How do you convert (-4, 3) into polar coordinates?

2 Answers
Jan 9, 2016

If (a,b) is a are the coordinates of a point in Cartesian Plane, u is its magnitude and alpha is its angle then (a,b) in Polar Form is written as (u,alpha).
Magnitude of a cartesian coordinates (a,b) is given bysqrt(a^2+b^2) and its angle is given by tan^-1(b/a)

Let r be the magnitude of (-4,3) and theta be its angle.
Magnitude of (-4,3)=sqrt(-4)^2+3^2)=sqrt(16+9)=sqrt25=5=r
Angle of (-4,3)=Tan^-1(3/-4)=Tan^-1(-3/4)=-36.869 degree

implies Angle of (-4,3)=-36.869 degree

But since the point is in second quadrant so we have to add 180 degree which will give us the angle.

implies Angle of (-4,3)=-36.869+180=143.131

implies Angle of (-4,3)=143.131=theta

implies (-4,3)=(r,theta)=(5,143.131)
implies (-4,3)=(5,143.131)
Note that the angle is given in degree measure.

Jan 9, 2016

Given that a point color(brown)(P-> (x,y)->(-4,3) " Cartesian")

Then color(blue)(P ->(5,143.13^o) " Polar ") to 2 decimal places

Explanation:

This is not a polar graph!!!
Tony BTony B

color(blue)("Where it all comes from")

We are give the coordinates of (-4,3)
Suppose we viewed this in the context of Cartesian form and use y=mx+c

Then c=0 and m=y/x=-3/4

So we would have y=-3/4x

Suppose the graph was only plotted over the range x->(0,-4)

Then the above graph would not be continuous but be a line from
(0,0) to (3,-4)

All we need now is the angle that that line makes to the x-axis and the length of that line.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Finding the Polar r value")

"Length "= sqrt(x^2+y^2) =sqrt(3^2+4^2) =5

color(blue)("So the polar "r=5)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Finding the Polar "theta" value")
The Polar angle theta is measured from the positive x-axis counterclockwise.

Let the angle from the line to the negative x-axis be phi

Then phi = tan^(-1)(y/x) = tan^(-1)(3/4)

But color(white)(..)phi+theta=180

so color(white)(..)theta =180 - tan^(-1)(3/4)

color(blue)( theta = 143.13 to 2 decimal places
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Putting at all together")

Given that a point P-> (x,y)->(-4,3) "Cartesian"

Then P ->(5,143.13^o) " Polar " to 2 decimal places