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Finding a Coterminal Angle Between 0° and 360°: A Comprehensive Guide

Finding coterminal angles is a fundamental concept in trigonometry. Understanding coterminal angles is crucial for simplifying trigonometric expressions and solving various problems related to angles and their properties. This comprehensive guide will delve into the intricacies of finding coterminal angles, focusing specifically on finding an angle between 0° and 360° that is coterminal with 371°. We'll explore the underlying principles, provide step-by-step solutions, and offer practical examples to solidify your understanding.

What are Coterminal Angles?

Coterminal angles are angles that share the same terminal side when drawn in standard position. Standard position means the angle's vertex is at the origin (0,0) of a coordinate plane, and one ray (the initial side) lies along the positive x-axis. The other ray (the terminal side) rotates counterclockwise for positive angles and clockwise for negative angles.

In simpler terms: Imagine a clock. If the minute hand moves 360 degrees, it returns to its original position. Any angle that ends up in the same position as another angle is its coterminal angle.

The 360° Rule: The Key to Finding Coterminal Angles

The key to finding coterminal angles lies in the fact that adding or subtracting multiples of 360° (or 2π radians in radians) to an angle doesn't change its terminal side's position. This is because a full rotation around the circle brings you back to the same point.

Therefore, if you have an angle 'θ', its coterminal angles are given by:

θ ± 360°n, where 'n' is any integer (0, ±1, ±2, ±3, ...).

Finding the Coterminal Angle of 371° Between 0° and 360°

Our objective is to find a coterminal angle for 371° that falls within the range of 0° to 360°. Since 371° is greater than 360°, we need to subtract multiples of 360° until we obtain an angle within the desired range.

Step 1: Determine the number of full rotations.

Divide 371° by 360°:

371° ÷ 360° ≈ 1.03

This tells us that 371° completes one full rotation (360°) and has an additional portion.

Step 2: Subtract the full rotations.

Subtract one full rotation (360°) from 371°:

371° - 360° = 11°

Step 3: Verify the Result.

The angle 11° is between 0° and 360°, fulfilling our requirement. Therefore, 11° is a coterminal angle of 371°.

Conclusion: The coterminal angle of 371° that lies between 0° and 360° is 11°.

Illustrative Examples: Exploring Different Scenarios

Let's explore a few more examples to solidify our understanding.

Example 1: Finding a coterminal angle for -100° between 0° and 360°.

Since -100° is negative (representing a clockwise rotation), we need to add multiples of 360°:

-100° + 360° = 260°

The coterminal angle of -100° between 0° and 360° is 260°.

Example 2: Finding a coterminal angle for 780° between 0° and 360°.

Divide 780° by 360°:

780° ÷ 360° ≈ 2.17

This means 780° completes two full rotations. Subtract two full rotations (2 * 360° = 720°) from 780°:

780° - 720° = 60°

The coterminal angle of 780° between 0° and 360° is 60°.

Example 3: Finding a coterminal angle for -450° between 0° and 360°.

We need to add multiples of 360° to -450° until we get an angle between 0° and 360°:

-450° + 360° = -90° -90° + 360° = 270°

The coterminal angle of -450° between 0° and 360° is 270°.

Applications of Coterminal Angles

Understanding coterminal angles is vital in several areas of mathematics and its applications:

• Trigonometry: Coterminal angles have the same trigonometric function values (sine, cosine, tangent, etc.). This allows simplification of trigonometric expressions and problem-solving.

• Circular Motion: In physics and engineering, analyzing circular motion frequently involves working with angles exceeding 360°. Finding coterminal angles simplifies calculations and provides a more manageable representation of the rotational position.

• Graphing Trigonometric Functions: When graphing trigonometric functions, understanding coterminal angles helps visualize the periodic nature of these functions.

Advanced Considerations: Radians and Generalization

While we've focused on degrees, coterminal angles can also be expressed in radians. The equivalent of 360° in radians is 2π. Therefore, to find a coterminal angle in radians, you add or subtract multiples of 2π.

The general formula for finding a coterminal angle 'θc' for a given angle 'θ' is:

θc = θ ± k * (2π) (for radians) θc = θ ± k * 360° (for degrees)

Where 'k' is any integer.

Conclusion: Mastering Coterminal Angles

Finding coterminal angles is a fundamental skill in trigonometry and related fields. By understanding the 360° (or 2π) rule and applying the methods outlined above, you can confidently determine coterminal angles for any given angle, ensuring accuracy and efficiency in your calculations. Remember to always check that your final answer falls within the specified range, which in this case was between 0° and 360°. This thorough understanding forms a solid foundation for tackling more advanced trigonometric problems and applications. Practice is key to mastering this concept. Work through various examples, experimenting with different angles, both positive and negative, to build confidence and proficiency in finding coterminal angles.