# 12.2: Polar Coordinates

Difficulty Level: At Grade Created by: CK-12

Cartesian coordinates \begin{align*}(x,y)\end{align*} are not the only way of labeling a point \begin{align*}P\end{align*} on a flat plane by a pair of numbers. Other ways exist, and they can be more useful in special situations.

One system (“polar coordinates") uses the length \begin{align*}r\end{align*} of the line \begin{align*}OP\end{align*} from the origin to \begin{align*}P\end{align*} (i. e. the distance of \begin{align*}P\end{align*} distance to the origin) and the angle that line makes with the x-axis. Angles are often denoted by Greek letters, and here we follow conventions by marking it with \begin{align*}\theta\end{align*}. Note that while in the cartesian system \begin{align*}x\end{align*} and \begin{align*}y\end{align*} play very similar roles, here roles are divided: \begin{align*}r\end{align*} gives distance and \begin{align*} \theta \end{align*} direction.

The two representations are closely related. From the definitions of the sine and cosine: \begin{align*} x = r \times \cos{\theta}\\ y = r \times \sin{\theta} \end{align*} This allows \begin{align*}(x,y)\end{align*} to be derived from polar coordinates. This relationship is illustrated below:

Converting from polar to cartesian coordinates.

To go in the reverse direction, we can use the Pythagorean theorem to find \begin{align*} r \end{align*}: \begin{align*} r^2 = x^2 + y^2 \end{align*} Once \begin{align*} r \end{align*} is known, the rest is easy: \begin{align*} \cos{\theta} = \frac{x}{r}\\ \sin{\theta} = \frac{y}{r} \end{align*} These relations fail only at the origin, where \begin{align*} x = y = r = 0\end{align*}. At that point, \begin{align*} \theta \end{align*} is undefined and one can choose for it whatever one pleases.

In three dimensional space, the cartesian labeling \begin{align*} (x,y,z) \end{align*} is nicely symmetric, but sometimes it is convenient to follow the style of polar coordinates and label distance and direction separately. Distance is easy: you take the line OP from the origin to the point and measure its length \begin{align*}r\end{align*}. You can even show from the theorem of Pythagoras that in this case \begin{align*} r^2 = x^2 +y^2 + z^2 \end{align*} All the points with the same value of \begin{align*}r\end{align*} form a sphere of radius \begin{align*}r\end{align*} around the origin \begin{align*}O\end{align*}. On a sphere we can label each point by latitude \begin{align*}\lambda\end{align*} (lambda, small Greek L) and longitude \begin{align*} \phi \end{align*} (phi, small Greek F), so that the position of any point in space is defined by the 3 numbers \begin{align*}(r, \lambda, \phi)\end{align*}.

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