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

Question

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

Ezekiel Alexander · Accepted Answer

The distance is # ~~ 8.88# The two points and the origin form a triangle with sides, #a = 2, b = 7,# and the angle between them #C = (7pi)/4 - (7pi)/8 = (7pi)/8#. Therefore, the distance between the two points will be the length of side, c, and we can use the Law of Cosines to find its length: #c = sqrt(a^2 + b^2 - 2(a)(b)cos(C)# #c = sqrt(2^2 + 7^2 - 2(2)(7)cos((7pi)/8))# #c ~~ 8.88#

Emma Aldridge · Answer

undefined To find the distance between two polar coordinates, you can use the formula:

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

where $ (r_1, \theta_1) $ and $ (r_2, \theta_2) $ are the polar coordinates.

For the given coordinates $ (2, \frac{7\pi}{4}) $ and $ (7, \frac{7\pi}{8}) $, the distance is:

$$ \text{Distance} = \sqrt{2^2 + 7^2 - 2 \cdot 2 \cdot 7 \cdot \cos\left(\frac{7\pi}{8} - \frac{7\pi}{4}\right)} $$

Simplify the expression inside the square root:

$$ \cos\left(\frac{7\pi}{8} - \frac{7\pi}{4}\right) = \cos\left(\frac{7\pi}{8} - \frac{14\pi}{8}\right) = \cos\left(-\frac{7\pi}{8}\right) $$

Since cosine is an even function, $ \cos(-\theta) = \cos(\theta) $, so:

$$ \cos\left(-\frac{7\pi}{8}\right) = \cos\left(\frac{7\pi}{8}\right) $$

Now, plug this back into the formula:

$$ \text{Distance} = \sqrt{2^2 + 7^2 - 2 \cdot 2 \cdot 7 \cdot \cos\left(\frac{7\pi}{8}\right)} $$

$$ \text{Distance} = \sqrt{4 + 49 - 28\cos\left(\frac{7\pi}{8}\right)} $$

$$ \text{Distance} = \sqrt{53 - 28\cos\left(\frac{7\pi}{8}\right)} $$

Now, calculate the cosine of $ \frac{7\pi}{8} $:

$$ \cos\left(\frac{7\pi}{8}\right) = \cos\left(\frac{\pi}{8}\right) $$

Since $ \cos(\theta) = \cos(-\theta) $, and $ \cos(\pi - \theta) = -\cos(\theta) $, we have:

$$ \cos\left(\frac{\pi}{8}\right) = -\cos\left(\pi - \frac{\pi}{8}\right) = -\cos\left(\frac{7\pi}{8}\right) $$

Substitute this back into the distance formula:

$$ \text{Distance} = \sqrt{53 + 28\cos\left(\frac{7\pi}{8}\right)} $$

$$ \text{Distance} = \sqrt{53 + 28(-\cos\left(\frac{\pi}{8}\right))} $$

$$ \text{Distance} = \sqrt{53 - 28\cos\left(\frac{\pi}{8}\right)} $$

Therefore, the distance between the polar coordinates $ (2, \frac{7\pi}{4}) $ and $ (7, \frac{7\pi}{8}) $ is $ \sqrt{53 - 28\cos\left(\frac{\pi}{8}\right)} $.