A coterminal angle is an angle that has the same terminal side as another angle, but a different initial side. In other words, coterminal angles are angles that share the same position on the coordinate plane, differing only by a multiple of 360 degrees or 2\u03C0 radians.
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Coterminal angles are important in the study of trigonometry and the unit circle, as they help to identify the position of an angle on the coordinate plane.
The difference between two coterminal angles is always a multiple of 360 degrees or 2\u03C0 radians.
Coterminal angles can be positive or negative, depending on the direction of rotation from the initial side.
Finding the coterminal angle of a given angle can be useful in simplifying trigonometric expressions and solving angle-related problems.
Coterminal angles are often used in the context of periodic functions, such as sine and cosine, where the period is 360 degrees or 2\u03C0 radians.
Review Questions
Explain the relationship between coterminal angles and the unit circle.
Coterminal angles are closely linked to the unit circle, as they represent the same position on the coordinate plane. The unit circle is divided into 360 degrees (or 2\u03C0 radians), and any angle that is a multiple of 360 degrees (or 2\u03C0 radians) from the initial angle will have the same terminal side, making them coterminal. This relationship is crucial in understanding the behavior of trigonometric functions and solving problems involving angles.
Describe how to find the coterminal angle of a given angle.
To find the coterminal angle of a given angle, you can add or subtract multiples of 360 degrees (or 2\u03C0 radians) until the resulting angle is within the range of 0 to 360 degrees (or 0 to 2\u03C0 radians). This process involves determining the appropriate number of full rotations to add or subtract to the original angle to obtain the coterminal angle. The coterminal angle will have the same terminal side as the original angle, but a different initial side.
Analyze the importance of coterminal angles in the context of periodic functions, such as sine and cosine.
Coterminal angles are crucial in the study of periodic functions like sine and cosine, as these functions repeat their values every 360 degrees (or 2\u03C0 radians). This means that any two angles that are coterminal will have the same sine and cosine values, as they represent the same position on the unit circle. Understanding coterminal angles allows for simplifying trigonometric expressions, solving angle-related problems, and analyzing the behavior of periodic functions more effectively.
Related terms
Standard Position: The standard position of an angle is when the initial side of the angle lies on the positive x-axis and the terminal side is in the first quadrant of the coordinate plane.
Reference Angle: The reference angle is the acute angle formed between the terminal side of an angle and the nearest coordinate axis.
Angle Addition: The process of adding two angles together, where the resulting angle has a terminal side that is the sum of the terminal sides of the original angles.