Determining Terminal Points for an Angle of -135 Degrees on the Unit Circle

Introduction

Angles are a fundamental concept in mathematics, particularly in trigonometry and geometry. A standard form of an angle can be specified in various ways, one common form being the angle measured in degrees. This article discusses the terminal points of an angle of -135 degrees on the unit circle. Understanding these points can help in visualizing and solving problems related to trigonometric functions.

Understanding the Unit Circle

The unit circle is a circle with a radius of one unit, centered at the origin (0,0) in the Cartesian coordinate system. Angles in standard position are measured from the positive x-axis, and the terminal side of the angle intersects the unit circle to give specific (x, y) coordinates. These coordinates are related to the trigonometric functions of the angle: sine and cosine.

Angle of -135 Degrees

Let's examine the angle of -135 degrees. This is a negative angle, indicating a clockwise rotation from the positive x-axis.

Locating the Terminal Point

To locate the terminal point of an angle, we need to understand its position on the unit circle:

The x-coordinate represents the value of the cosine of the angle. The y-coordinate represents the value of the sine of the angle.

Placing the Angle on the Unit Circle

Since -135 degrees is a negative angle, we move clockwise. -135 degrees is equivalent to turning -135 360 225 degrees. Therefore, it is the same as a 225-degree angle, which is located in the third quadrant.

Trigonometric Values

To find the exact values of the terminal coordinates of -135 degrees:

Cosine: The cosine value can be found by considering that 225 degrees is 45 degrees past 180 degrees. The cosine of an angle that is 45 degrees past 180 degrees is -cos(45°) -0.707. Sine: Similarly, the sine value is -sin(45°) -0.707.

Hence, the terminal point of an angle of -135 degrees on the unit circle is (-0.707, -0.707).

Visualizing on the Unit Circle

Note how the angle of -135 degrees aligns with the coordinates (-0.707, -0.707) on the unit circle, placing it in the third quadrant.

Applications

Understanding the terminal points of angles like -135 degrees is crucial in various applications:

Navigation: Navigators use angles to determine direction and position. Physics: In physics, angles are used to analyze and predict motion and forces. Engineering: Engineers often need to calculate angles and coordinates for design and construction.

Conclusion

The terminal points of an angle of -135 degrees are (-0.707, -0.707). By understanding the unit circle and trigonometric functions, we can effectively determine and use these points in various mathematical and real-world applications. For more detailed information on trigonometry and related concepts, refer to scholarly articles and resources on the subject.

Keywords

angle terminal coordinates, unit circle, sine cosine