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

Answer:

(a) $(3, -\frac{4\pi}{3})$

(b) $(-3, \frac{5\pi}{3})$

(c)$(3, \frac{8\pi}{3})$

Step-by-step 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 $(3, \frac{2\pi}{3})$. So, all polar coordinates of point are

$(3, \frac{2\pi}{3}+2n\pi)$ and $(-3, \frac{2\pi}{3}+(2n+1)\pi)$

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

Substitute n=-1 in $(3, \frac{2\pi}{3}+2n\pi)$, to find the point for which $r>0,-2\pi\leq \theta$.

$(3, \frac{2\pi}{3}+2(-1)\pi)$

$(3, -\frac{4\pi}{3})$

Therefore, the required point is $(3, -\frac{4\pi}{3})$.

(b) $r$

Substitute n=0 in $(-3, \frac{2\pi}{3}+(2n+1)\pi)$, to find the point for which $r>0,-2\pi\leq \theta$.

$(-3, \frac{2\pi}{3}+(2(0)+1)\pi)$

$(-3, \frac{2\pi}{3}+\pi)$

$(-3, \frac{5\pi}{3})$

Therefore, the required point is $(-3, \frac{5\pi}{3})$.

(c) $r>0,2\pi \leq \theta$

Substitute n=1 in $(3, \frac{2\pi}{3}+2n\pi)$, to find the point for which $r>0,2\pi \leq \theta$.

$(3, \frac{2\pi}{3}+2(1)\pi)$

$(3, \frac{2\pi}{3}+2\pi)$

$(3, \frac{8\pi}{3})$

Therefore, the required point is $(3, \frac{8\pi}{3})$.