Problem 75 - Algebra 2

Question

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 395^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle within \(0^{\circ}\) and \(360^{\circ}\) to the angle \(395^{\circ}\) is \(35^{\circ}\).

Step 1: Convert The Given Angle to Coterminal Angle

To find an angle between \(0^{\circ}\) and \(360^{\circ}\) coterminal with \(395^{\circ}\), subtract \(360^{\circ}\) from the given angle. This is because one complete revolution on a circle is \(360^{\circ}\) so by subtracting \(360^{\circ}\) from the given angle, an equivalent angle which lies in the first complete revolution (0 to \(360^{\circ}\)) can be determined.\n\nTherefore, the calculation is as follows: \n\(395^{\circ} - 360^{\circ} = 35^{\circ}\)

Step 2: Confirm The Result is within the Correct Range

Verify that the result obtained is between \(0^{\circ}\) and \(360^{\circ}\). This range is the requirement as set by the problem. Our result is \(35^{\circ}\), which falls within the range of \(0^{\circ}\) to \(360^{\circ}\). Hence, \(35^{\circ}\) is the angle we are looking for.

Key Concepts

Angle Measurement

When we talk about angle measurement, it's crucial to understand the basics of how angles are formed and measured, especially in degrees. Angles are a way of representing the amount of rotation from one line or plane to another.
This is measured in degrees (°), and a full revolution around a circle is 360 degrees.

• A straight angle is 180°.
• A right angle, which forms a square corner, is 90°.
• And a full circle, as previously mentioned, is 360°.

Understanding these basic angle measurements helps in determining relationships between different angles, such as coterminal angles, which are essentially angles that differ by full rotations (360° or multiples of it). By subtracting 360° from any given angle, you can find an equivalent angle that is within the standard range of a circle, i.e., between 0° and 360°.

Trigonometry

Trigonometry is a branch of mathematics that deals with the study of triangles, and more importantly, the relationships between their angles and sides. However, in the context of angles and their measurement, trigonometry extends beyond triangles to include circular functions.
These functions help explain how angles can be analyzed through their sine, cosine, and tangent relationships.

• Sine and cosine functions particularly highlight the circular nature of angle measures, as they relate directly to the coordinates on a unit circle.
• Using trigonometric functions can help us understand transformations or alterations in angles, such as when finding coterminal angles.

Understanding trigonometry is key to grasping more complex angle relationships, including the determination of angles that are coterminal, as these concepts are often used in various real-world applications.

Circle Revolution

Circle revolution is a concept that might initially seem complex, but it's quite straightforward. It refers to the complete rotation around a circle, which we know totals 360 degrees.
This fundamental property allows us to simplify angle calculations and find coterminal angles.

• When an angle extends beyond 360°, it's simply an indication that multiple complete circle revolutions have occurred.
• By subtracting multiples of 360° from such angles, we can deduce an equivalent angle that is easier to work with.

Coterminal angles are typically examined in a circle because a revolution of 360° from a given point will land you at the same position on the circle.
Understanding this concept aids in visualizing angles and their respective positions, making tasks like determining the coterminal angle of 395° very manageable. In this case, subtracting 360° leaves us with 35°, a straightforward measure within the initial full circle.