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# Converting Cartesian coordinates to polar coordinates

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Question: Convert the rectangular coordinates left parenthesis, minus, 3, square root of, 2, end square root, comma, minus, 3, square root of, 2, end square root, right parenthesis(−3 2 ​ ,−3 2 ​ ) into polar form. Express the angle using radians in terms of piπ over the interval 0, is less than or equal to, theta, is less than, 2, pi0≤θ<2π, with a positive value of rr.

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To convert the given rectangular coordinates $$(-3 \sqrt{2}, -3 \sqrt{2})$$ into polar form, we need to find the radius $$r$$ and the angle $$\theta$$. 1. **Calculate the radius $$r$$:** The radius $$r$$ is given by the distance from the origin to the point, which can be found using the Pythagorean theorem: $r = \sqrt{x^2 + y^2}$ Given $$x = -3\sqrt{2}$$ and $$y = -3\sqrt{2}$$: $r = \sqrt{(-3\sqrt{2})^2 + (-3\sqrt{2})^2}$ Simplify the squares: $r = \sqrt{(9 \cdot 2) + (9 \cdot 2)}$ $r = \sqrt{18 + 18}$ $r = \sqrt{36}$ $r = 6$ 2. **Calculate the angle $$\theta$$:** The angle $$\theta$$ is given by the arctangent of the ratio of $$y$$ to $$x$$: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$ Substitute $$x = -3\sqrt{2}$$ and $$y = -3\sqrt{2}$$: $\theta = \tan^{-1}\left(\frac{-3\sqrt{2}}{-3\sqrt{2}}\right)$ Simplify the ratio: $\theta = \tan^{-1}(1)$ The arctangent of 1 is $$\frac{\pi}{4}$$, but we need to consider the correct quadrant. Since both $$x$$ and $$y$$ are negative, the point is in the third quadrant. In the third quadrant, the angle is: $\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$ Therefore, the polar coordinates are: $(r, \theta) = (6, \frac{5\pi}{4})$

May 30, 2024, 6:55 a.m.

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