Converting Coordinates from Rectangular to Polar

Key Questions

• How do I convert Cartesian coordinates to polar coordinates?

  Let's look at the trig formulas SYR, CXR, TYX:

  `sin theta = y/r`
  `cos theta = x/r`
  `tan theta = y/x`

  Since we are given the Cartesian coordinates, we are given `x` and `y`. For polar coordinates, we need to figure out `r` and `theta`. `r` is easy, we just use Pythagorean:

  `r=sqrt(x^2+y^2)`

  To figure out `theta`, I like to use cosine because the range of arccosine is in quadrants I and II and adjusting `theta'` is easier. So,

  `theta'=cos^(-1)x/r`

  If `y>=0` then `theta=theta'`.
  If `y<0` then `theta=2 pi - theta'` (in radians) or `theta=360-theta'` (in degrees).

  Our final answer is `(r, theta)`.

  Let's look at a concrete example: Convert `(-3, 3sqrt3)` to polar coordinates:

  `r=sqrt((-3)^2+(3sqrt3)^2)=sqrt(36)=6`
  `theta'=cos^(-1)((-3)/6)=(2pi)/3`
  `y<0` so, `theta=2pi-(2pi)/3=(4pi)/3`

  So the polar coordinates are `(6, (4pi)/3)`.

  Judy O. · · Aug 19 2014
• How do I find the polar form of `a+bi`?

  I presume we're looking for a radius `r` and angle `theta` such that `a + bi = r(cos theta + i sin theta)`.

  Pythagoras theorem gives us `r = sqrt(a^2+b^2)`.

  Simple trigonometry gives us `tan theta = b/a`, so `theta = arctan (b/a)`.

  George C. · 1 · May 10 2015

• What are the polar coordinates of `(0, -2)`?
• What are the polar coordinates of `(-4, 0)`?
• What are the polar coordinates of `(3, 4)`?
• What are the polar coordinates of `(-2,0)`?
• How do I convert Cartesian coordinates to polar coordinates?
• How do I find the polar form of `a+bi`?
• How do I find the polar form of `3sqrt2 - 3sqrt2i`?
• How do you change (4, -1) from rectangular to cylindrical coordinates between [0, 2π)?
• How do you change (0,3,-3) from rectangular to spherical coordinates?
• How do you find the rectangular coordinates if you given the cylindrical coordinate `(5, pi/6, 5)`?
• How do you convert the cartesian coordinate (-5, -5) into polar coordinates?
• How do you convert the cartesian coordinate (6, 6) into polar coordinates?
• How do you convert the cartesian coordinate (19, 6) into polar coordinates?
• How do you convert the cartesian coordinate (−1, 5) into polar coordinates?
• How do you convert the cartesian coordinate (18, −6) into polar coordinates?
• How do you convert the cartesian coordinate (−2, −9) into polar coordinates?
• How do you convert the cartesian coordinate (0, −4) into polar coordinates?
• How do you convert the cartesian coordinate (3√3, 3) into polar coordinates?
• How do you convert the cartesian coordinate (2, -4) into polar coordinates?
• How do you convert the cartesian coordinate ( 8.1939 , -0.4145 ) into polar coordinates?
• How do you convert the Cartesian coordinates (-6, 13) to polar coordinates?
• How do you convert the Cartesian coordinates (10,10) to polar coordinates?
• How do you convert the Cartesian coordinates (-2,-2) to polar coordinates?
• How do you convert the Cartesian coordinates (1.1 m, 1.6 m) to polar coordinates?
• How do you convert the Cartesian coordinates (0, -5) to polar coordinates?
• How do you convert the Cartesian coordinates (6,-4) to polar coordinates?
• How do you convert the Cartesian coordinates (3,-9) to polar coordinates?
• How do you convert the Cartesian coordinates (-7,6) to polar coordinates?
• How do you convert the Cartesian coordinates (20,3) to polar coordinates?
• How do you convert the Cartesian coordinates (-9,-4) to polar coordinates?
• How do you convert the Cartesian coordinates (0,-10) to polar coordinates?
• How do you convert the Cartesian coordinates (3, - 3) to polar coordinates?
• How do you convert the Cartesian coordinates (3,2) to polar coordinates?
• How do you convert the Cartesian coordinates (-2,2) to polar coordinates?
• How do you convert the Cartesian coordinates (2√3, 2) to polar coordinates?
• How do you convert the Cartesian coordinates (3 sqrt 3, 3) to polar coordinates?
• How do you convert the Cartesian coordinates `(-1/2, -sqrt(3)/2)` to polar coordinates?
• How do you convert the Cartesian coordinates (-7sqrt3/2,7/2) to polar coordinates?
• How do you convert the Cartesian coordinates (-3, -2) to polar coordinates?
• How do you convert the Cartesian coordinates (-√3,-1) to polar coordinates?
• How do you convert the Cartesian coordinates (1 , sqrt(3)) to polar coordinates?
• How do you convert the Cartesian coordinates (-1,1) to polar coordinates?
• How do you convert `(-sqrt3,-1)` into polar coordinates?
• How do you convert `(5.7, -1.2)` into polar coordinates?
• How do you convert `(-3, 3)` into polar coordinates?
• How do you find the polar coordinate for (3, -5)?
• How do you find the polar coordinate for ( 1/3 , 1/4 )?
• How do you find the polar coordinate for (-5,0)?
• How do you find the polar coordinate for (32, -60)?
• How do you find the polar coordinate for (-4, -4)?
• How do you find the polar coordinate for (-12, -12)?
• How do you find the polar coordinate of (-1,1)?
• How do you convert (5, 0) into polar coordinates?
• How do you convert (0,-5) into polar coordinates?
• How do you write polar coordinates of `(-sqrt 3, 3)`?
• How do you write polar coordinates of `(3, -3)`?
• How do you find the polar coordinates of the rectangular point (-4,4)?
• How do you find the polar coordinate of `(2sqrt3, 2)`?
• How do you find the polar coordinate of `(3, -2)`?
• What are the polar coordinates of `(−1, 5)`?
• What are the polar coordinates of `(18, −6)`?
• What are the polar coordinates of `(−2, −9)`?
• What are the polar coordinates of `(0, −4)`?
• What are the polar coordinates of `(6, -6)`?
• What are the polar coordinates of `(1, sqrt3)`?
• How do you write the polar coordinates for the point: `(sqrt(3), 1)`?
• How do you convert (0,2) into polar coordinates?
• How do you convert (6, -6) into polar coordinates?
• How do you convert (-4, 3) into polar coordinates?
• How do you write the polar coordinates for the point with rectangular coordinates (0, 3)?
• How do you convert (5, -4) into polar coordinates?
• How do you convert (15, -3) into polar coordinates?
• How do you convert (9, 0) into polar coordinates?
• How do you convert (-5, 6) into polar coordinates?
• How do you convert (0, 9) into polar coordinates?
• How do you convert `(7sqrt3, -7)` into polar coordinates?
• How do you convert `(-2, 2sqrt3)` into polar coordinates?
• How can (4, -7) be converted into polar coordinates?
• How can `(sqrt3,-1)` be converted into polar coordinates?
• How do you convert (0, 2) into polar coordinates?
• How do you convert `(-2, -2sqrt{3})`into polar coordinates?
• How do you convert `( 1 , - sqrt3 )`into polar coordinates?
• How do you convert `( -2 , 0 )`into polar coordinates?
• How do you convert `-2i`into polar coordinates?
• How do you convert (1, -4) into polar coordinates?
• How do you convert (28,21) rectangular to polar coordinates?
• How do you convert `(-8,0)` into polar forms?
• How do you convert `(-8,0)` into polar forms?
• How do you convert `(2,2)` into polar forms?
• How do you convert `(sqrt(3), 1)` into polar forms?
• How do you convert `(8.3, 4.2)` into polar forms?
• How do you find the polar coordinate of the following point (-2, 5)?
• How do you find the polar coordinate of the following point (-3, -2)?
• How do you find the polar coordinate of the following point (0, -3)?
• How do you find the polar coordinate of the following point (4, 0)?
• What is the polar form of `( -6,36 )`?
• How do you express the Cartesian coordinates (4, 0) as polar coordinates?
• How do you express the Cartesian coordinates (0, - 3) as polar coordinates?
• How do you express the Cartesian coordinates (-square root 3, -1) as polar coordinates?
• How do you express the Cartesian coordinates (3, - 4) as polar coordinates?
• How do you express the Cartesian coordinates `(sqrt(3)/2, 1/2)` as polar coordinates?
• How do you convert the Cartesian coordinates (-6, 13) to polar coordinates?
• How do you find the corresponding rectangular coordinates for the point `( 4, (3pi)/2 )`?
• How do you find rectangular coordinates for the point with polar coordinates (4, 4pi/3)?
• What are the polar coordinates `(sqrt(7), 40.89^@)` in rectangular coordinates ?
• Question #8a89a
• Question #00d88
• The rectangular coordinates of a point are given `(4, -4sqrt3)`, how do you find polar coordinates of the point?
• How do you convert the point (3,-3, 7) from rectangular coordinates to cylindrical coordinates?
• Write the complex numbers `4cis120^@` and `9cis((3pi)/2)` in the form `a+ib`?
• Question #976bb
• Convert the equation from rectangular to polar form and draw its graph? `xy=4`
• Question #4703a
• Question #26517
• Convert `(x+2)^2 + (y+3)^2 = 13` into polar form?