POLAR COORDINATES!
An Introduction to Polar Coordinates
The polar coordinate plane is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. We've been exposed to this in math classes before, but we'll include a quick refresher on some of the basics of the coordinate system to bring everyone up to speed.
A point on the polar plane is expressed as (r, θ), where r is the distance from the fixed point (generally the origin) and θ is the angle from the fixed direction (again, generally the origin). It is fairly easy to convert between polar and Cartesian coordinates with simple trig: to find the x Cartesian coordinate from a polar coordinate, you just need to do rcos(θ): x=rcos(θ). To find the y Cartesian coordinate, the process is very similar: y=rsin(θ). This beautiful image may help:
Now, what about changing from Cartesian to polar? From the diagram above, it's fairly easy to tell that r can be found through the Pythagorean Theorem: r=√(x2+y2). This image also makes clear how to find θ, again through the use of trig. If given x and y, you can solve for θ through the following: θ=tan-1(y/x). There are at least five PatrickJMT videos on this topic, but here is one that is very brief and straightforward. Let's move on to polar coordinates in calculus now!
Finding Areas with Polar Coordinates
Finding areas in the polar plane is similar to finding areas in the cartesian plane in that integrals are used in both cases. However, the limits of integration and the equation are slightly different.
Cartesian coordinates: Polar Coordinates:
As you can see, we are no longer working with values of x and y, but instead with values of theta.
Also, the integral looks slightly different because there is a 1/2 out front and an r2 inside the integral. This is because the area of a circle is given by πr2 and we’re summing little tiny segments of different circles. However, because each part is not an entire circle, we need to put it into proportion, and we do this by multiplying πr2 by dθ/(2π), because dθ/(2π) is the ratio of the little tiny change in θ to the total number of radians in a circle. This gives us r2dθ/2, or (1/2)r2dθ. Then, since we’re summing all the little areas, we throw in a definite integral sign with limits from the initial angle θ to the final angle θ and we’re done.
In terms of finding areas once you know the equation, the hardest part is visualizing the graph. PatrickJMT does a wonderful job explaining how to find an area with polar coordinates here:
Even more complicated is finding areas between two polar curves. PatrickJMT does it again in this thrilling sequel:
Derivatives in Polar Coordinates/GeoGebra
One crucial element of understanding the mystical overlap between polar coordinates and Calculus is the process of finding dy/dx in polar coordinates. First, you must realize that x=rcos(theta) and y=rsin(theta). Just like y is often seen as a function of x, r in this case can be viewed as a function of theta. Therefore, y can really equal f(theta)sin(theta) and x can really equal f(theta)cos(theta). Taking the derivative of that, you must use product rule to get dy/d(theta) and dx/d(theta). For example, dy/d(theta) would equal f prime(theta)sin(theta)+f(theta)cos(theta). Then you replace f prime (theta) with dr/dtheta since r is the function and you put dy over dx and you get:
Another interesting and vital skill is knowing how to use Geogebra to plot polar points and functions. First, you’ll want to go into grid options and change grid type to polar. You can show the grid and you’ll start to see the magical unit circle-like grid.
You can always plot a point in polar by just entering coordinates in style (r;theta) with a semicolon separating the coordinates. To graph a polar function simply create a slider with value t. Then create point a with polar coordinates (r(t);t) where r(t) is your function of t. To make the function visible, find the locus of all points by creating a locus with point A.
If this looks complicated at first, don't worry, it is. Good luck!
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