Question

What is the Cartesian form of `( -8 , ( - 15pi)/4 )`?

Answer 1

Use the polar conversion formulae `x = r cos theta, y = r sin theta`. `(-4sqrt2, -4sqrt2)`

Explanation:

In order to perform the conversion from polar to cartesian coordinates, it helps us to remind ourselves of what the various coordinates represent.

On the xy-plane, the x coordinate simply denotes the x value (i.e. its position along the x-axis), and the y coordinate denotes the y-value (its position along the y-axis)

The polar coordinates of a given point on the xy-plane, however, are represented by `(r,theta)`, with `r` denoting the distance from the origin, and `theta` the angle formed between the x-axis and the line connecting the given point to the origin.

For a graphical representation, look here:

https://www.mathsisfun.com/polar-cartesian-coordinates.html

Normally we prefer to deal with a positive r-value.
The length of the segment, we know from the distance formula, will be `sqrt(x^2+y^2)`. We can then make a right triangle with `r` (the segment) as our hypotenuse, the `x` coordinate as the length of one side, and the `y` coordinate as the length of the other side.

If the angle formed between the x-axis and r is represented by `theta`, then we know that

`sin theta = y/r, cos theta = x/r`

Multiplying both sides by r we get

`y = r sin theta, x = r cos theta`

Recall several important properties of trigonometric functions to make our conversion process easier:

`sin(-theta) = -sin(theta), cos(-theta) = cos(theta), sin(theta+2npi) = sin(theta), cos(theta+2npi)=cos(theta), sin(pi/4) = cos(pi/4) = (sqrt2)/2 = 1/sqrt2`
Using this, we can perform our conversion:

`x = -8cos(-15pi/4) = -8 cos(-7pi/4) = -8cos (7pi/4) = -8sqrt2/2 = -8/sqrt2 = -4sqrt2`

`y = -8sin(-15pi/4) = -8sin (-7pi/4) = 8 sin(7pi/4) = 8*(-1/sqrt2) = -4sqrt2`

The cartesian coordinate form of our point is `(x,y) = (-4sqrt2,-4sqrt2)`

To confirm, we will convert these back into polar coordinates.

`r = sqrt(x^2+y^2) = sqrt (32+32) = sqrt(64) = +-8`
`theta = tan^-1(y/x) = tan^-1(1) = pi/4 + npi`, which gives us `-7pi/4` when `n=-2`, and `-15pi/4` when `n=-4`,