Question

How do you find the polar coordinates given the rectangular coordinates (3,8)?

Answer 1

Please see the explanation.

Explanation:

The polar coordinate system is an ordered pair `(r,theta)`

To convert from Cartesian, `(x, y)` to r:

`r = sqrt(x^2 + y^2)`

To convert from Cartesian, `(x, y)` to `theta`:

If `x > 0 and y >= 0`, then use: `theta = tan^-1(y/x)" [1]"`
If `x = 0 and y > 0`, then use: `theta = pi/2" [2]"`
If `x = 0 and y < 0`, then use: `theta = (3pi)/3" [3]"`
If `x < 0`, then use: `theta = pi + tan^-1(y/x)" [4]"`
If `x > 0 and y < 0`, then use: `theta = 2pi + tan^-1(y/x)" [5]"`

For your point, `(3, 8)`:

`r = sqrt(3^2 + 8^2) = sqrt(73) " length units"`

Use equation [1]:

`theta = tan^-1(8/3)`

`theta ~~ 1.249 " radians"`

The polar point is `(sqrt(73)" length units", 1.249" radians")`