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

2 Answers
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 (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.

Mar 11, 2018

(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))