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

Mar 11, 2018

(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 $\left(x , y\right)$ 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)

$\therefore$ the cartesian coordinates $\left(4 , 9\right)$ are (10, 66°) in Polar Coordinates.

Mar 11, 2018

$\left(\sqrt{97} , \arctan \left(\frac{4}{9}\right)\right)$

#### Explanation:

The line joining the origin to the given point, that is the line joining
$\left(0 , 0\right)$ to $\left(4 , 9\right)$

is the hypotenuse of a right angled triangle with

the line along the x axis from
$\left(0 , 0\right)$ to $\left(4 , 0\right)$

and the line along the y axis from
$\left(0 , 0\right)$ to $\left(0 , 9\right)$

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 $\frac{4}{9}$
that is,
$\theta = \arctan \left(\frac{4}{9}\right)$

so, the point has polar form $\left(r , \theta\right)$

$\left(r , \theta\right) = \left(\sqrt{97} , \arctan \left(\frac{4}{9}\right)\right)$