Finding Coterminal Angle D A Detailed Solution

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In the realm of trigonometry, angles often extend beyond the familiar 0 to 360-degree range. To navigate these rotations, we introduce the concept of coterminal angles: angles that share the same initial and terminal sides. This article delves into a fascinating problem involving coterminal angles, specifically focusing on determining the measure of angle D, which lies between 0° and 360° and is coterminal with a -920° angle. Understanding coterminal angles is crucial for simplifying trigonometric calculations and visualizing angular relationships. We will explore the underlying principles, step-by-step solutions, and practical applications of coterminal angles to help you master this essential concept.

Understanding Coterminal Angles

Before we dive into the specific problem, let's solidify our understanding of coterminal angles. Coterminal angles are angles that, when drawn in standard position (with the initial side on the positive x-axis), share the same terminal side. Imagine a ray rotating around the origin; it can rotate more than 360° in either the clockwise (negative) or counterclockwise (positive) direction. Each full rotation of 360° brings the ray back to its original position, resulting in coterminal angles. This means that coterminal angles differ by multiples of 360°.

To find coterminal angles, we simply add or subtract multiples of 360° from the given angle. For instance, if we have an angle of θ, then θ + 360°n (where n is an integer) will be coterminal with θ. The integer 'n' signifies the number of full rotations, positive for counterclockwise rotations and negative for clockwise rotations. This principle allows us to find an infinite number of coterminal angles for any given angle. In this article, we will specifically focus on finding the coterminal angle within the range of 0° to 360°, which is often referred to as the principal angle.

The Significance of Principal Angles

The principal angle plays a vital role in trigonometry, particularly when evaluating trigonometric functions. Because trigonometric functions are periodic, their values repeat every 360°. Therefore, the trigonometric functions of an angle and its coterminal angles are identical. This property enables us to simplify calculations by reducing any angle to its principal angle within the 0° to 360° range. For example, sin(θ) = sin(θ + 360°n), where n is an integer. This simplifies computations and allows us to work with angles within a single revolution, making analysis and problem-solving more manageable. The concept of principal angles is also foundational for understanding the unit circle and its applications in trigonometry and beyond.

Problem Statement: Finding Angle D

Now, let's revisit the problem at hand: Angle D has a measure between 0° and 360° and is coterminal with a -920° angle. What is the measure of angle D? This problem challenges us to apply our understanding of coterminal angles to find the specific angle D within the given range. The key here is to recognize that -920° represents multiple full rotations in the clockwise direction. Our task is to add multiples of 360° to -920° until we arrive at an angle within the 0° to 360° range. This will give us the measure of angle D. This type of problem is common in trigonometry and requires a solid grasp of angular relationships and arithmetic operations.

Breaking Down the Problem

To solve this problem effectively, we need to systematically add multiples of 360° to the given angle (-920°) until we obtain an angle between 0° and 360°. We can achieve this by dividing -920 by 360 to determine the number of full rotations and then adding an appropriate multiple of 360° to bring the angle into the desired range. This process involves a combination of arithmetic and conceptual understanding of angles and rotations. The goal is to find the smallest positive coterminal angle, which corresponds to angle D.

Step-by-Step Solution

Let's walk through the solution step by step:

  1. Identify the given angle: The given angle is -920°.
  2. Determine the number of rotations: Divide -920 by 360 to find out how many full rotations are represented by this angle: -920 / 360 ≈ -2.56. This indicates that the angle represents more than two full rotations in the clockwise direction.
  3. Add multiples of 360°: Since we have a negative angle, we need to add multiples of 360° to find a coterminal angle within the range of 0° to 360°. We can start by adding 360° multiple times:
    • -920° + 360° = -560° (Still negative)
    • -560° + 360° = -200° (Still negative)
    • -200° + 360° = 160° (Positive and within the desired range)
  4. Identify the coterminal angle: We found that -920° + 360° * 3 = 160°. Therefore, 160° is a coterminal angle of -920° and lies between 0° and 360°.
  5. State the answer: The measure of angle D is 160°.

Verification

To ensure our solution is correct, we can verify that 160° is indeed coterminal with -920°. We can do this by subtracting 160° from -920° and checking if the result is a multiple of 360°: -920° - 160° = -1080°. Since -1080° = -3 * 360°, it confirms that 160° is coterminal with -920°.

Alternative Method: Using Remainders

Another approach to solving this problem involves using remainders. We can find the remainder when -920 is divided by 360. However, since we are looking for a positive coterminal angle, we need to adjust the remainder accordingly.

  1. Divide and find the remainder: Divide -920 by 360: -920 ÷ 360 = -2 with a remainder of -200.
  2. Adjust the remainder: Since the remainder is negative, we add 360° to it to get a positive coterminal angle: -200° + 360° = 160°.
  3. State the answer: The measure of angle D is 160°.

This method provides a more direct way to find the coterminal angle by focusing on the remainder after dividing by 360°.

Conclusion

In conclusion, the measure of angle D, which is coterminal with a -920° angle and lies between 0° and 360°, is 160°. We arrived at this solution by adding multiples of 360° to -920° until we obtained an angle within the desired range. We also explored an alternative method using remainders, which provided a more direct approach. Understanding coterminal angles is a fundamental aspect of trigonometry, allowing us to simplify calculations and analyze angular relationships effectively. This problem highlights the importance of mastering these concepts for success in more advanced mathematical studies.

By grasping the principles of coterminal angles and practicing problem-solving techniques, you can confidently tackle a variety of trigonometric challenges. Remember that coterminal angles are angles that share the same terminal side, and they differ by multiples of 360°. This understanding will serve as a valuable tool in your mathematical journey.

Practice Problems

To further enhance your understanding, try solving these practice problems:

  1. Find the angle between 0° and 360° that is coterminal with a 450° angle.
  2. Determine the principal angle of a -600° angle.
  3. What is the measure of the angle between 0° and 360° that is coterminal with a 1000° angle?
  4. Find the coterminal angle of -800° that lies between 0° and 360°.

By working through these problems, you will reinforce your knowledge of coterminal angles and improve your problem-solving skills in trigonometry. Happy calculating!