What is the distance between the following polar coordinates?: # (7,(19pi)/12), (4,(13pi)/8) #

Question

What is the distance between the following polar coordinates?: #   (7,(19pi)/12), (4,(13pi)/8)   #

Evelyn Agard · Accepted Answer

#color(blue)(" Distance bet. the polar coordinates " d ~~ 3.0788#

Distance between two 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 = 7, r_2 = 4, theta_1 = ((19pi)/12)^c, theta_2 = ((13pi)/8)^c#

#d = sqrt (7^2 +4^2 - (2 * 7 * 4* cos ((13pi)/8 - (19pi)/12))#

#color(blue)(d )= sqrt (65 - 56 cos ((pi)/24)) color(blue)(~~ 3.0788)#

Samantha Adkison · Answer

undefined To find the distance between two polar coordinates $(r_1, \theta_1)$ and $(r_2, \theta_2)$, you can use the formula:
$$ \text{Distance} = \sqrt{(r_1 \cos \theta_1 - r_2 \cos \theta_2)^2 + (r_1 \sin \theta_1 - r_2 \sin \theta_2)^2} $$

Plugging in the values from the given coordinates $(7, \frac{19\pi}{12})$ and $(4, \frac{13\pi}{8})$, we get:
$$ \text{Distance} = \sqrt{(7 \cos \frac{19\pi}{12} - 4 \cos \frac{13\pi}{8})^2 + (7 \sin \frac{19\pi}{12} - 4 \sin \frac{13\pi}{8})^2} $$

Calculating the trigonometric functions and then finding the square root gives us the distance between these polar coordinates.