The problem involves converting and finding alternate representations of a point in polar coordinates $(-5, 4\pi)$ and determining its equivalent coordinates for different conditions of $r > 0$ and $-2\pi \leq \theta < 0$.

Part (a): Convert $(-5, 4\pi)$ to $r > 0$ and $-2\pi \leq \theta < 0$

The initial polar point is $(-5, 4\pi)$, where:

• $r = -5$
• $\theta = 4\pi$

To make $r > 0$, we must reflect $r$ to positive and adjust $\theta$ accordingly: $r > 0 \implies r = 5$ For polar coordinates, adding or subtracting $2\pi$ from $\theta$ represents the same angle. Since the current $\theta = 4\pi$ is too large, subtract multiples of $2\pi$ to bring it within $-2\pi \leq \theta < 0$: $\theta = 4\pi - 2\pi = 2\pi$ $\theta = 2\pi - 2\pi = 0$ [ The Equivalent point (5.