To convert rectangular coordinates (x,y)(x,y)(x,y) to polar coordinates (r,θ)(r,\theta)(r,θ), use the following formulas:

• Calculate the radius rrr (the distance from the origin to the point):

r=x2+y2r=\sqrt{x^2+y^2}r=x2+y2​

• Calculate the angle θ\theta θ (the angle measured counterclockwise from the positive x-axis):

θ=tan⁡−1(yx)\theta =\tan^{-1}\left(\frac{y}{x}\right)θ=tan−1(xy​)

Note that θ\theta θ should be adjusted based on the quadrant in which the point (x,y)(x,y)(x,y) lies, since the arctangent function alone only returns values between −π2-\frac{\pi}{2}−2π​ and π2\frac{\pi}{2}2π​. Thus, the polar coordinates are expressed as:

(r,θ)(r,\theta)(r,θ)

Summary of steps:

• Compute r=x2+y2r=\sqrt{x^2+y^2}r=x2+y2​
• Compute θ=tan⁡−1(y/x)\theta =\tan^{-1}(y/x)θ=tan−1(y/x), adjusting for quadrant
• Write the point as (r,θ)(r,\theta)(r,θ)

Example:

For the point (3,3)(3,3)(3,3),

r=32+32=18=32r=\sqrt{3^2+3^2}=\sqrt{18}=3\sqrt{2}r=32+32​=18​=32​

θ=tan⁡−1(33)=tan⁡−1(1)=π4\theta =\tan^{-1}\left(\frac{3}{3}\right)=\tan^{-1}(1)=\frac{\pi}{4}θ=tan−1(33​)=tan−1(1)=4π​

So the polar coordinates are (32,π4)\left(3\sqrt{2},\frac{\pi}{4}\right)(32​,4π​)

. This method covers the fundamental conversion from rectangular to polar coordinates.