Coterminal angles

Definition

Coterminal angles are angles that share the same initial side and terminal side but differ by full rotations. This means that if you take an angle and add or subtract multiples of 360 degrees (or $2\pi$ radians), you create an angle that is coterminal with the original angle. Understanding coterminal angles is essential for working with angle measures in degrees and radians, as it helps identify equivalent positions on a unit circle and simplifies calculations in trigonometry.

5 Must Know Facts For Your Next Test

1. Coterminal angles can be found by adding or subtracting $360^\circ$ or $2\pi$ radians from the given angle.
2. Every angle has infinitely many coterminal angles because you can keep adding or subtracting $360^\circ$ or $2\pi$ radians.
3. For example, $30^\circ$, $390^\circ$, and $-330^\circ$ are all coterminal angles because they end at the same position on the unit circle.
4. Coterminal angles help in simplifying problems in trigonometry by allowing you to work with angles in standard position between $0^\circ$ and $360^\circ$ or between $0$ and $2\pi$ radians.
5. When converting angles between degrees and radians, identifying coterminal angles can make it easier to visualize their position on the unit circle.

Review Questions

• How do you determine if two angles are coterminal? Provide an example.
  • To determine if two angles are coterminal, you can check if their difference is a multiple of $360^\circ$ or $2\pi$ radians. For example, consider the angles $45^\circ$ and $405^\circ$. The difference is $405^\circ - 45^\circ = 360^\circ$, which is a multiple of $360^\circ$. Therefore, these two angles are coterminal.
• Explain how understanding coterminal angles can simplify trigonometric problems.
  • Understanding coterminal angles is crucial for simplifying trigonometric problems because it allows you to work with angles that are within the first full rotation of the unit circle. Since trigonometric functions like sine, cosine, and tangent repeat every full rotation, using coterminal angles can reduce complex calculations. For instance, instead of calculating the sine of $720^\circ$, you can find its coterminal angle of $0^\circ$, making it easier to determine that sin($720^\circ$) = sin($0^\circ$) = 0.
• Evaluate the importance of coterminal angles in understanding periodicity in trigonometric functions.
  • Coterminal angles highlight the periodic nature of trigonometric functions, which repeat their values at regular intervals. For instance, both sin($30^\circ$) and sin($390^\circ$) yield the same result because they represent the same point on the unit circle. Recognizing coterminal angles helps reinforce that sine, cosine, and tangent functions have periods of $360^\circ$ or $2\pi$ radians. This understanding is fundamental for solving equations involving these functions and analyzing their graphs.

Related terms

reference angle: The acute angle formed by the terminal side of an angle and the x-axis, used to simplify trigonometric calculations.

unit circle:

A circle with a radius of one centered at the origin of a coordinate plane, used to define trigonometric functions.

radian:

A measure of angle defined as the ratio of the length of an arc to its radius, where one complete revolution corresponds to $2\pi$ radians.