How do you plot functions and curves in polar coordinates?

1 Polar vs. Cartesian coordinates

The most common way of locating points on a plane is using Cartesian coordinates, which are based on two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). A point in Cartesian coordinates is given by a pair of numbers (x, y) that indicate how far the point is from the origin along each axis. For example, the point (3, 4) is 3 units to the right and 4 units up from the origin.

Polar coordinates, on the other hand, are based on an angle and a radius. The angle, usually denoted by theta (θ), is measured from the positive x-axis in a counterclockwise direction. The radius, usually denoted by r, is the distance from the origin to the point. A point in polar coordinates is given by a pair of numbers (r, θ) that indicate how far and in what direction the point is from the origin. For example, the point (5, π/4) is 5 units away from the origin and at a 45-degree angle from the positive x-axis.

2 Converting between polar and Cartesian coordinates

To plot functions and curves in polar coordinates, you may need to convert them to or from Cartesian coordinates. This can be done using some trigonometric formulas that relate the x and y values to the r and θ values. Here are the formulas:

x = r cos θ

y = r sin θ

r = √(x^2 + y^2)

θ = tan^-1 (y/x)

For example, to convert the Cartesian point (3, 4) to polar coordinates, you can use the formulas to find r and θ:

r = √(3^2 + 4^2) = √25 = 5

θ = tan^-1 (4/3) = 0.927 radians or 53.13 degrees

So, the polar coordinates of (3, 4) are (5, 0.927) or (5, 53.13°).

3 Plotting polar functions

A polar function is a function that relates the radius r to the angle θ. For example, r = 2 + cos θ is a polar function that describes a cardioid, a heart-shaped curve. To plot a polar function, you can use a graphing calculator, an online tool, or a software program that supports polar graphs. Alternatively, you can convert the polar function to a Cartesian function using the formulas above, and plot it on a regular xy-plane.

For example, to convert r = 2 + cos θ to a Cartesian function, you can substitute x = r cos θ and y = r sin θ and simplify:

r = 2 + cos θ

x = r cos θ

y = r sin θ

x = (2 + cos θ) cos θ

y = (2 + cos θ) sin θ

x^2 + y^2 = (2 + cos θ)^2

x^2 + y^2 = 4 + 4 cos θ + cos^2 θ

x^2 + y^2 = 4 + 4x + (x^2 + y^2)/2

x^2 + y^2 - 2x - 2 = 0

This is a Cartesian function that can be plotted on an xy-plane using standard methods.

4 Plotting polar curves

A polar curve is a curve that is defined by a set of points in polar coordinates. For example, a circle with radius 3 and center at (0, π/2) is a polar curve that can be represented by the equation r = 3 sin θ. To plot a polar curve, you can use the same methods as for plotting polar functions, or you can use a table of values to find the corresponding Cartesian points and plot them on an xy-plane.

For example, to plot the circle r = 3 sin θ, you can use a table of values like this:

θ | r | x | y

0 | 0 | 0 | 0

π/6 | 1.5 | 1.3 | 0.75

π/4 | 2.1 | 1.5 | 1.5

π/3 | 2.6 | 1.5 | 2.25

π/2 | 3 | 0 | 3

... | ... | ... | ...

You can then plot the points (x, y) on an xy-plane and connect them with a smooth curve to get the circle.

5 Tips and tricks

Plotting functions and curves in polar coordinates can be made easier and more accurate with the following tips and tricks: using radians instead of degrees; a polar grid instead of a regular grid; taking advantage of symmetry and periodicity; and using a calculator or software program to simplify calculations and conversions. Radians are more convenient for calculations and conversions, while a polar grid consists of concentric circles and radial lines that divide the plane into sectors. Furthermore, symmetry and periodicity can reduce the amount of work needed, such as plotting points for θ in [0, π] and reflecting them across the x-axis. Finally, a calculator or software program can handle complex numbers and trigonometric functions to simplify calculations and conversions.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?