Problem 12 - Algebra 2

Question

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) coterminal with each given angle. $$ 385^{\circ} $$

Step-by-Step Solution

Answer

The measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) coterminal with \(385^{\circ}\) is \(25^{\circ}\).

Step 1 - Identify the given angle

The given angle is \(385^{\circ}\).

Step 2 - Find the measure of given angle, less than \(360^{\circ}\)

To find the measure of the angle equivalent to \(385^{\circ}\) but less than \(360^{\circ}\), subtract \(360^{\circ}\) from \(385^{\circ}\). The result is \(385^{\circ} - 360^{\circ} = 25^{\circ}\).

Step 3 - Confirm the angle is between \(0^{\circ}\) and \(360^{\circ}\)

Verify that the calculated angle, \(25^{\circ}\), falls within the bounds of \(0^{\circ}\) and \(360^{\circ}\). Since it does, \(25^{\circ}\) is the angle coterminal with \(385^{\circ}\).

Key Concepts

Angle Measurement

Angles are fundamental in the study of geometry, and understanding their measurement is crucial. Angles are typically measured in degrees, symbolized by the degree sign "°". A complete circle is divided into 360 parts, each part representing one degree. This means an angle showing a full circle is 360°. Measuring angles allows us to understand their size and how they relate to other angles.

In different scenarios, you may encounter angles larger than 360°. These are not uncommon and can result from multiple rotations around a circular path. Rather than seeing these as confusing, it's helpful to comprehend that any angle can be simplified by determining a coterminal angle—an angle that shares the same initial and terminal sides. Coterminal angles help us see angles within the standard range of 0° to 360°.

Angle Subtraction

When dealing with angles beyond the limit of 360°, angle subtraction becomes a handy tool. Subtraction clears the clutter by simplifying the angle within the typical range. For instance, take the angle 385°, which exceeds 360° by 25°.

To find a corresponding coterminal angle between 0° and 360°, simply subtract 360° from the given angle:

• Given angle: 385°
• Subtract 360°: 385° - 360° = 25°

Now we have simplified the angle to 25°, neatly returning it to a more familiar range within a singular rotation. This method doesn’t alter the angle's original direction or end point; it merely expresses it in a more manageable form.

Degrees in a Circle

Understanding degrees in a circle is essential for grasping how angles work. A circle measures 360 degrees, and each degree signifies a fractional part of that circle. This 360° framework allows for easy manipulation and identification of angles.

This foundational concept is why coterminal angles often involve such operations as subtraction or addition of 360°. For any angle beyond 360°, such as 385°, subtracting 360° effectively reduces the angle to its simplest form.

This reduction aligns with our standard measuring practice, ensuring angles are always discussed within the complete rotation of a circle. It's an effective way to consistently discuss and compare angles, regardless of their size or number of rotations.