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

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.

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))#