Unit Circle

Definition

The unit circle is a circle with a radius of 1 unit, centered at the origin (0, 0) of the coordinate plane. It is a fundamental concept in trigonometry, as it provides a way to define and visualize the trigonometric functions.

5 Must Know Facts For Your Next Test

1. The unit circle has a radius of 1 unit and is centered at the origin (0, 0) of the coordinate plane.
2. Angles on the unit circle are measured in radians, where one full revolution around the circle corresponds to $2\pi$ radians.
3. The coordinates of points on the unit circle are given by $(\cos(\theta), \sin(\theta))$, where $\theta$ is the angle in radians.
4. The trigonometric functions (sine, cosine, tangent, etc.) can be defined using the coordinates of points on the unit circle.
5. The unit circle is used to study the periodic nature of trigonometric functions and their properties, such as their domain, range, and transformations.

Review Questions

• Explain how the unit circle is used to define the trigonometric functions.
  • The unit circle provides a way to define the trigonometric functions, such as sine and cosine, using the coordinates of points on the circle. Specifically, the x-coordinate of a point on the unit circle corresponds to the cosine of the angle, and the y-coordinate corresponds to the sine of the angle. This allows the trigonometric functions to be defined in terms of the coordinates of points on the unit circle, which is a fundamental concept in trigonometry.
• Describe how the unit circle is used to measure angles in radians.
  • The unit circle is used to measure angles in radians, which is the ratio of the arc length to the radius of the circle. One full revolution around the unit circle corresponds to $2\pi$ radians. This means that the angle of $\theta$ radians on the unit circle can be represented by the point $(\cos(\theta), \sin(\theta))$. Understanding the relationship between angles and the unit circle is crucial for working with trigonometric functions and their applications.
• Analyze how the properties of the unit circle are used to study the periodic nature of trigonometric functions.
  • The unit circle is a powerful tool for understanding the periodic nature of trigonometric functions. Because the coordinates of points on the unit circle repeat every $2\pi$ radians, the trigonometric functions defined using these coordinates also exhibit a periodic behavior with a period of $2\pi$. This allows for the study of the domain, range, and transformations of trigonometric functions, as well as their applications in various fields, such as engineering, physics, and mathematics.

Related terms

Angle Measure: The unit circle is used to measure angles in radians, which is the ratio of the arc length to the radius of the circle.

Trigonometric Functions:

The trigonometric functions (sine, cosine, tangent, etc.) can be defined using the coordinates of points on the unit circle, which allows for the study of their properties and relationships.

Coordinate Plane: The unit circle is typically drawn on the coordinate plane, with the x-axis representing the horizontal axis and the y-axis representing the vertical axis.