Convert the rectangular coordinates $$ (4,-4 \sqrt{3}) $$ to polar coordinates $$ (r, 8) $$, with $$ r0 $$ and $$ 0 \leq 8 \leq 2 \pi $$. Choose the correct answer for the question: a. $$ r=4,8=\frac{5 \pi}{4} $$

Question

Convert the rectangular coordinates $$ (4,-4 \sqrt{3}) $$ to polar coordinates $$ (r, 8) $$, with $$ r>0 $$ and $$ 0 \leq 8 \leq 2 \pi $$. Choose the correct answer for the question: a. $$ r=4,8=\frac{5 \pi}{4} $$

UpStudy AI · Accepted Answer

To convert the rectangular coordinates $$ (4, -4\sqrt{3}) $$ to polar coordinates $$ (r, \theta) $$, we use the following formulas:

1. The radius $$ r $$ is given by:
   $$
   r = \sqrt{x^2 + y^2}
   $$
   where $$ x = 4 $$ and $$ y = -4\sqrt{3} $$.

2. The angle $$ \theta $$ is given by:
   $$
   \theta = \tan^{-1}\left(\frac{y}{x}\right)
   $$

Let's calculate $$ r $$:

$$
r = \sqrt{4^2 + (-4\sqrt{3})^2} = \sqrt{16 + 48} = \sqrt{64} = 8
$$

Now, let's calculate $$ \theta $$:

$$
\theta = \tan^{-1}\left(\frac{-4\sqrt{3}}{4}\right) = \tan^{-1}(-\sqrt{3})
$$

The value of $$ \tan^{-1}(-\sqrt{3}) $$ corresponds to an angle in the fourth quadrant. The reference angle where $$ \tan(\theta) = \sqrt{3} $$ is $$ \frac{\pi}{3} $$. Therefore, in the fourth quadrant, the angle is:

$$
\theta = 2\pi - \frac{\pi}{3} = \frac{6\pi}{3} - \frac{\pi}{3} = \frac{5\pi}{3}
$$

Thus, the polar coordinates are:

$$
(r, \theta) = (8, \frac{5\pi}{3})
$$

However, the problem states that we need to express the angle as $$ 8 $$ in the form $$ (r, 8) $$. Since $$ 8 $$ is not the angle we calculated, we need to ensure that the angle is expressed correctly.

The correct polar coordinates are:

$$
(r, \theta) = (8, \frac{5\pi}{3})
$$

Since $$ r > 0 $$ and $$ 0 \leq \theta < 2\pi $$, we have:

- $$ r = 8 $$
- $$ \theta = \frac{5\pi}{3} $$

The answer provided in the question $$ a. \, r=4, \theta=\frac{5\pi}{4} $$ is incorrect based on our calculations. The correct polar coordinates are:

$$
\boxed{(8, \frac{5\pi}{3})}
$$
The correct polar coordinates are $$ r=8 $$ and $$ \theta=\frac{5\pi}{3} $$.

Convert the rectangular coordinates to polar coordinates ,
with and .
Choose the correct answer for the question:
a.

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Convert the rectangular coordinates to polar coordinates , with and . Choose the correct answer for the question: a.