Question

Find the rectangular coordinates of the point with the polar coordinates ordered pair 7 comma 2 pi divided by 3.

Answer 1

`\left(-(7)/(2),(7√(3))/(2)\right)`.

Explanation:

The given point is

`\left(7,(2\pi)/(3)\right)`

We need to find the rectangular coordinates of the given point.

If a polar coordinate is
`(r,\theta)`, then

`x=r\cos theta`

`y=r\sin theta`

In the given point
`\left(7,(2\pi)/(3)\right)`,

`r=7,\theta=(2\pi)/(3)`

Now,

`x=7\cos (2\pi)/(3)`

`x=7\cos \left(\pi-(\pi)/(3)\right)`

`x=-7\cos \left((\pi)/(3)\right)`

`x=-7\left((1)/(2)\right)`

`x=-(7)/(2)`

and,

`y=7\sin (2\pi)/(3)`

`y=7\sin \left(\pi-(\pi)/(3)\right)`

`y=7\sin \left((\pi)/(3)\right)`

`y=7\left((√(3))/(2)\right)`

`y=(7√(3))/(2)`

Therefore, the required point is
`\left(-(7)/(2),(7√(3))/(2)\right)`.