What is the distance between the following polar coordinates?: # (3,(5pi)/4), (1,(pi)/8) #

Question

What is the distance between the following polar coordinates?: #   (3,(5pi)/4), (1,(pi)/8)   #

Paisley Johnson · Accepted Answer

#color(chocolate)("Distance between the given two polar coordinates " d ~~ 3.94#

Distance between tow points knowing the polar coordinates is given by the formula using cosine rule

#d = sqrt(r_1 ^2 + r_2 ^2 - 2 r_1 r_2 cos (theta_2 - theta_1)#

#"Given ' r_1 = 3, r_2 = 1, theta_1 = ((5pi)/4)^c, theta_2 = (pi/8)^c#

#d = sqrt (3^2 + 1^2 - (2 * 3 * 1 * cos (pi/8 - (5pi)/4))#

#color(chocolate)(d )= sqrt (10 - 6 cos ((-9pi)/8)) color(chocolate)(~~ 3.94#

Luke Alvarez · Answer

undefined To find the distance between the polar coordinates $(3, \frac{5\pi}{4})$ and $(1, \frac{\pi}{8})$, you can use the formula for the distance $d$ between two points $(r_1, \theta_1)$ and $(r_2, \theta_2)$ in polar coordinates:

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

Substitute the given values:

$ r_1 = 3 $, $ \theta_1 = \frac{5\pi}{4} $

$ r_2 = 1 $, $ \theta_2 = \frac{\pi}{8} $

$$ d = \sqrt{3^2 + 1^2 - 2(3)(1) \cos\left(\frac{\pi}{8} - \frac{5\pi}{4}\right)} $$

Now, calculate the angle difference and the cosine value:

$$ \frac{\pi}{8} - \frac{5\pi}{4} = -\frac{15\pi}{8} $$

$$ \cos\left(-\frac{15\pi}{8}\right) = \cos\left(\frac{\pi}{8}\right) $$

Finally, substitute back into the distance formula:

$$ d = \sqrt{9 + 1 - 6 \cos\left(\frac{\pi}{8}\right)} $$

Evaluate $ \cos\left(\frac{\pi}{8}\right) $ and then calculate the distance $ d $.