Converting Between Rectangular and Polar Points

Up until now, you have worked purely with a rectangular coordinate system, in which there are two perpendicular axes, and points are specified according to their coordinates, (x, y). What’s interesting is that whenever you plot a point on an x-y plane, that point can also be represented as a vector, that is, once you find the distance from the origin to that point and the angle it makes with the x-axis. For example, if you were to plot the point (3, 7) (shown below), using the Pythagorean theorem and the trigonometric function tangent, you can find the magnitude and angle of the terminal side:

Using the Pythagorean theorem and inverse tangent to find the angle θ that the terminal side makes relative to the x-axis, the point (3, 7) can be written as:

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Therefore, you can now record x-y coordinates as polar coordinates. This new coordinate system, known as polar coordinates deals mainly with angles and radius length. Furthermore, unlike rectangular coordinates, true polar coordinates utilize something other than the typical Cartesian plane; instead, you use polar coordinate graph paper (shown below). For applications regarding angles and radii of vectors, the polar plane below makes it easier to plot such information.

Polar plane in degrees (left) and radians (right). Each concentric ring represents one radius unit of length.

The polar coordinates of a point P are r and θ, usually written in the form P(r, θ) or as r∠θ reads ‘r at an angle of θ’. It’s important to note that r and θ are not independent/dependent variables, as in x and y coordinates – neither depends on the other.

Two ways to represent polar coordinates.

In the video below, you will be shown how to convert a point that’s in rectangular coordinate to polar coordinates. You’ll also be shown how to graph the polar coordinate on a polar plane; a full section on graphing polar equations is covered later in this unit.

The same four formulas introduced in the video above can be applied when converting from polar coordinates to rectangular. This is illustrated in Part 2 below:

Important Formulas:

  • First block: Use these to go from polar to Cartesian
  • Second block: Use these to go from Cartesian to polar.