Pergunta
![Suppose a point has polar coordinates (4,-(5pi )/(3)) with the angle measured in radians.
Find two additional polar representations of the point.
Write each coordinate In simplest form with the angle in [-2pi ,2pi ]
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Suppose a point has polar coordinates (4,-(5pi )/(3)) with the angle measured in radians. Find two additional polar representations of the point. Write each coordinate In simplest form with the angle in [-2pi ,2pi ] square
Solução

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ElisaVeterano · Tutor por 10 anos
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To find two additional polar representations of the point with polar coordinates $(4,-\frac {5\pi }{3})$, we can add or subtract multiples of $2\pi$ to the angle until it falls within the desired range of $[-2\pi,2\pi ]$.<br /><br />First, let's add $2\pi$ to the angle:<br />$-\frac {5\pi }{3} + 2\pi = -\frac {5\pi }{3} + \frac {6\pi }{3} = \frac {\pi }{3}$<br /><br />So, one additional polar representation of the point is $(4,\frac {\pi }{3})$.<br /><br />Next, let's subtract $2\pi$ from the angle:<br />$-\frac {5\pi }{3} - 2\pi = -\frac {5\pi }{3} - \frac {6\pi }{3} = -\frac {11\pi }{3}$<br /><br />Since the angle is negative, we can add $2\pi$ to it until it falls within the desired range:<br />$-\frac {11\pi }{3} + 2\pi = -\frac {11\pi }{3} + \frac {6\pi }{3} = -\frac {5\pi }{3}$<br /><br />So, another additional polar representation of the point is $(4,-\frac {5\pi }{3})$.<br /><br />Therefore, the two additional polar representations of the point with polar coordinates $(4,-\frac {5\pi }{3})$ are $(4,\frac {\pi }{3})$ and $(4,-\frac {5\pi }{3})$.
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