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

plot the point $\\left(2,\\frac{4\\pi}{3}\\right)$, given in polar coordinates, and find other polar coordinates $(r,\\theta)$ of the point for which the following are true\n(a) $r>0,-2\\pi\\leq\\theta<0$\n(b) $r<0,0\\leq\\theta<2\\pi$\n(c) $r>0,2\\pi\\leq\\theta<4\\pi$\n(a) what are the coordinates of the point for which $r>0,-2\\pi\\leq\\theta<0$\n$\\left(2,-\\frac{2\\pi}{3}\\right)$\n(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)\n(b) what are the coordinates of the point for which $r<0,0\\leq\\theta<2\\pi$\n$\\left(-2,\\frac{\\pi}{3}\\right)$\n(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.)\n(c) what are the coordinates of the point for which $r>0,2\\pi\\leq\\theta<4\\pi$\n(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)

plot the point $\\left(2,\\frac{4\\pi}{3}\\right)$, given in polar coordinates, and find other polar coordinates $(r,\\theta)$ of the point for which the following are true\n(a) $r>0,-2\\pi\\leq\\theta<0$\n(b) $r<0,0\\leq\\theta<2\\pi$\n(c) $r>0,2\\pi\\leq\\theta<4\\pi$\n(a) what are the coordinates of the point for which $r>0,-2\\pi\\leq\\theta<0$\n$\\left(2,-\\frac{2\\pi}{3}\\right)$\n(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)\n(b) what are the coordinates of the point for which $r<0,0\\leq\\theta<2\\pi$\n$\\left(-2,\\frac{\\pi}{3}\\right)$\n(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.)\n(c) what are the coordinates of the point for which $r>0,2\\pi\\leq\\theta<4\\pi$\n(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)

Answer

Explanation:

Step1: Understand the polar coordinate transformation

For a polar coordinate ((r,\theta)), other representations are ((r,\theta + 2n\pi)) ((n\in\mathbb{Z})) when (r>0) and ((-r,\theta+(2n + 1)\pi)) ((n\in\mathbb{Z})) when (r<0)

Step2: Solve for part (c)

Given the point ((2,\frac{4\pi}{3})), we want (r>0) and (2\pi\leq\theta<4\pi). We use the formula ((r,\theta+2n\pi)) with (n = 1) (since we want to increase the angle by (2\pi)) [ \theta=\frac{4\pi}{3}+2\pi=\frac{4\pi + 6\pi}{3}=\frac{10\pi}{3} ]

Answer:

(\left(2,\frac{10\pi}{3}\right))