(-1, 0)

Definition

The coordinate (-1, 0) represents a specific point on the Cartesian plane, particularly located on the unit circle at an angle of $rac{3 ext{π}}{2}$ radians or 270 degrees. This point is significant in circular functions as it illustrates the connection between angles and their corresponding coordinates on the unit circle, serving as a reference for various trigonometric calculations. Understanding this coordinate helps in grasping how angles are represented and how they relate to sine, cosine, and other circular functions.

5 Must Know Facts For Your Next Test

1. The coordinate (-1, 0) corresponds to the leftmost point on the unit circle, indicating that the cosine value at this point is -1.
2. At the angle of 180 degrees (or $ ext{π}$ radians), which corresponds to (-1, 0), the sine value is 0.
3. This point is crucial for understanding periodicity in trigonometric functions, where values repeat every $2 ext{π}$ radians.
4. The coordinates on the unit circle help visualize how different angles relate to their sine and cosine values.
5. Knowing that (-1, 0) represents an angle can aid in solving equations involving trigonometric identities and transformations.

Review Questions

• How does the coordinate (-1, 0) help illustrate the relationship between angles and trigonometric functions?
  • The coordinate (-1, 0) directly illustrates how angles translate into specific points on the unit circle. This point corresponds to an angle of 180 degrees or $ ext{π}$ radians, where the cosine value is -1 and sine is 0. By recognizing this connection, it's easier to understand how angles influence the output of sine and cosine functions and how these functions behave within their periodic nature.
• Discuss the significance of understanding coordinates like (-1, 0) in solving trigonometric equations.
  • Understanding coordinates such as (-1, 0) is vital for solving trigonometric equations because it provides a clear reference point for angles. When working with equations involving sine and cosine, knowing that this coordinate corresponds to an angle can help simplify calculations and lead to solutions. It also helps in recognizing symmetries and transformations in trigonometric graphs, allowing for deeper insight into problem-solving.
• Evaluate how the properties of the unit circle, including points like (-1, 0), affect the broader understanding of circular functions.
  • The properties of the unit circle, especially points like (-1, 0), significantly enhance our understanding of circular functions by illustrating fundamental concepts such as periodicity and symmetry. This specific point shows that every angle has both sine and cosine values derived from its position on the circle. By analyzing these relationships through various points around the circle, we can gain insights into more complex functions and identities in trigonometry, reinforcing how foundational these principles are in mathematics.

Related terms

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

Sine Function: A trigonometric function that represents the y-coordinate of a point on the unit circle as a function of an angle.

Cosine Function: A trigonometric function that represents the x-coordinate of a point on the unit circle as a function of an angle.