Polar Coordinates

Polar coordinates are used to describe the location of a point in a 2D plane using a radial distance and an angle from the positive x-axis. This system is often used to simplify calculus involving circular curves and other problems where symmetry about a point is involved.

The Essentials

In a Cartesian coordinate system, a point is represented simply by \((x,y)\). In polar coordinates the same pair could be written as \((r,\theta)\), where \(r\) represents the radial distance from the origin to the point and \(\theta\) represents the angle measured from the positive x-axis

To better understand the relationship between these two coordinate systems you can visualize a right triangle formed by the point.

To represent polar coordinates in Cartesian form:

\[ cos(\theta) = \frac{x}{r}\implies x = rcos(\theta) \]
\[ sin(\theta) = \frac{y}{r}\implies y = rsin(\theta) \]

To represent Cartesian coordinates in Polar form:

\[ r^2 = x^2 + y^2 \implies r = \sqrt{x^2+y^2} \]
\[ tan(\theta) = \frac{y}{x}\implies\theta = tan^{-1}\left(\frac{y}{x}\right) \]
Right Triangle Representation of Polar Coordinates

Figure 1. Right Triangle Representation of Polar Coordinates

Example

Convert the following Cartesian coordinates into polar form

\[ (12, 5) \]
\[ r = \sqrt{(12^2)+(5^2)} \]
\[ r = \sqrt{169} = 13 \]
\[ \theta = tan^{-1}\left(\frac{5}{12}\right) \]
\[ \theta = 22.61^\circ \]
\[ (12,5) \implies (13, 22.61^\circ) \]
\[ (-3,7) \]
\[ r = \sqrt{(-3^{2})+(7^2)} \]
\[ r = \sqrt{58}=7.62 \]
\[ \theta = tan^{-1}\left(\frac{7}{-3}\right) \]
\[ \theta = -66^\circ + 180^\circ = 113.20^\circ \]

Convert the following polar coordinates into Cartesian form

\[ (6, \pi) \]
\[ x = 6cos(180^\circ) \]
\[ x = -6 \]
\[ y = 6sin(180^\circ) \]
\[ y = 0 \]
\[ (6, \pi) \implies (-6, 0) \]

Practice

Pair each set of coordinates with their respective polar/Cartesian conversion

  1. \( (-4, -6) \)
  2. \( (4, 76.48^\circ) \)
  3. \( (11, 6.72) \)
  4. \( (4, -\pi/2) \)
  1. \( (0, -4) \)
  2. \( (7.21, 236.31^\circ) \)
  3. \( (12.89, 31.42^\circ) \)
  4. \( (0.94, 3.89) \)

Solution:

  1. 1 \( \to \) b
  2. 2 \( \to \) d
  3. 3 \( \to \) c
  4. 4 \( \to \) a

More Resources