Which statement about a unit circle is true?

Which statement about a unit circle is true?

• A. It has a radius of any length
• B. It can be located anywhere on the coordinate plane
• C. It is always centered at (1, 0)
• D. It always has a radius of 1

Answer: (D)

A unit circle is defined specifically by having a radius of exactly 1 unit. This definition is fundamental to the concept of the unit circle in mathematics, particularly in trigonometry and geometry. The significance of the radius being 1 is that it allows for the simplification of trigonometric functions, as the coordinates of any point on the unit circle can be represented as (cosθ, sinθ), where θ represents the angle measured in radians from the positive x-axis. The other statements provided do not align with the definition of a unit circle. For instance, while a circle can be located anywhere on the coordinate plane and can have varying radii, the unit circle is specifically characterized by its radius of 1 and does not refer to locations or arbitrary lengths. The center of a unit circle is not restricted to any specific point like (1, 0); it can be centered at the origin (0, 0) for the most common representation, but that does not preclude other placements in the coordinate system. This clear definition of the radius and properties of the unit circle is essential for graphing and understanding the relationships within trigonometric functions.

A unit circle is defined specifically by having a radius of exactly 1 unit. This definition is fundamental to the concept of the unit circle in mathematics, particularly in trigonometry and geometry. The significance of the radius being 1 is that it allows for the simplification of trigonometric functions, as the coordinates of any point on the unit circle can be represented as (cosθ, sinθ), where θ represents the angle measured in radians from the positive x-axis.

The other statements provided do not align with the definition of a unit circle. For instance, while a circle can be located anywhere on the coordinate plane and can have varying radii, the unit circle is specifically characterized by its radius of 1 and does not refer to locations or arbitrary lengths. The center of a unit circle is not restricted to any specific point like (1, 0); it can be centered at the origin (0, 0) for the most common representation, but that does not preclude other placements in the coordinate system. This clear definition of the radius and properties of the unit circle is essential for graphing and understanding the relationships within trigonometric functions.