CONVERT RECTANGULAR COORDINATES INTO POLAR FORM

Example 1 :

Convert to polar coordinates on the interval 0 < θ < 2π

(a)  (-1, 1)    (b)  (1, √3)

Solution :

(a)  (-1, 1) 

x  =  -1 and y  =  1

If r  =  √2 then  θ  =  -π/4  ==>  (√2, -π/4)

If r  =  -√2 then  θ  =  -π/4 + π  ==>  (√2, 3π/4)

If r  =  √2 then  θ  =  -π/4 + 2π  ==>  (√2, 7π/4)

(b)  (1, √3)

x  =  1 and y  =  √3

If r  =  2 then  θ  =  π/4  ==>  (2, π/4)

If r  =  -2 then  θ  =  π/4 + π  ==>  (-2, 5π/4)

If r  =  2 then  θ  =  π/4 + 2π  ==>  (2, 9π/4)

If we have any restriction like the angle must between 0 to 360, then we have to ignore the last option.

Example 2 :

Convert the given rectangular coordinate to polar coordinate between 0 ≤ θ ≤ 2π

(1, -√3) 

x  =  1 and y  =  -√3

If r  =  2 then  θ  =  -π/3  ==>  (2, -π/3)

If r  =  -2 then  θ  =  -π/3 + π  ==>  (-2, 2π/3)

If r  =  2 then  θ  =  -π/3 + 2π  ==>  (2, 5π/3)

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