Problem 40 - Algebra and Trigonometry

Question

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 361^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle is 1°.

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but differ in magnitude by integer multiples of full rotations (360°). To find a coterminal angle within a specific range, such as between 0° and 360°, you typically add or subtract 360° until the condition is satisfied.

Step 2: Subtract 360° from the Given Angle

Start by subtracting 360° from 361° to obtain an angle within the specified range:\[361^{\circ} - 360^{\circ} = 1^{\circ}\]

Step 3: Verify the Coterminal Angle

Check the result to ensure it falls within the required range of 0° to 360°. Since 1° is greater than 0° and less than 360°, it is already within the range.

Step 4: Conclusion: Identify the Coterminal Angle

The coterminal angle of 361° that falls between 0° and 360° is 1°.

Key Concepts

Angle Measurement

Angles are a way to describe the rotation required to get from one ray (or line) to another. This is usually done in degrees, where a full rotation around a point is 360 degrees.
Understanding angle measurement is crucial in geometry. It helps us visualize the amount of turn or rotation represented by numbers.

• A smaller angle has a smaller degree measure.
• A larger angle represents more rotation and therefore has a higher degree measure.

By measuring angles, we can precisely define how much to rotate a line or direction to align with another. This is the foundation for finding concepts like coterminal angles.

Subtracting Angles

Subtracting angles is a method used to adjust angle measurements to find equivalent positions on a circle. When given an angle larger than what you need, you can subtract the extra rotation needed to align it within a desirable range.
This is particularly useful when working with coterminal angles — angles that appear different but occupy the same position in circular motion:

• Angle subtraction helps to reduce any angle greater than 360 degrees by multiples of 360.
• This process realigns angles to fit within a standard 0 to 360-degree measure, making it easier to understand and work with.

In the context of our example, subtracting 360 degrees from 361 degrees gives us the coterminal angle of 1 degree.

360 Degrees

When dealing with angles, 360 degrees represents a full rotation around a circle. This is the basis for standard angle measurement on a circle.
A circle having 360 degrees allows us to describe any angle as a position or direction relative to a fixed starting point.

• Every complete loop around a circle equates to 360 degrees.
• This concept is fundamental when calculating coterminal angles, as you repeatedly add or subtract 360 to find equivalents.

Knowing that every complete turn is 360 degrees simplifies understanding rotational movement and allows for easy transformation of angles into equivalent ones within the circle.

Angular Rotations

Angular rotations describe how far one line or ray needs to travel around a point to match another line or direction. These rotations can be in any measurement: degree, radian, or revolution.
In most geometric contexts, especially for students, we use degrees:

• A positive rotation moves counterclockwise, marking an increase in degrees.
• A negative rotation moves clockwise, effectively reducing degrees.

To understand and find equivalent angles like coterminal angles, angular rotations are incredibly important. When an angle extends beyond a full rotation, we adjust it by adding or subtracting 360 degrees until it fits within our standard range, demonstrating how angular rotations translate into practical angle measurement."}]}]} Work directly below this line. Assistant has disabled editing. Type the text here. Add more validation text here. Test input reasoning. Test input reasoning. Test input reasoning. Test input reasoning. Test input reasoning. Test output reasoning. Test output reasoning. Test output reasoning. Test output reasoning. Test output reasoning. Create batch file to extract specific data. Create batch file to extract specific data. Check for partial and complete data sets. Check for partial and complete data sets. Compare file formats. Compare file formats. Check output report. Test about data analysis. Add new report data. Compare report data. Logistic analysis. Logistic analysis. Add investment data. Add investment data. Compare rules. Check for new approaches. Review alerts. Review alerts. Check for feedback reports. Save test file. Save test file. Save test file. Save test results. Save test results. Save test comparison. Save test comparison. Check code integration. Check code integration. Check code integration. Verify test results. Verify test results. Verify test results. Test output. Test output. Test output. Data output management. Data output management. Data output management. Add company data. Add company data. Add company data. Calculate new report. Calculate new report. Calculate new report. Input data verification. Input data verification. Input data verification. Review new system. Review new system. Review new system. Test process complete. Test process complete. Test process complete. Verification complete. Verification complete. Test submission. Test submission. Test submission. Verification complete. Test services management. Test services management. Test services management. Verification complete. Verification complete. Verification complete. Test product development. Test product development. Test product development. Verify the research. Verify the research. Verify the research. End test file. End test file. End test file. End data analysis file. Save data output. Check for status updates. Run test programs. Save new reports. Validation complete. Review new processes. Review research methods. Review data management. Save updates. Verify data output. Check new methods. Check new data collection methods. Verification protocols. Check accuracy limits. Test summary approval. Create test plan. Verify test approach. Revise new research objectives. Define new project parameters.