Question

What is the Cartesian form of `(-10,(14pi)/8))`?

Answer 1

See a solution process below:

Explanation:

The formula for converting Polar Coordinates to Cartesian coordinates is:

For, `(r, theta)`; `x = r xx cos(theta)`; `y = r xx sin(theta)`

Substituting the coordinates from the problem gives:

For, `(-10, (14pi)/8)`:

`x = -10 xx cos((14pi)/8) = -10 xx 0.707 = -70.7`

`y = -10 xx sin((14pi)/8) = -10 xx -0.707 = 70.7`

`(-10, (14pi)/8) =(-70.7, 70.7)`

Answer 2

The cartesian form is `(-5sqrt(2), 5sqrt(2))`.

Explanation:

To convert polar to rectangular we use:

`x=rcos(theta)`
`y=rsin(theta)`

We're given the polar point `(-10,(14pi)/8)`.

First let's simplify `(14pi)/8` to `(7pi)/4`.

From what we have we know:

`x=(-10)cos((7pi)/4)`
`y=(-10)sin((7pi)/4)`

`(7pi)/4` is a Unit Circle angle so we know its sine and cosine.

`cos((7pi)/4)=sqrt(2)/2`
`sin((7pi)/4)=-sqrt(2)/2`

so:

`x=(-10)(sqrt(2)/2)=-5sqrt(2)`
`y=(-10)(-sqrt(2)/2)=5sqrt(2)`

so the cartesian form is `(-5sqrt(2), 5sqrt(2))`.