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
Accepted Solution
A:
Answer:(a) [tex](3, -\frac{4\pi}{3})[/tex](b) [tex](-3, \frac{5\pi}{3})[/tex](c)[tex](3, \frac{8\pi}{3})[/tex]Step-by-step explanation:All polar coordinates of point (r, θ ) are[tex](r,\theta+2n\pi)[/tex] and [tex](-r,\theta+(2n+1)\pi)[/tex]where, θ is in radian and n is an integer.The given point is [tex](3, \frac{2\pi}{3})[/tex]. So, all polar coordinates of point are[tex](3, \frac{2\pi}{3}+2n\pi)[/tex] and [tex](-3, \frac{2\pi}{3}+(2n+1)\pi)[/tex](a) [tex]r>0,-2\pi\leq \theta <0[/tex]Substitute n=-1 in [tex](3, \frac{2\pi}{3}+2n\pi)[/tex], to find the point for which [tex]r>0,-2\pi\leq \theta <0[/tex].[tex](3, \frac{2\pi}{3}+2(-1)\pi)[/tex][tex](3, -\frac{4\pi}{3})[/tex]Therefore, the required point is [tex](3, -\frac{4\pi}{3})[/tex].(b) [tex]r<0,0\leq \theta <2\pi[/tex]Substitute n=0 in [tex](-3, \frac{2\pi}{3}+(2n+1)\pi)[/tex], to find the point for which [tex]r>0,-2\pi\leq \theta <0[/tex].[tex](-3, \frac{2\pi}{3}+(2(0)+1)\pi)[/tex][tex](-3, \frac{2\pi}{3}+\pi)[/tex][tex](-3, \frac{5\pi}{3})[/tex]Therefore, the required point is [tex](-3, \frac{5\pi}{3})[/tex].(c) [tex]r>0,2\pi \leq \theta <4\pi[/tex]Substitute n=1 in [tex](3, \frac{2\pi}{3}+2n\pi)[/tex], to find the point for which [tex]r>0,2\pi \leq \theta <4\pi[/tex].[tex](3, \frac{2\pi}{3}+2(1)\pi)[/tex][tex](3, \frac{2\pi}{3}+2\pi)[/tex][tex](3, \frac{8\pi}{3})[/tex]Therefore, the required point is [tex](3, \frac{8\pi}{3})[/tex].