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?
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• 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?
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• 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?
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• Question #976bb
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• Question #4703a
• Question #26517
• Convert #(x+2)^2 + (y+3)^2 = 13# into polar form?