Polar Coordinates: Learn It 3

Converting Between Polar Coordinates to Rectangular Coordinates

When given a set of polar coordinates, we may need to convert them to rectangular coordinates. To do so, we can recall the relationships that exist among the variables [latex]x,y,r[/latex], and [latex]\theta[/latex].

[latex]\begin{gathered} \cos \theta =\frac{x}{r}\to x=r\cos \theta \\ \sin \theta =\frac{y}{r}\to y=r\sin \theta \end{gathered}[/latex]

Dropping a perpendicular from the point in the plane to the x-axis forms a right triangle. An easy way to remember the equations above is to think of [latex]\cos \theta[/latex] as the adjacent side over the hypotenuse and [latex]\sin \theta[/latex] as the opposite side over the hypotenuse.

Comparison between polar coordinates and rectangular coordinates. There is a right triangle plotted on the x,y axis. The sides are a horizontal line on the x-axis of length x, a vertical line extending from thex-axis to some point in quadrant 1, and a hypotenuse r extending from the origin to that same point in quadrant 1. The vertices are at the origin (0,0), some point along the x-axis at (x,0), and that point in quadrant 1. This last point is (x,y) or (r, theta), depending which system of coordinates you use.

converting polar to rectangular coordinates

To convert polar coordinates [latex]\left(r,\theta \right)[/latex] to rectangular coordinates [latex]\left(x,y\right)[/latex], let

[latex]\cos \theta =\frac{x}{r}\to x=r\cos \theta[/latex]

[latex]\sin \theta =\frac{y}{r}\to y=r\sin \theta[/latex]

How To: Given polar coordinates, convert to rectangular coordinates.

  1. Given the polar coordinate [latex]\left(r,\theta \right)[/latex], write [latex]x=r\cos \theta[/latex] and [latex]y=r\sin \theta[/latex].
  2. Evaluate [latex]\cos \theta[/latex] and [latex]\sin \theta[/latex].
  3. Multiply [latex]\cos \theta[/latex] by [latex]r[/latex] to find the x-coordinate of the rectangular form.
  4. Multiply [latex]\sin \theta[/latex] by [latex]r[/latex] to find the y-coordinate of the rectangular form.

 

Write the polar coordinates [latex]\left(3,\frac{\pi }{2}\right)[/latex] as rectangular coordinates.

Write the polar coordinates [latex]\left(-2,0\right)[/latex] as rectangular coordinates.

Write the polar coordinates [latex]\left(-1,\frac{2\pi }{3}\right)[/latex] as rectangular coordinates.

Converting from Rectangular Coordinates to Polar Coordinates

To convert rectangular coordinates to polar coordinates, we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.

converting rectangular to polar coordinates

Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships:

[latex]\begin{gathered}\cos \theta =\frac{x}{r}\text{ or }x=r\cos \theta \\ \sin \theta =\frac{y}{r}\text{ or }y=r\sin \theta \\ {r}^{2}={x}^{2}+{y}^{2} \\ \tan \theta =\frac{y}{x} \end{gathered}[/latex]

Convert the rectangular coordinates [latex]\left(3,3\right)[/latex] to polar coordinates.