Subjects geometry

Cartesian To Polar

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Cartesian To Polar


1. The problem is to convert a point or equation from Cartesian coordinates $(x,y)$ to polar coordinates $(r,\theta)$. 2. The formulas for conversion are: $$r = \sqrt{x^2 + y^2}$$ $$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$ 3. Here, $r$ is the distance from the origin to the point, and $\theta$ is the angle measured counterclockwise from the positive $x$-axis to the point. 4. To convert, first calculate $r$ using the Pythagorean theorem. Then find $\theta$ using the inverse tangent function, adjusting for the correct quadrant if necessary. 5. For example, if the Cartesian point is $(3,4)$: $$r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ $$\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 0.93 \text{ radians}$$ 6. So the polar coordinates are approximately $(5, 0.93)$ radians. 7. This method applies to any point or equation conversion from Cartesian to polar coordinates.