(0, 1)

Definition

(0, 1) represents a specific point on the unit circle, which is a circle with a radius of one centered at the origin (0, 0) in a Cartesian coordinate system. This point corresponds to an angle of 90 degrees or $$\frac{\pi}{2}$$ radians and is significant in circular functions as it defines the maximum value of the sine function, which equals 1 at this position. Understanding this point helps connect the concepts of angles, radians, and the values of trigonometric functions on the unit circle.

5 Must Know Facts For Your Next Test

1. (0, 1) is located at the topmost point of the unit circle and directly relates to the sine function's maximum value.
2. The coordinates (0, 1) indicate that at an angle of $$\frac{\pi}{2}$$ radians, the x-coordinate is 0 and the y-coordinate is 1.
3. This point can be used to find other key values on the unit circle by rotating around the circle by specific angles.
4. The sine value at (0, 1) is always equal to 1, while the cosine value at this point is always equal to 0.
5. The coordinates (0, 1) play a crucial role in solving problems related to periodic functions and their graphs.

Review Questions

• Explain how the coordinates (0, 1) relate to the sine and cosine functions on the unit circle.
  • (0, 1) corresponds to an angle of $$\frac{\pi}{2}$$ radians on the unit circle. At this point, the sine function reaches its maximum value of 1, while the cosine function equals 0. This illustrates how specific points on the unit circle reflect values for these trigonometric functions based on their respective angles.
• Discuss why understanding the significance of (0, 1) is essential for solving trigonometric equations.
  • Knowing that (0, 1) represents the maximum sine value allows for easier solving of equations involving sine functions. When setting equations equal to values like 1, recognizing that this occurs at an angle of $$\frac{\pi}{2}$$ radians informs us about possible solutions. This connection simplifies understanding how sine behaves within its periodic nature.
• Analyze how transformations or rotations around the unit circle impact other points when starting from (0, 1).
  • Starting from (0, 1), if we rotate clockwise by various angles such as $$\frac{\pi}{4}$$ or $$\frac{\pi}{2}$$ radians, we can determine new coordinates that relate to other trigonometric values. For instance, rotating by $$\frac{\pi}{4}$$ yields approximately (0.707, 0.707), reflecting both sine and cosine values for this angle. This analysis shows how movements from (0, 1) connect to different positions on the unit circle and their corresponding trigonometric function values.

Related terms

Unit Circle: A circle with a radius of one centered at the origin of a coordinate plane, used to define trigonometric functions based on angles.

Sine Function: A trigonometric function that describes the relationship between an angle and the ratio of the length of the opposite side to the hypotenuse in a right triangle.

Radians: A unit of angular measure defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.