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.