What is the distance between the following polar coordinates?: # (5,(17pi)/12), (9,(11pi)/8) #

Question

What is the distance between the following polar coordinates?: #   (5,(17pi)/12), (9,(11pi)/8)   #

Isabella Ackerman · Accepted Answer

#D~~4.095# units

Polar coordinates of point A is: #r_1=5# and #theta_1=(17pi)/12# Polar coordinates of point B is: #r_2=9# and #theta_2=(11pi)/8# Distance between points A and B can be found using: #D=sqrt(r_1^2+r_2^2-2r_1r_2cos(theta_1-theta_2)# #D=sqrt(5^2+9^2-2*5*9cos(((17pi)/12)-((11pi)/8))# #D~~4.095# units

Charles Arocha · 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:

$$ r_1 = 5, \theta_1 = \frac{17\pi}{12}, r_2 = 9, \theta_2 = \frac{11\pi}{8} $$

$$ d = \sqrt{5^2 + 9^2 - 2(5)(9)\cos\left(\frac{11\pi}{8} - \frac{17\pi}{12}\right)} $$

$$ d = \sqrt{25 + 81 - 90\cos\left(\frac{11\pi}{8} - \frac{17\pi}{12}\right)} $$

$$ d = \sqrt{106 - 90\cos\left(\frac{11\pi}{8} - \frac{17\pi}{12}\right)} $$

Now, calculate the difference of the angles:

$$ \frac{11\pi}{8} - \frac{17\pi}{12} = \frac{33\pi}{24} - \frac{34\pi}{24} = -\frac{\pi}{24} $$

$$ d = \sqrt{106 - 90\cos\left(-\frac{\pi}{24}\right)} $$

$$ d = \sqrt{106 - 90\cos\left(\frac{\pi}{24}\right)} $$

$$ d \approx \sqrt{106 - 90 \times 0.9659} $$

$$ d \approx \sqrt{106 - 86.931} $$

$$ d \approx \sqrt{19.069} $$

$$ d \approx 4.366 $$

Therefore, the distance between the given polar coordinates is approximately $4.366$.