Y = r sin(θ)

Definition

The equation y = r sin(θ) represents the relationship between the rectangular coordinates (x, y) and the polar coordinates (r, θ) in a two-dimensional coordinate system. It defines the y-coordinate of a point in the plane given the polar radius (r) and the polar angle (θ).

5 Must Know Facts For Your Next Test

1. The equation y = r sin(θ) is used to convert between rectangular coordinates (x, y) and polar coordinates (r, θ).
2. The polar radius (r) represents the distance from the origin to the point, while the polar angle (θ) represents the angle between the positive x-axis and the line segment connecting the origin to the point.
3. The y-coordinate of a point in the plane is determined by the product of the polar radius (r) and the sine of the polar angle (θ).
4. The equation y = r sin(θ) is one of the fundamental relationships in the conversion between rectangular and polar coordinate systems.
5. Understanding the equation y = r sin(θ) is crucial for working with polar coordinates and analyzing graphs in the polar coordinate system.

Review Questions

• Explain the relationship between the rectangular coordinates (x, y) and the polar coordinates (r, θ) using the equation y = r sin(θ).
  • The equation y = r sin(θ) establishes a direct connection between the rectangular coordinates (x, y) and the polar coordinates (r, θ). The y-coordinate of a point in the plane is determined by the product of the polar radius (r) and the sine of the polar angle (θ). This relationship allows for the conversion between the two coordinate systems, enabling the representation and analysis of points, shapes, and functions in either rectangular or polar form.
• Describe how the equation y = r sin(θ) can be used to graph a point in the polar coordinate system.
  • To graph a point in the polar coordinate system using the equation y = r sin(θ), one must first determine the polar radius (r) and the polar angle (θ) of the point. The polar radius (r) represents the distance from the origin to the point, while the polar angle (θ) represents the angle between the positive x-axis and the line segment connecting the origin to the point. Substituting these values into the equation y = r sin(θ) allows you to calculate the y-coordinate of the point, which, along with the known polar radius (r), fully defines the location of the point in the polar coordinate system.
• Analyze how changes in the polar radius (r) and polar angle (θ) affect the value of y calculated using the equation y = r sin(θ).
  • $$y = r \sin(\theta)$$ The equation y = r sin(θ) demonstrates that the y-coordinate of a point in the polar coordinate system is directly proportional to the polar radius (r) and the sine of the polar angle (θ). An increase in the polar radius (r) will result in a proportional increase in the y-coordinate, while an increase in the polar angle (θ) will cause the sine function to vary, leading to changes in the y-coordinate. By understanding how these parameters affect the equation, one can predict and analyze the behavior of points, shapes, and functions represented in the polar coordinate system.

Related terms

Polar Coordinates:

A coordinate system that specifies the location of a point in a plane by a distance from a reference point (the pole) and an angle from a reference direction (the polar axis).

Polar Radius (r): The distance from the origin (pole) to a point in the plane, also known as the radial coordinate.

Polar Angle (θ): The angle between the positive x-axis and the line segment connecting the origin (pole) to the point in the plane, also known as the angular coordinate.