What is the distance between the following polar coordinates?: # (1,(-5pi)/12), (3,(pi)/8) #

Question

What is the distance between the following polar coordinates?: #   (1,(-5pi)/12), (3,(pi)/8)   #

Isaiah Allen · Accepted Answer

#3.284#

Write each polar point as a cartesian point using parametric equations.

#x=rcos(theta)#

#y=rsin(theta)#

For #(1,-5pi/12)#

#x(t) = 1 cos(-5pi/12) ~~ .259#

#y(t)= 1 sin(-5pi/12) ~~ -.966#

For #(3,pi/8)#

#x= 3 cos(pi/8) ~~ 2.772#

#y=3 sin(pi/8) ~~ 1.148#

So now we have the points in cartesian form:

#(.259,-.966), (2.772,1.148)#

Use distance formula:

#d=sqrt((2.772-.259)^2+(1.148-(-.966))^2)~~3.284#

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:

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

Substituting the given values:

$$ r_1 = 1, \theta_1 = -\frac{5\pi}{12}, r_2 = 3, \theta_2 = \frac{\pi}{8} $$

$$ \text{Distance} = \sqrt{1^2 + 3^2 - 2(1)(3)\cos\left(\frac{\pi}{8} + \frac{5\pi}{12}\right)} $$

$$ \text{Distance} = \sqrt{10 - 6\cos\left(\frac{\pi}{8} + \frac{5\pi}{12}\right)} $$

$$ \text{Distance} = \sqrt{10 - 6\cos\left(\frac{\pi}{8} - \frac{5\pi}{12}\right)} $$

$$ \text{Distance} \approx \sqrt{10 - 6\cos\left(\frac{3\pi}{8}\right)} $$

$$ \text{Distance} \approx \sqrt{10 - 6\left(-\frac{\sqrt{2}}{2}\right)} $$

$$ \text{Distance} \approx \sqrt{10 + 3\sqrt{2}} $$

So, the distance between the given polar coordinates is approximately $ \sqrt{10 + 3\sqrt{2}} $.