Math Snap

STEP 1

1. Polar coordinates are given by $(r, \theta)$, where $r$ is the radial distance from the origin and $\theta$ is the angle measured from the positive x-axis.
2. Cartesian coordinates are given by $(x, y)$.
3. The conversion formulas from Cartesian to polar coordinates are:
- $r = \sqrt{x^2 + y^2}$
- $\theta = \tan^{-1}\left(\frac{y}{x}\right)$
4. Adjust $\theta$ to ensure $0 \leq \theta < 2\pi$.

STEP 2

1. Verify the Cartesian coordinates.
2. Convert Cartesian coordinates to polar coordinates.
3. Adjust $\theta$ if necessary to ensure it lies within the specified range.

STEP 3

Verify the Cartesian coordinates provided in the options. Note that the given Cartesian coordinates for the point are $(4, -4)$.

STEP 4

Calculate the radial distance $r$ using the formula $r = \sqrt{x^2 + y^2}$.
$$x = 4, \quad y = -4$$ $$r = \sqrt{4^2 + (-4)^2}$$ $$r = \sqrt{16 + 16}$$ $$r = \sqrt{32}$$ $$r = 4\sqrt{2}$$

STEP 5

Calculate the angle $\theta$ using the formula $\theta = \tan^{-1}\left(\frac{y}{x}\right)$.
$$\theta = \tan^{-1}\left(\frac{-4}{4}\right)$$ $$\theta = \tan^{-1}(-1)$$ $$\theta = -\frac{\pi}{4}$$

Adjust $\theta$ to ensure it lies within the range $0 \leq \theta < 2\pi$.
Since $\theta = -\frac{\pi}{4}$ is negative, add $2\pi$ to adjust:
$$\theta = -\frac{\pi}{4} + 2\pi$$ $$\theta = \frac{7\pi}{4}$$ The polar coordinates of the point $(4, -4)$ are:
$$\boxed{\left(4\sqrt{2}, \frac{7\pi}{4}\right)}$$