1. Let's start by stating the problem: We want to understand how to convert between Cartesian coordinates $(x,y)$ and polar coordinates $(r,\theta)$. 2. The formulas for conversion are: - From Cartesian to polar: $$r = \sqrt{x^2 + y^2}$$ $$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$ - From polar to Cartesian: $$x = r \cos\theta$$ $$y = r \sin\theta$$ 3. Important rules: - $r$ is the distance from the origin to the point, so it is always non-negative. - $\theta$ is the angle measured counterclockwise from the positive $x$-axis. - When calculating $\theta$, consider the quadrant of $(x,y)$ to get the correct angle. 4. Example: Convert Cartesian point $(3,4)$ to polar coordinates. - Calculate $r$: $$r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ - Calculate $\theta$: $$\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 0.93 \text{ radians}$$ 5. So the polar coordinates are approximately $(5, 0.93)$ radians. 6. To convert back, use: $$x = 5 \cos(0.93) \approx 3$$ $$y = 5 \sin(0.93) \approx 4$$ This confirms the conversion is correct.