Polar Equations and Cartesian Intersections: A Comprehensive Analysis of r 2 sin θ

Polar equations can often be converted into Cartesian coordinates to explore their geometric properties. One such interesting case is the polar equation r 2 sin θ. This article delves into the intersection of this polar equation with the line x 1, providing a detailed analysis based on various mathematical techniques.

Introduction to Polar and Cartesian Coordinates

In the polar coordinate system, a point is represented as (r, θ), where r is the radial distance from the origin and θ is the angle measured in radians from the positive x-axis. In Cartesian coordinates, the same point is represented as (x, y). The relationships between these coordinates are given by:

x r cos(θ) y r sin(θ)

Converting the Polar Equation to Cartesian Form

Given the polar equation r 2 sin θ, we can substitute the Cartesian relations to explore the shape and properties of the curve in Cartesian coordinates.

To find the points where the curve intersects the line x 1, we start by expressing r cos(θ) 1 based on the given polar equation. We substitute r 2 sin θ into this equation:

2 sin θ cos(θ) 1

Using the double-angle identity, sin(2θ) 2 sin θ cos(θ), we get:

sin(2θ) 1

For 0 ≤ θ ≤ 2π, the solutions to this equation are:

2θ π/2, 5π/2 θ π/4, 5π/4

Substituting these values back into the polar equation:

When θ π/4, r 2 sin(π/4) 2 × (1/√2) √2 When θ 5π/4, r 2 sin(5π/4) 2 × (-1/√2) -√2

Converting these to Cartesian coordinates:

(x, y) (√2 cos(π/4), √2 sin(π/4)) (1, 1) (x, y) (-√2 cos(5π/4), -√2 sin(5π/4)) (1, -1)

Therefore, the curve intersects the line x 1 at two points: (1, 1) and (1, -1).

Visualizing the Curve with Graphing Utilities

To further illustrate and verify these intersections, it is beneficial to use graphing utilities such as Desmos. By plotting the polar equation r 2 sin θ, you can observe the figure of eight pattern formed by the curve. This visualization confirms the two intersection points with the line x 1.

Conclusion

The polar equation r 2 sin θ intersects the line x 1 at two points. These intersections can be determined through a combination of algebraic manipulation and trigonometric identities. The use of polar and Cartesian coordinate systems provides a deeper understanding of the geometric properties of the curve. For further exploration, online graphing tools like Desmos offer invaluable insights and confirmations.