Polar Coordinates: Apply It

  • Plot points using polar coordinates.
  • Convert between polar coordinates and rectangular coordinates.
  • Transform equations between polar and rectangular forms.
  • Identify and graph polar equations by converting to rectangular equations.

GPS Navigation and Location Services

Modern GPS navigation systems work behind the scenes with both rectangular and polar coordinate systems. While your phone displays locations using latitude and longitude (similar to rectangular coordinates), GPS satellites and navigation algorithms often convert between coordinate systems to calculate distances, directions, and optimal routes.

Polar to Rectangular:

  • [latex]x = r\cos\theta[/latex]
  • [latex]y = r\sin\theta[/latex]

Rectangular to Polar:

  • [latex]r = \sqrt{x^2 + y^2}[/latex]
  • [latex]\tan\theta = \frac{y}{x}[/latex]
A search and rescue helicopter is located 8 miles from base at a bearing of [latex]\frac{\pi}{3}[/latex] radians (60°) from north. Convert this polar location [latex]\left(8, \frac{\pi}{3}\right)[/latex] to rectangular coordinates for mapping software.

A drone is positioned 12 miles from its launch point at an angle of [latex]\frac{3\pi}{4}[/latex] radians. Find the rectangular coordinates of the drone’s location.
A delivery vehicle is at position [latex](6, 8)[/latex] miles from the distribution center (using east and north as positive directions). What distance and bearing should the navigation system display?

A GPS tracker shows a hiker at rectangular position [latex](-5, 12)[/latex] miles from the trailhead. Find the distance and bearing (angle from the positive x-axis) in polar form.
When converting from rectangular to polar coordinates using [latex]\tan^{-1}\left(\frac{y}{x}\right)[/latex], always check which quadrant your point is in. The inverse tangent function only returns angles in [latex]\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)[/latex], so you may need to add [latex]\pi[/latex] for points in Quadrants II or III.