Problem 13 - Algebra 2

Question

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) coterminal with each given angle. $$ 575^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle of \(575^{\circ}\) between \(0^{\circ}\) and \(360^{\circ}\) is \(215^{\circ}\).

Step 1: Identifying the Properties

Notice that an angle is coterminal with another if they share the same terminal side but can have different rotations (positive or negative, smaller or larger than \(360^{\circ}\)). The goal is to find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(575^{\circ}\).

Step 2: Get the coterminal angle

Since the angle \(575^{\circ}\) is greater than \(360^{\circ}\), subtract \(360^{\circ}\) from it to find a coterminal angle that is within \(0^{\circ}\)-\(360^{\circ}\). Follow the formula: Angle - \(360^{\circ} \times\) (Number of rotations). Here, \(575^{\circ} - 360^{\circ} = 215^{\circ}\).

Step 3: Confirming the result

The result, \(215^{\circ}\), is between \(0^{\circ}\) and \(360^{\circ}\). Hence, it is a coterminal angle of \(575^{\circ}\) within the required range.

Key Concepts

Angle Measurement

Angle measurement is an essential concept in trigonometry, providing a way to quantify the rotation around a point. Angles can be expressed in several units such as degrees, radians, or even gradians. In most practical and educational contexts, degrees are commonly used. A full circle rotation is measured as 360 degrees.
Here's a breakdown of angle measurement concepts:

• Degrees are a unit that divides a complete circle into 360 parts.
• Radians relate the angle to the radius of a circle, where a full circle is expressed as 2π radians.
• Gradians divide a circle into 400 parts and are mainly used in surveying.

Understanding these different units of angle measurement helps in various applications, like converting between them or finding specific angle measures as demonstrated in coterminal angle calculations.

Angle Reduction

Angle reduction helps in simplifying angles to find equivalent measures usually within one full rotation, which is between 0 to 360 degrees. The process involves manipulation of the original angle by adding or subtracting full circles (360 degrees), depending on the size of the angle you start with. This method is crucial for identifying coterminal angles, as angles outside the 0 to 360 degree range can be difficult to comprehend directly.
Let's explore some key points of angle reduction:

• To reduce a positive angle greater than 360 degrees, subtract 360 until it falls within the 0 to 360 degree range.
• Negative angles are increased by adding 360 degrees until they reach a positive measure within the desired interval.
• The angle achieved through this reduction is known as a coterminal angle, meaning it shares the same terminal side on the circle.

In the case of the angle 575 degrees, by subtracting 360, we found that 215 degrees is coterminal with it. This makes visualization and trigonometric calculations easier.

Trigonometry

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. It extends beyond triangles to describe periodic phenomena and analyze wave patterns through the function of angles. A strong foundation in understanding angle measurement and manipulation is critical when learning about trigonometry.
Key components of trigonometry related to angles include:

• Understanding sine, cosine, and tangent functions, which rely heavily on angle measures.
• Utilizing the unit circle, where each point on a circle of radius 1 unit corresponds to the sine and cosine of an angle.
• Implementing the concept of coterminal angles helps in simplifying expressions involving trigonometric functions.

By utilizing coterminal angles, complex angle measures are transformed into more manageable forms, aiding in the calculation of trigonometric functions efficiently.