Question

What is the Cartesian form of `(-4,(-21pi)/4))`?

Answer 1

`(2sqrt2, -2sqrt2)`

Explanation:

We have the coordinate `(-4, (-21pi)/4)` in polar form.

Coordinates in polar form have the standard form `(color(green)(r), color(purple)(Θ))`.

To convert from polar form to Cartesian form, we use the following formulas:

• `color(red)(x) = color(green)(r)coscolor(purple)(Θ)`
• `color(blue)(y) = color(green)(r)sincolor(purple)(Θ)`

Now, let's plug stuff in. We know `color(green)(r) = -4` and `color(purple)(Θ) = (-21pi)/4`

`color(red)(x) = (-4)*cos((-21pi)/4)`

`color(red)(x) = (-4)*(-sqrt2/2)`

`color(red)(x) = 2sqrt2`

`color(blue)(y) = (-4)*sin((-21pi)/4)`

`color(blue)(y) = (-4)*(sqrt2/2)`

`color(blue)(y) = -2sqrt2`

We get `color(red)(x) = 2sqrt2` and `color(blue)(y) = -2sqrt2`, making our Cartesian coordinate `(2sqrt2, -2sqrt2).`

Answer 2

`(2\sqrt2, -2\sqrt2)`

Explanation:

Cartesian coordinates `(x, y)` of given point `(r, \theta)\equiv(-4, {-21\pi}/4)` are given as

`x=r\cos\theta`

`=-4\cos(-{21\pi}/4)`

`=-4\cos({5\pi}/4)`

`=-4(-1/\sqrt2)`

`=2\sqrt2`

`y=r\sin\theta`

`=-4\sin(-{21\pi}/4)`

`=4\sin({5\pi}/4)`

`=4(-1/\sqrt2)`

`=-2\sqrt2`

hence the cartesian coordinates of given point are

`(x, y)\equiv(2\sqrt2, -2\sqrt2)`