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Chapter 11: Problem 2

Plot the points with polar coordinates \(\left(2, \frac{\pi}{6}\right)\) and \(\left(-3,-\frac{\pi}{2}\right) .\) Give two alternative sets of coordinate pairs for both points.

Short Answer

Expert verified
Question: Plot the given polar coordinates on a plane and provide alternative coordinate pairs for each point: \((2, \frac{\pi}{6})\) and \((-3, -\frac{\pi}{2})\). Answer: The cartesian coordinates of the given points are \((\sqrt{3}, 1)\) and \((0,3)\). Alternative coordinate pairs for the first point are \((2, \frac{13\pi}{6})\) and \((2, -\frac{11\pi}{6})\). Alternative coordinate pairs for the second point are \((-3, \frac{3\pi}{2})\) and \((-3, -\frac{5\pi}{2})\).

Step by step solution

01

Plot the given points

To plot the given points in polar coordinates, we need to find the corresponding cartesian coordinates \((x, y)\). To convert polar coordinates \((r, \theta)\) to cartesian coordinates \((x, y)\), we use the equations \(x = r \cos{(\theta)}\) and \(y = r \sin{(\theta)}\) For point 1: \((2, \frac{\pi}{6})\) \(x = 2 \cos{(\frac{\pi}{6})} = \sqrt{3}\) \(y = 2 \sin{(\frac{\pi}{6})} = 1\) So, the cartesian coordinates for point 1 are \((\sqrt{3}, 1)\). Plot this point on the plane. For point 2: \((-3, -\frac{\pi}{2})\) \(x = -3 \cos{(-\frac{\pi}{2})} = 0\) \(y = -3 \sin{(-\frac{\pi}{2})} = 3\) So, the cartesian coordinates for point 2 are \((0,3)\). Plot this point on the plane.
02

Find alternative coordinate pairs

We can find alternative coordinate pairs for the given points by adding or subtracting multiples of \(2\pi\) to their angles. This is because polar coordinates are periodic with a period of \(2\pi\). For point 1: \((2, \frac{\pi}{6})\) Alternative pair 1: \((2, \frac{\pi}{6} + 2\pi) = (2, \frac{13\pi}{6})\) Alternative pair 2: \((2, \frac{\pi}{6} - 2\pi) = (2, -\frac{11\pi}{6})\) For point 2: \((-3, -\frac{\pi}{2})\) Alternative pair 1: \((-3, -\frac{\pi}{2} + 2\pi) = (-3, \frac{3\pi}{2})\) Alternative pair 2: \((-3, -\frac{\pi}{2} - 2\pi) = (-3, -\frac{5\pi}{2})\)

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