Question

Plot the point (3, 2pi/3)​, given in polar​ coordinates, and find other polar coordinates (r, θ )of the point for which the following are true. ​

(a) r>0, -2 pi < 0 ​
(b) r < 0, 0 < 2 pi ​
(c) r > 0, 2 pi < 4 pi

Answer 1

(a)
`(3, -(4\pi)/(3))`

(b)
`(-3, (5\pi)/(3))`

(c)
`(3, (8\pi)/(3))`

Explanation:

All polar coordinates of point (r, θ ) are

`(r,\theta+2n\pi)` and
`(-r,\theta+(2n+1)\pi)`

where, θ is in radian and n is an integer.

The given point is
`(3, (2\pi)/(3))`. So, all polar coordinates of point are

`(3, (2\pi)/(3)+2n\pi)` and
`(-3, (2\pi)/(3)+(2n+1)\pi)`

(a)
`r>0,-2\pi\leq \theta <0`

Substitute n=-1 in
`(3, (2\pi)/(3)+2n\pi)`, to find the point for which
`r>0,-2\pi\leq \theta <0`.

`(3, (2\pi)/(3)+2(-1)\pi)`

`(3, -(4\pi)/(3))`

Therefore, the required point is
`(3, -(4\pi)/(3))`.

(b)
`r<0,0\leq \theta <2\pi`

Substitute n=0 in
`(-3, (2\pi)/(3)+(2n+1)\pi)`, to find the point for which
`r>0,-2\pi\leq \theta <0`.

`(-3, (2\pi)/(3)+(2(0)+1)\pi)`

`(-3, (2\pi)/(3)+\pi)`

`(-3, (5\pi)/(3))`

Therefore, the required point is
`(-3, (5\pi)/(3))`.

(c)
`r>0,2\pi \leq \theta <4\pi`

Substitute n=1 in
`(3, (2\pi)/(3)+2n\pi)`, to find the point for which
`r>0,2\pi \leq \theta <4\pi`.

`(3, (2\pi)/(3)+2(1)\pi)`

`(3, (2\pi)/(3)+2\pi)`

`(3, (8\pi)/(3))`

Therefore, the required point is
`(3, (8\pi)/(3))`.