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Chapter 11: Problem 25

Express the following Cartesian coordinates in polar coordinates in at least two different ways. $$(-4,4 \sqrt{3})$$

Short Answer

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Question: Convert the Cartesian coordinates (-4, 4鈭�3) into polar coordinates in two different ways. Answer 1: (8, 240掳) Answer 2: (8, 600掳)

Step by step solution

01

Find the magnitude (r)

To find the magnitude of the polar coordinates (r), we can use the Pythagorean theorem which states that $$r^2 = x^2 + y^2$$. Using the given Cartesian coordinates \((-4, 4\sqrt{3})\), we can calculate r as follows: $$r^2 = (-4)^2 + (4\sqrt{3})^2$$ $$r^2 = 16 + 48$$ $$r^2 = 64$$ $$r = \sqrt{64}$$ $$r = 8$$
02

Find the angle (胃)

To find the angle (胃) of the polar coordinates, we can use the inverse tangent function (arctan). This is given by the formula \(胃 = \arctan \frac{y}{x}\). Using the given Cartesian coordinates \((-4, 4\sqrt{3})\), we can calculate 胃 as follows: $$胃 = \arctan{\frac{4\sqrt{3}}{-4}}$$ $$胃 = \arctan{-\sqrt{3}}$$ $$胃 = 240^\circ$$
03

Polar Coordinate Representation 1

Our first representation of the Cartesian coordinates \((-4, 4\sqrt{3})\) in polar coordinates is as follows: $$(8, 240^\circ)$$
04

Polar Coordinate Representation 2

Polar coordinates have an infinite number of equivalent representations because adding or subtracting multiples of \(360^\circ\) to the angle results in the same position. Therefore, we can represent the given Cartesian coordinates in another way using polar coordinates. We can add \(360^\circ\) to the angle: $$胃 = 240^\circ + 360^\circ$$ $$胃 = 600^\circ$$ Our second representation of the Cartesian coordinates \((-4, 4\sqrt{3})\) in polar coordinates is as follows: $$(8, 600^\circ)$$

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