Question

What is the polar form of `( 4,9 )`?

Answer 1

`(10, 66°)` in Polar Coordinates

Explanation:

To solve this question you must imagine a triangle that has a base distance of `4` and a height of `9`.

We want to do this so that we can find out how far and at what angle the polar coordinates are and so that we can convert the current cartesian coordinates `(x,y)` to the polar coordinates `(r,θ)`.

The triangle

Base`=x = 4`
Height`=y=9`
Hypotenuse`=r`
The angle between `r` and `x = θ`

Use Pythagoras Theorem to find the hypotenuse (`r`)

`r=sqrt(4^2+9^2)`

`r=sqrt(97)=9.848857802`

`r = 10` (1d.p)

Use the Tangent Function to find the desired angle

`tan(θ)=9/4`

`θ = tan^-1(9/4)=66.03751103`

`θ = 66°` (1d.p)

`:.` the cartesian coordinates `(4,9)` are `(10, 66°)` in Polar Coordinates.

Answer 2

`(sqrt(97), arctan (4/9))`

Explanation:

The line joining the origin to the given point, that is the line joining
`(0, 0)` to `(4, 9)`

is the hypotenuse of a right angled triangle with

the line along the x axis from
`(0, 0)` to `(4, 0)`

and the line along the y axis from
`(0, 0)` to `(0, 9)`

forming the other two sides that enclose the right angle.

The side on the x axis has length 4, and that on the y axis has length 9, so that, by the Pythagorean relationship, the length of the hypotenuse is

`sqrt(4^2 + 9^2) = sqrt(16 + 81) = sqrt (97)`

That is the modulus of the polar form, conventionally denoted by `r`.

The tangent of the angle (in radians, conventionally denoted by `theta`) between the hypotenuse and the positive x axis is `4/9`
that is,
`theta = arctan (4/9)`

so, the point has polar form `(r, theta)`

`(r, theta) = (sqrt(97), arctan (4/9))`