What is the distance between the following polar coordinates?: # (10,(17pi)/12), (4,(15pi)/8) #

Question

What is the distance between the following polar coordinates?: #   (10,(17pi)/12), (4,(15pi)/8)   #

Scarlett Allison · Accepted Answer

Distance between two points is #D= 10.27# unit

Polar coordinates of two points are #r_1=10 , theta_1=(17pi)/12 = # #255^0 and r_2=4 , theta_2=(15pi)/8 = 337.5^0# Cartesian coordinate of 1st point is # x_1= r_1cos theta_1#or #x_1= 10 cos 255 ~~ -2.588 and y_1= r_1sin theta_1 = # #y_1= 10 sin 255 ~~ -9.659 :. (-2.588,-9.659)# Cartesian coordinate of 2nd point is # x_2= r_2cos theta_2#or #x_2= 4 cos 337.5 ~~ 3.696 and y_2= r_2sin theta_2 = # #y_2= 4 sin 337.5 ~~ -1.53 :. (3.696,-1.531)# Distance between two points is #D=sqrt( (x_1 − x_2)^2 + (y_1 − y_2)^2) =# #D=sqrt( (-2.588 − 3.696)^2 + (-9.659 +1.531)^2) :. # #D= 10.27# unit [Ans]

Ezra Adams · Answer

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

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

Substituting the given values:

$$ r_1 = 10, \theta_1 = \frac{17\pi}{12}, r_2 = 4, \theta_2 = \frac{15\pi}{8} $$

we have:

$$ \text{Distance} = \sqrt{10^2 + 4^2 - 2(10)(4)\cos\left(\frac{15\pi}{8} - \frac{17\pi}{12}\right)} $$