Determining the Quadrant for Angles: -200 Degrees and -215 Degrees Explained

Understanding how angles are distributed across the four quadrants of a coordinate plane is fundamental in trigonometry and coordinate geometry. This article will guide you through determining the quadrants for angles -200 degrees and -215 degrees, providing a comprehensive explanation and offering tips on how to approach similar problems.

Understanding the Quadrants in the Coordinate Plane

The standard coordinate plane is divided into four quadrants based on the signs of the x and y coordinates:

Quadrant I: Positive x and positive y Quadrant II: Negative x and positive y Quadrant III: Negative x and negative y Quadrant IV: Positive x and negative y

Angles increase counterclockwise from the positive x-axis starting at 0 degrees, with 360 degrees completing a full circle.

Determining the Quadrant for -200 Degrees

Let's start with the angle -200 degrees. To determine its position, we can convert it to a positive angle by adding 360 degrees:

Step 1: Convert the Angle to a Positive Angle

Using the formula:

[ -200^circ 360^circ 160^circ ]

This means that -200 degrees is coterminal with 160 degrees.

Step 2: Identify the Quadrant

Referencing the ranges for each quadrant:

0 degrees to 90 degrees: First Quadrant 90 degrees to 180 degrees: Second Quadrant 180 degrees to 270 degrees: Third Quadrant 270 degrees to 360 degrees: Fourth Quadrant

Since 160 degrees falls within the range of 90 degrees to 180 degrees, it lies in the second quadrant.

Determining the Quadrant for -215 Degrees

Next, let's determine the quadrant for -215 degrees.

Step 1: Convert the Angle to a Positive Angle

Again, we'll use the formula:

[ -215^circ 360^circ 145^circ ]

This means that -215 degrees is coterminal with 145 degrees.

Step 2: Identify the Quadrant

Using the same quadrant ranges as before:

145 degrees falls within the range of 90 degrees to 180 degrees, so it lies in the second quadrant.

Additional Tips for Coterminal Angles

Coterminal angles are angles that share the same terminal side. For any angle A, the angle A 360n (where n is any integer) will be coterminal with A. This periodicity is a key concept in trigonometry and helps in solving problems related to angles and quadrants.

Example:

-200° 360 160° (n 1) -200° 720 520° (n 2) -200° 1080 880° (n 3) -200° - 360 -560° (n -1)

Similarly, for -215°:

-215° 360 145° (n 1) -215° 720 505° (n 2) -215° 1080 865° (n 3)

Conclusion

In summary, the angle -200 degrees is in the second quadrant, and -215 degrees is also in the second quadrant. By converting these angles to their positive counterparts, we can easily identify their locations in the coordinate plane. Understanding coterminal angles is crucial for solving various trigonometric problems and working with angles in a coordinate system.

Key Takeaways

Coterminal angles share the same terminal side. To determine the quadrant of a negative angle, convert it to a positive angle using 360 degrees. Angles from 90 degrees to 180 degrees lie in the second quadrant.

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