Introduction to Polar Coordinates

Polar coordinates graph image: Wikimedia Commons CC-SA 3.0 by Pbroks13
r = 2 sin (4*theta) Image – CC-SA 3.0 by Pbroks13
Frequently used in analytical geometry is the standard 2-dimensional x, y coordinate system called the Cartesian coordinate system (named after famous mathematician, René Descartes). It’s time to branch out to a different system, the polar coordinates system. In fact, there are any of a number of ways of locating points in 2-D space.

Conversion from Cartesian Coordinates

The polar coordinates system utilizes an angle and a radius. It is relatively simple to change from the x-y system to an r-θ system. Drawing a circle centered at the origin on an x-y plane and then drawing a right triangle with the radius of the circle equaling r, then by definition, the side adjacent to the angle divided by the hypotenuse (longest side) of the triangle equals the “cosine” of the angle.

The side opposite the angle divided by the hypotenuse equals the “sine” of the angle. Since the side opposite the angle is the same as y, and the side adjacent to the triangle is the same as x for the point where the hypotenuse touches the circle, we get,

x = r cos(θ) and y = r sin(θ). See the image.

Advantages of Polar Coordinates

One major advantage of changing coordinate systems is the matter of simplification. Choosing a coordinate system that makes life simpler is always a wise choice. Simpler? Sure. What, for instance, is the equation of a circle in Cartesian coordinates centered at the origin with a radius of 5?

x2 + y2 = 25

Now that’s not hard, right? But if you didn’t know that was a circle, you’d have to plot a collection of points, (x, y) after doing a number of calculations.

Is the equation for a circle of radius 5 centered at the origin easier in polar coordinates? Its equation is,

r = 5

In fact, many other curves that have a smooth, angular-generated configuration are simplified using polar coordinates.

As an example, consider how you would write this polar coordinate equation for a spiral in Cartesian coordinates…

r = θ

We need to introduce x and y, both. Let’s use the r term to introduce x and the theta term to introduce y. Then, x/cos(θ) is the left-hand side of the equation. The right hand side is not so easy. Since y = r sin(θ), we get sin(θ) = y/r. We haven’t succeeded in solving for pure θ.

We have to use an arc-function. If y/r = sin(θ), then arcsin(y/r) = θ. So the equation becomes,

x/cos(θ) = arcsin(y/r)

and we still haven’t gotten rid of r (which adds further complications). Let’s give it up for this discussion. Clearly, this is not so easy! See the advantage of the polar coordinate system when it comes to the spiral and similar equations?

Practical Applications Advantages

Polar coordinates are useful when studying the motions of the human body. This is because the human body utilizes pivotal joint movements.

When discussing orbits, where radial symmetry is involved, or there is a point source, such as the generation of ripples in water, polar coordinates simplify matters.

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