plot the point $(2,\\frac{4\\pi}{3})$, given in polar coordinates, and find other polar coordinates…

plot the point $(2,\\frac{4\\pi}{3})$, given in polar coordinates, and find other polar coordinates $(r,\\theta)$ of the point for which the following are true (a) $r>0$, $-2\\pi\\leq\\theta<0$ (b) $r<0$, $0\\leq\\theta<2\\pi$ (c) $r>0$, $2\\pi\\leq\\theta<4\\pi$ (a) what are the coordinates of the point for which $r>0$, $-2\\pi\\leq\\theta<0$? $(2,-\\frac{2\\pi}{3})$ (type an ordered pair. type an exact answer, using $\\pi$ as needed. use integers or fractions for any numbers in the expression. simplify your answer ) (b) what are the coordinates of the point for which $r<0$, $0\\leq\\theta<2\\pi$?

plot the point $(2,\\frac{4\\pi}{3})$, given in polar coordinates, and find other polar coordinates $(r,\\theta)$ of the point for which the following are true (a) $r>0$, $-2\\pi\\leq\\theta<0$ (b) $r<0$, $0\\leq\\theta<2\\pi$ (c) $r>0$, $2\\pi\\leq\\theta<4\\pi$ (a) what are the coordinates of the point for which $r>0$, $-2\\pi\\leq\\theta<0$? $(2,-\\frac{2\\pi}{3})$ (type an ordered pair. type an exact answer, using $\\pi$ as needed. use integers or fractions for any numbers in the expression. simplify your answer ) (b) what are the coordinates of the point for which $r<0$, $0\\leq\\theta<2\\pi$?

Answer

Explanation:

Step1: Recall polar coordinate transformation rules

For a polar coordinate ((r,\theta)), if (r>0), (\theta + 2k\pi) ((k\in\mathbb{Z})) represents the same point. If (r<0), the point ((r,\theta)) is equivalent to ((-r,\theta+\pi))

Step2: Solve part (b)

Given the original point ((2,\frac{4\pi}{3})). For (r < 0), we use the rule ((r,\theta)=(-r,\theta+\pi)). Here (r=- 2), and (\theta=\frac{4\pi}{3}+\pi=\frac{4\pi + 3\pi}{3}=\frac{7\pi}{3})

Answer:

(\left(-2,\frac{7\pi}{3}\right))