What is the distance between the following polar coordinates?: # (5,(7pi)/4), (9,(11pi)/8) #

Question

What is the distance between the following polar coordinates?: #   (5,(7pi)/4), (9,(11pi)/8)   #

Charles Arocha · Accepted Answer

#color(violet)("Distance " d ~~ 8.4592#

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 = 5, r_2 = 9, theta_1 = ((7pi)/4)^c, theta_2 = ((11pi)/8)^c#

#d = sqrt (5^2 + 9^2 - (2 * 5 * 9 * cos ((11pi)/8 - (7pi)/4))#

#color(violet)(d )= sqrt (106 - 90 cos ((-3pi)/8)) color(violet)(~~ 8.4592#

Paisley Johnson · Answer

undefined To find the distance between two polar coordinates, 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:

$$r_1 = 5, \theta_1 = \frac{7\pi}{4}$$
$$r_2 = 9, \theta_2 = \frac{11\pi}{8}$$

$$d = \sqrt{5^2 + 9^2 - 2(5)(9)\cos\left(\frac{11\pi}{8} - \frac{7\pi}{4}\right)}$$

$$d = \sqrt{25 + 81 - 90\cos\left(\frac{11\pi}{8} - \frac{7\pi}{4}\right)}$$

$$d = \sqrt{106 - 90\cos\left(\frac{11\pi}{8} - \frac{7\pi}{4}\right)}$$

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

$$d \approx \sqrt{106 - 90(0.9239)}$$

$$d \approx \sqrt{106 - 83.55}$$

$$d \approx \sqrt{22.45}$$

$$d \approx 4.74$$

So, the distance between the given polar coordinates is approximately $4.74$.