Question

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

Answer 1

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 by`sqrt(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.

Answer 2

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!!!

`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