What is the distance between the following polar coordinates?: # (4,(-7pi)/12), (2,(pi)/8) #

Question

What is the distance between the following polar coordinates?: #   (4,(-7pi)/12), (2,(pi)/8)   #

Cameron Alexander · Accepted Answer

#D~~5.4535#

The distance formula for polar coordinates can be derived from the distance formula for rectangular coordinates

#D=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

Instead of using #x# and #y# values, though, we would just plug in their polar equivalents

#x=rcos(theta)#

#y=rsin(theta)#

Plugging those and using a couple of trigonometric identities, you get the following in purely polar coordinates

#D=sqrt(r_1^2+r_2^2-2r_1r_2cos(theta_1-theta_2))#

Plugging in the polar coordinates you have been given, we get

#D=sqrt((4)^2+(2)^2-2(4)(2)cos(-(7pi)/12-pi/8))#

#D=sqrt(16+4-16cos(-(17pi)/24))#

#D~~sqrt(20-16(-0.60876))#

#D~~5.4535#

Paisley Johnson · Answer

undefined To find the distance between two polar coordinates $ (r_1, \theta_1) $ and $ (r_2, \theta_2) $, you can use the formula:

$$
d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_2 - \theta_1)}
$$

Using the given polar coordinates:

$ r_1 = 4 $, $ \theta_1 = -\frac{7\pi}{12} $

$ r_2 = 2 $, $ \theta_2 = \frac{\pi}{8} $

Substituting these values into the formula:

$$
\begin{split}
d & = \sqrt{(4)^2 + (2)^2 - 2(4)(2)\cos\left(\frac{\pi}{8} - \left(-\frac{7\pi}{12}\right)\right)} \
& = \sqrt{16 + 4 - 16\cos\left(\frac{\pi}{8} + \frac{7\pi}{12}\right)} \
& = \sqrt{20 - 16\cos\left(\frac{\pi}{8} + \frac{7\pi}{12}\right)}
\end{split}
$$

To evaluate $ \cos\left(\frac{\pi}{8} + \frac{7\pi}{12}\right) $:

$$
\begin{split}
\cos\left(\frac{\pi}{8} + \frac{7\pi}{12}\right) & = \cos\left(\frac{3\pi}{24} + \frac{14\pi}{24}\right) \
& = \cos\left(\frac{17\pi}{24}\right)
\end{split}
$$

Since $ \cos $ is periodic with a period of $ 2\pi $, we can subtract $ 2\pi $ to bring the angle within one full period:

$$
\begin{split}
\cos\left(\frac{17\pi}{24}\right) & = \cos\left(\frac{17\pi}{24} - 2\pi\right) \
& = \cos\left(\frac{17\pi}{24} - \frac{48\pi}{24}\right) \
& = \cos\left(-\frac{31\pi}{24}\right)
\end{split}
$$

Since $ \cos(-x) = \cos(x) $, we have:

$$
\cos\left(-\frac{31\pi}{24}\right) = \cos\left(\frac{31\pi}{24}\right)
$$

So, $ d = \sqrt{20 - 16\cos\left(\frac{31\pi}{24}\right)} $. You can approximate this value numerically.