Transforming Polar to Cartesian Coordinates: Unraveling the Equation r^2 sin θ = 5
In the fascinating world of mathematics, the interplay between different coordinate systems is a captivating topic. Today, we'll dive into the process of transforming a polar equation into its Cartesian (rectangular) counterpart. This exploration will not only enhance your understanding of coordinate geometry but also shed light on the fascinating curves that emerge from such equations.
(1) Equation in Cartesian Coordinates
Let's consider the polar equation:
r^2 sin θ = 5
To find the equivalent Cartesian equation, we need to express the variables r and θ in terms of the Cartesian coordinates x and y.
In the Cartesian coordinate system, the relationship between the polar and Cartesian coordinates is given by:
x = r cos θ
y = r sin θ
Substituting these expressions into the polar equation, we get:
(x^2 + y^2) sin θ = 5
Dividing both sides by sin θ (assuming sin θ ≠ 0), we arrive at the Cartesian equation:
x^2 + y^2 = 5 / sin θ
This equation represents a curve in the Cartesian coordinate plane.
(2) Associated Curve
The curve associated with the polar equation r^2 sin θ = 5 is a hyperbola.
To see this, let's consider the Cartesian equation we derived earlier:
x^2 + y^2 = 5 / sin θ
This equation can be rearranged to the standard form of a hyperbola:
(x^2 / a^2) - (y^2 / b^2) = 1
where:
a = √(5 / sin θ)b = √(5 / sin θ)
The resulting curve is a hyperbola with its center at the origin (0, 0) and its major and minor axes aligned with the x and y axes, respectively.
The shape and orientation of the hyperbola will depend on the value of sin θ. When sin θ > 0, the hyperbola will be in the first and third quadrants. When sin θ < 0, the hyperbola will be in the second and fourth quadrants.
By exploring the transformation from polar to Cartesian coordinates and the associated curve, you have gained valuable insights into the interconnectedness of different coordinate systems and the rich tapestry of mathematical expressions.