Problem 25 - Algebra 2 and Trigonometry

Question

In \(18-27,\) for each given angle, find a coterminal angle with a measure of \(\theta\) such that \(0 \leq \theta < 360\). $$ 540^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle is \(180^\circ\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles in standard position that share the same terminal side. To find a coterminal angle, we add or subtract multiples of \(360^\circ\) until the resulting angle falls within the desired range \(0^\circ \leq \theta < 360^\circ\).

Step 2: Subtract 360° from Given Angle

Given the angle \(540^\circ\), subtract \(360^\circ\) to find a coterminal angle.\[ 540^\circ - 360^\circ = 180^\circ \]

Step 3: Verify the Range

Check if the resulting angle \(180^\circ\) is within the specified range \(0^\circ \leq \theta < 360^\circ\). Since \(180^\circ\) is within this range, it is the required coterminal angle.

Key Concepts

Angle Measurements

When we discuss angle measurements, we're referring to the size of an angle, which is usually measured in degrees or radians. In most everyday contexts, we use degrees. Angles are formed by two rays meeting at a common endpoint known as the vertex. Understanding how to measure angles correctly is crucial, especially when working with concepts like coterminal angles. In geometry and trigonometry, angles play a significant role in defining shapes, motion, and rotation. Each full rotation around a circle is 360 degrees, which is why we often refer to angles in terms of this full circle.

• Angles can be positive or negative, depending on their direction of rotation. Positive angles typically rotate counterclockwise, while negative angles rotate clockwise.
• Coterminal angles are those that share the same terminal side after a full rotation, differing by full circles (360 degrees).

Standard Position

An angle is said to be in standard position when its vertex is at the origin of a coordinate plane and its initial side lies along the positive x-axis. This setup allows for a clear and consistent way to measure angles and compare angle sizes. The terminal side of the angle is then determined based on the measure given or required. By placing the angle in this way, it becomes easier to calculate coterminal angles. You simply add or subtract full circle rotations to determine other angles sharing the same terminal side.

• Standard position simplifies identifying angle direction and size.
• It is a foundational concept for understanding rotation and transformations in mathematics.
• It aids in visualizing and solving problems related to angles and their positions in the coordinate plane.

Degrees in a Circle

Degrees in a circle refer to how a circle encompasses a total of 360 degrees. This value arises from historical and mathematical reasoning, providing a comprehensive measure of a full rotation. Each degree represents an equal part of this complete rotation. The concept of degrees is key in measuring various angles, from tiny slivers to full rotations. When finding coterminal angles, the idea that a full circle equals 360 degrees becomes crucial. We adjust angles by whole multiples of 360 degrees to find equivalent positions that will still lay on the same terminal side.

• The 360-degree system facilitates subdivision into smaller angles, such as 30°, 45°, 90°, contributing to practical applications in geometry.
• Understanding degrees helps link angular movements to navigational and engineering contexts.
• This system supports identifying angles across disciplines and is a critical tool in disciplines like physics and astronomy.