(Note that this expression results from the Pythagorean theorem:, which we showed to be related to a circle of radius c in our discussion of trigonometry.) What about in polar coordinates? First, we can assume r is the dependent variable and θ is the independent variable a function would then be written r( θ). To express this relation in rectangular coordinates, we would need to write the following: For instance, consider the equation of a circle of radius c.

The best approach to ensuring that you chose the correct angle is to convert the polar coordinates back to rectangular coordinates to see if they match if not, you've either erred in calculating r or the angle θ.Īs mentioned previously, certain algebraic relations have much simpler expressions in polar coordinates than in their rectangular counterparts. We will not delve too deeply into this issue, however. Thus, the location ( r, θ) is equivalent (in some sense, at least) to ( r, θ + 2 π), and so on. In other words, given the location ( r, θ), if you go 'around the circle' any number of complete revolutions, you end up at the same point. That is, in the case of our point P, it is located at ( r, θ + 2 nπ), where n is any integer. No inconsistencies arise if we say θ has a range (-∞,∞) the only concern is that each point is defined by more than one set of coordinates. But these ranges need not be limited as such-they are just what is necessary to define every point on the plane.

Now, the inverse tangent must be interpreted carefully, because the angle θ has the range, depending on which way it is measured. Note that we can use inverse trig functions to represent θ. (Note that unlike the constant unit vectors i and j that define locations in rectangular coordinates, the unit vector in polar coordinates changes direction with θ.) We have already related r to the rectangular coordinates x and y, but we can also do so for θ. These coordinates are called polar coordinates. We can find any point on the plane using the coordinates ( r, θ), where. Now, we have uniquely identified the location of the point P: it is located at a distance from the origin and an angle θ from the horizontal ( x) axis. Recall from trigonometry that we can use an angle measured relative to the x-axis to get a specific triangle-which, not coincidentally, corresponds with a specific point on a circle centered on the origin. But that still leaves a set of points on a circle, not the specific point P. Thus, we can narrow down the location of P by way of one number, the distance from the origin. Here, following the results of our investigation into vectors, Note that P resides on a circle that has a radius of length. (Note that ( x, y) can also represent a vector r, which is also shown on the graph.) In this article, we will introduce a simple alternative coordinate system: polar coordinates.Ĭonsider some point P with rectangular coordinates ( x, y). Although this coordinate system may be the most intuitive, it is not always the most convenient one for representing functions or points. This system uses fixed axes (in the context of vectors, the unit vectors have a constant direction). You should already be familiar with rectangular (or Euclidean) coordinates: in three dimensions, we generally use x, y, and z as labels for our axes. Plot algebraic relations in polar coordinates.

Polar plot mathematica