, we'll focus on degrees.
Degrees are a unit of measure used to describe the size of an angle. A full circle is measured as 360 degrees, which is why we often add or subtract 360 degrees to find coterminal angles.
When you measure an angle, you're essentially determining the amount of rotation from the initial side of the angle (often the positive x-axis in standard position) to the terminal side. Different angles that share a terminal side but have been rotated multiple times around the circle are referred to as coterminal, as they "end up" at the same place despite potentially starting with different angle measures.

Standard Position

Understanding the standard position of an angle is essential in the study of geometric and trigonometric concepts. An angle is said to be in standard position if its initial side is placed on the positive x-axis of a coordinate plane.
This setup makes it easier to measure angles and find coterminal angles, as the initial setup is consistent for any given problem. When the terminal side of an angle is rotated from the positive x-axis, it informs whether the rotation is clockwise or counterclockwise, impacting whether the angle is measured as positive or negative.

• A counterclockwise rotation results in a positive angle measurement.
• A clockwise rotation results in a negative angle measurement.

Integer Rotations

Integer rotations involve revolving an angle around the circle multiple times, where each complete revolution is considered a 'rotation.' This concept is key to understanding coterminal angles.
An integer rotation involves adding or subtracting multiples of 360 degrees (a full circle) to or from an existing angle. This is because 360 degrees represents a full rotation around a circle. By adjusting the number of rotations, we can determine angles that appear equivalent to the original.

• To find coterminal angles, we use the formula: \( \theta + 360^{\circ}k \), where \( \theta \) is the original angle and \( k \) is any integer.
• If \( k \) is positive, additional full rotations are added, leading to larger angles.
• If \( k \) is negative, rotations are subtracted, resulting in smaller (or more negative) angles.

Through integer rotations, we can easily convert any angle to its coterminal equivalents, simplifying many geometric and trigonometric operations.