Question

How do you convert the polar coordinate ( 4.5541 , 1.2352 ) into cartesian coordinates?

Answer 1

Polar coordinates are in the form `(r, theta)`, Cartesian (also called rectangular) coordinates are in the form `(x, y)`. In this case the Cartesian coordinates are `(1.50, 4.30)`.

Explanation:

Polar coordinates are stated in the form `(r, theta)`, a distance from the origin and an angle, in radians, counterclockwise from the positive `x` axis. In this case, we have a point `4.5541` units from the origin, at an angle of `1.2352` `rad`.

Cartesian coordinates, sometimes also called 'rectangular coordinates', are expressed in the form `(x, y)` where `x` is the distance along the `x` axis and `y` is the distance up the `y` axis.

To convert from one to the other, we use trigonometry. The `r` value from the polar coordinates is the hypotenuse of a right-angled triangle, and the `x` and `y` coordinates are the adjacent and opposite sides respectively, from the perspective of the angle `theta`.

Knowing the angle and a side allows us to use the definitions of sine and cosine to find the `x` and `y` coordinates:

`sin theta = (opposite)/("hypotenuse") to` `opposite = "hypotenuse"* sin theta`

In this case, `y = 4.5541*sin(1.2352) = 4.5541*0.9442=4.30`

(Be sure to ensure that your calculator is on the 'radians', not 'degrees' setting when calculating the sine and cosine in this context.)

By similar reasoning:

`cos theta = (adjacent)/("hypotenuse") to` `adjacent = "hypotenuse"* cos theta`

In this case, `x = 4.5541*cos(1.2352) = 4.5541*0.3293=1.50`

So that you don't have to remember all this discussion every time, you can memorize (but should understand):

`x = r cos theta`

`y=rsintheta`

Combine these, and the Cartesian coordinates for the point are `(1.50, 4.30)`.