What is the distance between the following polar coordinates?: # (2,(pi)/8), (7,(3pi)/8) #

Question

What is the distance between the following polar coordinates?: #   (2,(pi)/8), (7,(3pi)/8)   #

Aurora Alvarado · Accepted Answer

# sqrt( 53 - 14sqrt 2 ) #

As scene of the picture, i have drawn each of the polar coordinates, with there respective lengths and angles

We can see that #alpha = (3pi)/8 - pi/8 = pi/4 #

Now we can use the cosine rule to find #x# our distance:

#A^2 = B^2 + C^2 -2BCcosa#

# => x^2 = 7^2 + 2^2 - (2*7*2*cos(pi/4))#

#=> x^2 = 53 - 14sqrt2 #

As #cos(pi/4) = sqrt2 /2 #

Hence yielding:

# sqrt( 53 - 14sqrt 2 ) #

Adam Adkins · Answer

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

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

Substitute the given values:

$ r_1 = 2 $, $ \theta_1 = \frac{\pi}{8} $

$ r_2 = 7 $, $ \theta_2 = \frac{3\pi}{8} $

Then, plug these values into the formula and calculate:

$$ d = \sqrt{2^2 + 7^2 - 2(2)(7)\cos\left(\frac{3\pi}{8} - \frac{\pi}{8}\right)} $$

$$ d = \sqrt{4 + 49 - 28\cos\left(\frac{\pi}{2}\right)} $$

$$ d = \sqrt{53 - 0} $$

$$ d = \sqrt{53} $$

So, the distance between the polar coordinates $ (2, \frac{\pi}{8}) $ and $ (7, \frac{3\pi}{8}) $ is $ \sqrt{53} $ units.