By now you're used to the Cartesian coordinate system, the x-y grid that we generally use to graph functions and plot points. Each point has a set of coordinates (x, y), a recipe for finding the point starting from the origin.
Aside from its convenience—axes 90˚ apart, evenly-spaced grid points—there's nothing unique about the Cartesian plane. In fact, there are an infinite number of different coordinate systems, mostly silly ones, that we could use. Though less convenient, any of them will work to represent any mathematical problem.
For example, if you're reading this section, you've probably dealt with balls rolling down ramps in physics. Recall that you redefined your coordinate system; you tilted it to make the x-axis parallel to the ramp and the y-axis perpendicular.
This helped you to resolve vectors and made the problem much easier to deal with.
But the fact is, nature couldn't care less about your coordinate system. Whatever system you choose, the ball will still roll down the ramp just the same.
Drawn below are a Cartesian grid (left) and a couple of goofy coordinate systems no one uses, but they can locate a point just the same. The same point on each square is represented with an ordered pair of coordinates in each system.
Polar coordinates, defined below, come in handy when we're describing things that are centrosymmetric (have a center of symmetry, like a circle) or that rotate in a circle, like a wheel or a spinning molecule.
In the Cartesian coordinate system, we move over (left-right) x units, and y units in the up-down direction to find our point.
In the polar system, think of a clock with one hand. We move out a distance r, sometimes called the modulus, along the hand from the origin, then rotate the hand upward (counterclockwise) by an angle θ to reach the point.
The modulus of a vector is its length.
In mathematics, modulus is another word for absolute value.
Here is an example of a polar graph →
The point (r, θ) = (3, 60˚) is plotted by moving a distance 3 to the right along the zero-degree line, then rotating that line segment by 60˚ counterclockwise to reach the point.
Any point on a plane can be located in this manner, just like with Cartesian (x, y) coordinates.
The point in the center is usually called the "pole."
Converting between polar and Cartesian coordinates is really pretty simple. We just use a little trigonometry and the Pythagorean theorem.
x and y are related to the polar angle θ through the sine and cosine functions (purple box).
The radius, r, is just the hypotenuse of a right triangle, so r^{2} = x^{2} + y^{2}.
And because tan(θ) = ^{y}/_{x}, we relate θ to x and y using the inverse tangent function.
When calculating θ using the inverse tangent function, remember that the range of tan^{-1} is restricted to [^{-π}/_{2}, ^{π}/_{2}], the first and 4^{th} quadrants, so you'll have to add or subtract 90˚ to get the correct angles in the 2^{nd} and 3^{rd} quadrants. I'll do some examples showing this below.
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We can express all kinds of functions in polar coordinates, but some are more suitable for them than others. For example, the vertical and horizontal lines in the first graph below are what we could call "native" to the rectilinear (made up of lines and 90˚ angles) Cartesian coordinate system. That's why their equations are so simple. The equation of a circle of radius R, centered at the origin, however, is x^{2} + y^{2} = R^{2} in Cartesian coordinates, but just r = R in polar coordinates. The circle is a native figure in polar coordinates.
To convert this function to polar coordinates, we begin with the conversions x = r·cos(θ) and y = r·sin(θ):
Plugging those into our linear equation gives:
Now we solve for r to get r as a function of θ:
... and that's our line in polar coordinates:
Notice that the equation for this line is more complicated in polar coordinates. Lines aren't the natural thing to plot in polar coordinates.
To convert this function to polar coordinates, we begin with the squares of the conversions x = r·cos(θ) and y = r·sin(θ):
Plugging those into our circle equation gives:
If we factor out the r^{2}, and use the Pythagorean identity on what remains
... we have
We can simplify this equation, and we need only take the positive square root, to get
Notice that this is the simplest possible equation for such a circle. Polar coordinates are great for circles.
The function is
The first step is just to substitute x = r·cos(θ) and y = r·sin(θ) for x and y:
Now do the squaring, remembering that the binomial-squared has three terms,
Gathering terms with r^{2} gives us a Pythagorean relationship that we can evaluate to 1
The simplified equation is:
which gives us the polar function:
Convert these polar functions in the form of r(θ) to Cartesian equations in the form of y(x).
Convert these Cartesian functions in the form of y(x) to polar coordinates in the form of r(θ). You might want to plot these functions on a calculator to make sure they meet your expectations.
Here are just a few examples of the graphs of functions in polar coordinates. Notice how some functions are much simpler in polar than in Cartesian coordinates. In the exercises above, you might have noticed that some polar functions are very complicated in the Cartesian world.
The graphs of the sine and cosine functions in polar coordinates are shown here. They are simply circles.
The black curve is the sine function, the red r = cos(θ).
What you don't see in these graphs is that as the polar angle, θ, is traced around through 360˚, the sin or cos circle is traced twice. If you plot the polar graph of the function r(θ) = sin^{2}(θ), you'll see a figure-8 shape, the upper loop traced as θ runs from 0˚ to 180˚, and the lower loop as the lower half between 180˚ and 360˚.
The polar graphs of r(θ) = sin^{2}(θ) and r(θ) = cos^{2}(θ) are shown below.
These heart shaped (cardioid has the same root as cardiac) figures are the graphs of functions like r = 1 - cos(θ) and r = 1 - n·sin(θ), where n is a number.
Two functions are graphed here, and it's not difficult to see what tweaking the number in front of the sine function does. The analogous cosine function has its "dimple" or loop facing to the right.
These functions are very difficult to represent in Cartesian coordinates, but very simple in polar.
Some functions are quite fanciful in polar coordinates. Flower functions like these are formed by increasing the frequency factor (4 in this case). Just as it increases the number of periods per unit length in a Cartesian graph of a trig. function, it increases the number of loops in the polar graph.
You might want to try to graph some of these polar functions on your calculator:
You can graph polar functions on your TI-84 (or similar) calculator. Put the calculator in polar mode, then enter r(θ) (the variable button will automatically switch to θ), then enter the window values, x, y and the range of θ, and graph. Sometimes a small θ step size will cause your calculator to graph a complicated function very slowly, so be careful with that.
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