Plot the point $P$ whose polar coordinates are $\left( -2,-\dfrac{3\pi }{4}\right)$. Then find three other polar coordinates of the same point with the properties:

(a) $r>0$ and $0<\theta <2\pi$

(b) $r>0$ and $-2\pi <\theta <0$

(c) $r<0$ and $0<\theta <2\pi$

Figure 31

Solution  The point $\left( -2,-\dfrac{3\pi }{4}\right)$ is located by first drawing the angle $-\dfrac{3\pi }{4}$. Then $P$ is on the extension of the terminal side of $\theta$ through the pole at a distance $2$ units from the pole, as shown in Figure 31.

(a) The point $P=( r,\theta ) ,$ $r>0,$ $0<\theta<2\pi$ is $\left( 2,\dfrac{\pi }{4}\right)$, as shown in Figure 32(a).

(b) The point $P=( r,\theta ) ,$ $r>0,$ $-2\pi <\theta<0$ is $\left( 2,-\dfrac{7\pi }{4}\right)$, as shown in Figure 32(b).

(c) The point $P=( r,\theta ) ,$ $r<0,$ $0<\theta<2\pi$ is $\left( -2,\dfrac{5\pi }{4}\right)$, as shown in Figure 32(c).