(1, 0)

Definition

(1, 0) is a coordinate point on the Cartesian plane that represents a specific location on the unit circle, where the x-coordinate is 1 and the y-coordinate is 0. This point is significant because it corresponds to an angle of 0 radians (or 0 degrees) in trigonometric terms and serves as a reference point for defining the sine and cosine functions. Additionally, it marks the starting position for measuring angles in both the unit circle and circular functions, allowing for easy determination of other key points on the circle.

5 Must Know Facts For Your Next Test

1. (1, 0) corresponds to the cosine of 0 radians, which equals 1, and the sine of 0 radians, which equals 0.
2. This point is also where the terminal side of an angle that measures 0 radians intersects with the unit circle.
3. In terms of transformations, (1, 0) can be translated by adding or subtracting values from its coordinates to produce new points on or off the unit circle.
4. The position of (1, 0) makes it fundamental for understanding periodicity in trigonometric functions as they repeat every $2\pi$ radians.
5. (1, 0) plays a key role in determining values for other angles on the unit circle using symmetry properties.

Review Questions

• How does the point (1, 0) relate to the definitions of sine and cosine in trigonometry?
  • (1, 0) serves as a fundamental reference point on the unit circle where the cosine of 0 radians equals 1 and the sine equals 0. This relationship establishes how cosine represents the x-coordinate and sine represents the y-coordinate for any angle measured from this point. Understanding this connection is crucial for grasping how all other points on the unit circle derive their sine and cosine values based on their angular measurements.
• Discuss how (1, 0) is used in determining other key points on the unit circle as angles increase.
  • (1, 0) acts as the starting reference point for measuring angles in standard position. As angles increase from this position, we can find other points by applying cosine and sine values corresponding to those angles. For example, at $\frac{\pi}{2}$ radians, we move counterclockwise to (0, 1), which allows us to visualize how different angles relate back to (1, 0) using symmetry and periodicity on the unit circle.
• Evaluate the significance of (1, 0) in understanding periodic functions and their transformations within trigonometry.
  • (1, 0) is pivotal in understanding periodic functions because it marks where trigonometric functions start their cycles. Since sine and cosine are periodic with a period of $2\pi$, knowing that they return to (1, 0) after every complete cycle helps predict their behavior. Moreover, transformations like vertical shifts or horizontal translations can be applied to this point to derive new functions or analyze graph changes effectively. This deepens comprehension of how transformations affect all related functions derived from this base point.

Related terms

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

Sine Function: A trigonometric function that gives the y-coordinate of a point on the unit circle corresponding to a given angle.

Cosine Function: A trigonometric function that gives the x-coordinate of a point on the unit circle corresponding to a given angle.