Converting Polar Equation r = -4csc(theta) to Cartesian Equation

In this article, we will learn how to convert the polar equation r = -4csc(theta) into a Cartesian equation. This is a valuable skill for high school mathematics students, as it helps them understand the relationship between polar and Cartesian coordinate systems and apply it to various mathematical problems.

Step 1: Understand the Polar Equation

The given polar equation is r = -4csc(theta), where r represents the radial distance from the origin, and theta represents the angle between the positive x-axis and the line connecting the origin to the point.

The negative sign in front of the csc(theta) term indicates that the point lies in the second or third quadrant of the coordinate plane.

Step 2: Convert the Polar Equation to Cartesian Coordinates

To convert the polar equation to a Cartesian equation, we need to express the x and y coordinates in terms of r and theta.

The Cartesian coordinates (x, y) can be expressed in terms of the polar coordinates (r, theta) using the following formulas:

x = r * cos(theta) y = r * sin(theta)

Substituting the given polar equation r = -4csc(theta) into these formulas, we get:

x = -4csc(theta) * cos(theta) y = -4csc(theta) * sin(theta)

Step 3: Simplify the Cartesian Equation

To simplify the Cartesian equation, we can use the trigonometric identities:

csc(theta) = 1 / sin(theta) cos(theta) / sin(theta) = cot(theta)

Substituting these identities, we get:

x = -4 * cot(theta) y = -4 / sin(theta)

Therefore, the Cartesian equation corresponding to the polar equation r = -4csc(theta) is:

(x^2 + y^2) = 16 * cot^2(theta) + 16 / sin^2(theta)

This Cartesian equation represents a hyperbola in the second and third quadrants of the coordinate plane.

By following these steps, you can convert any polar equation into its corresponding Cartesian equation. This skill is essential for solving various mathematical problems that involve the transformation between polar and Cartesian coordinate systems.