Polar Coordinate System

By Esha Dupuguntla

This website will help you better understand everything about the Polar Coordinate System, and what the difference is between the Polar and Rectangular System.

What is the Polar Coordinate System?

The Polar Coordinate System is a type of Coordinate system that consists of polar coordinates, AKA coordinates that look like this: (r, θ). It allows us to think of a point on a plane as a distance r from the origin and an angle θ from an initial axis. In this system, the origin is referred to as the pole, and the initial axis is called the polar axis.

What is the Rectangular Coordinate System?

The Rectangular Coordinate System, also known as the Cartesian Coordinate System, consists of rectangular coordinates that look like this: (x,y). We use the Cartesian plane to plot points and lines, and visualize various algebraic relationships. The Rectangular Coordinate System has a horizontal x axis and a vertical y axis. To plot a point you start at the origin, travel a horizontal x distance using the x axis, then travel along the y axis a distance of y to get to your point (x,y).

Coordinate Conversion

To visualize the relationship between polar and rectangular coordinates, imagine that the polar axis is the same as the positive x axis and that the pole is the same as the origin. Now, visualize a point (x,y) on a circle with radius r. Looking at the image below, you can see that x,y, and r all form a right triangle in which r is the hypotenuse. Using the Pythagorean Theorem, we can form the equation x^2+y^2=r^2. Basic trigonometry is used to establish the relationship between polar and rectangular coordinates.

Formulas for Coordinate Conversion:

1. x = rcosθ; y = rsinθ

2. tanθ = y/x; r^2=x^2 + y^2

Polar to Rectangular Conversion:

Question: Convert this point into rectangular coordinates: (2, 𝛑)

Solution: First, use the formula x=rcosθ to find the rectangular x coordinate: x = rcosθ = 2cos𝛑 = -2

Now you have the x coordinate. Then, use y = rsinθ as such: y = rsinθ = 2sin𝛑 = 0

That's it! You have converted (2, 𝛑) into the rectangular coordinate (-2,0). But how do you convert from rectangular to polar? Let's find out!

Rectangular to Polar Conversion:

Question: Convert this point into polar coordinates: (-1,1)

Solution: Use tanθ = y/x to solve for θ: tanθ = 1/-1 = -1

Now, use inverse tangent to solve for theta: θ = 3𝛑/4 or 7𝛑/4. Eliminate 7𝛑/4 because we want the angle to be in the second quadrant since that's where our Rectangular coordinate is. So, θ = 3𝛑/4.

Now, use the equation x^2+ y^2=r^2 and solve for r: r^2=(-1^2)+(1)^2, r=√2

You have converted your coordinate from (-1,1) to (√2,3𝛑/4).

Equation Conversion

To convert equations, you can use the trig formulas given for coordinate conversion.

Polar Equations to Rectangular Form:

Question: r = 2

Solution: Using x^2+ y^2=r^2, you can convert r=2 into a rectangular equation as such: r =2 => r^2 = 2^2 = x^2+y^2=2^2

Rectangular to Polar Equations:

Question: y = x^2

Solution: Use the trig relationships already established: y = x^2 => rsinθ = (rcosθ)^2 => r=secθtanθ

You have converted your equation from y = x^2 to r=secθtanθ.