Questions: Step 4 (b) (-8, 3π/2) For a new point, we are asked to find two other pairs of polar coordinates to represent the point, one with the radius r>0 and one with r<0. We know from a previous part that we can add 2π to the angle of the given polar coordinates to find a second point. We also know that (-r, θ) represents the same point as (r, θ+σ). We use these facts to find two pairs of polar coordinates of the given point as follows. (-8, 3π/2) r>0 (r, θ)=(8, π/2), r<0 (r, θ)=( )

Step 4
(b) (-8, 3π/2)

For a new point, we are asked to find two other pairs of polar coordinates to represent the point, one with the radius r>0 and one with r<0.
We know from a previous part that we can add 2π to the angle of the given polar coordinates to find a second point. We also know that (-r, θ) represents the same point as (r, θ+σ).

We use these facts to find two pairs of polar coordinates of the given point as follows.

(-8, 3π/2)
r>0 (r, θ)=(8, π/2),
r<0 (r, θ)=( )

Transcript text: Step 4 (b) $\left(-8, \frac{3 \pi}{2}\right)$ For a new point, we are asked to find two other pairs of polar coordinates to represent the point, one with the radius $r>0$ and one with $r<0$. We know from a previous part that we can add $2 \pi$ to the angle of the given polar coordinates to find a second point. We also know that ( $-r, \theta$ ) represents the same point as $(r, \theta+\sigma)$. We use these facts to find two pairs of polar coordinates of the given point as follows. \[ \begin{array}{c} \left(-8, \frac{3 \pi}{2}\right) \\ r>0 \quad(r, \theta)=\left(8, \frac{\pi}{2},\right. \\ r<0 \quad(r, \theta)=(\square) \end{array} \]

Solution

Solution Steps

Step 1: Find a point with r > 0

Given the point $(-8, \frac{3\pi}{2})$, we want to find an equivalent representation with $r > 0$. We can achieve this by adding $\pi$ to the angle and changing the sign of the radius. So we have $(-(-8), \frac{3\pi}{2} + \pi) = (8, \frac{5\pi}{2})$. Since $\frac{5\pi}{2}$ is equivalent to $\frac{\pi}{2}$, we have $(8, \frac{\pi}{2})$.

Step 2: Find a point with r < 0

Now, we want to find an equivalent representation with $r < 0$. We can add $2\pi$ to the original angle: $(-8, \frac{3\pi}{2} + 2\pi) = (-8, \frac{7\pi}{2})$. Or we can simply use the given coordinates since $r$ is already negative: $(-8, \frac{3\pi}{2})$. For simplicity, let's use $(-8, \frac{3\pi}{2})$.

Final Answer

The equivalent polar coordinates are: $r > 0$: $(8, \frac{\pi}{2})$ $r < 0$: $(-8, \frac{3\pi}{2})$

Thus, we have \[ \begin{array}{c} \left(-8, \frac{3 \pi}{2}\right) \\ r>0 \quad(r, \theta)=\left(8, \frac{\pi}{2}\right) \\ r<0 \quad(r, \theta)=\left(-8, \frac{3 \pi}{2}\right) \end{array} \] So the answer is $\boxed{(-8, \frac{3 \pi}{2})}$.

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