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# 12.2: Polar Coordinates

Difficulty Level: At Grade Created by: CK-12

Cartesian coordinates $(x,y)$ are not the only way of labeling a point $P$ 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 $r$ of the line $OP$ from the origin to $P$ (i. e. the distance of $P$ 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 $\theta$. Note that while in the cartesian system $x$ and $y$ play very similar roles, here roles are divided: $r$ gives distance and $\theta$ direction.

The two representations are closely related. From the definitions of the sine and cosine: $x = r \times \cos{\theta}\\y = r \times \sin{\theta}$ This allows $(x,y)$ 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 $r$: $r^2 = x^2 + y^2$ Once $r$ is known, the rest is easy: $\cos{\theta} = \frac{x}{r}\\\sin{\theta} = \frac{y}{r}$ These relations fail only at the origin, where $x = y = r = 0$. At that point, $\theta$ is undefined and one can choose for it whatever one pleases.

In three dimensional space, the cartesian labeling $(x,y,z)$ 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 $r$. You can even show from the theorem of Pythagoras that in this case $r^2 = x^2 +y^2 + z^2$ All the points with the same value of $r$ form a sphere of radius $r$ around the origin $O$. On a sphere we can label each point by latitude $\lambda$ (lambda, small Greek L) and longitude $\phi$ (phi, small Greek F), so that the position of any point in space is defined by the 3 numbers $(r, \lambda, \phi)$.

Feb 27, 2012

Jan 21, 2015

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