What is the distance between the following polar coordinates?: # (3,(pi)/4), (2,(7pi)/8) #

Question

What is the distance between the following polar coordinates?: #   (3,(pi)/4), (2,(7pi)/8)   #

Emma Aldridge · Accepted Answer

Distance between points is #4.19 # unit.

Polar coordinates of point A is #r_1= 3 ,theta_1=pi/4# Polar coordinates of point B is #r_2=2, theta_2=(7pi)/8# Distance between points #A and B# is #D= sqrt(r_1^2+r_2^2-2r_1r_2cos(theta1-theta2)# or # D = sqrt((9+4-2*3*2*cos (pi/4-(7pi)/8))) ~~ 4.19 # unit Distance between points is #4.19 # unit [Ans]

Aurora Alvarado · Answer

undefined To find the distance between two polar coordinates (r1, θ1) and (r2, θ2), you can use the formula:

distance = √(r1^2 + r2^2 - 2 * r1 * r2 * cos(θ2 - θ1))

Given the coordinates (3, π/4) and (2, 7π/8),

r1 = 3, θ1 = π/4
r2 = 2, θ2 = 7π/8

Substituting these values into the formula:

distance = √(3^2 + 2^2 - 2 * 3 * 2 * cos(7π/8 - π/4))

Now, calculate the cosine of the difference of the angles:

cos(7π/8 - π/4) = cos(π/8)

Substitute this into the distance formula:

distance = √(9 + 4 - 12 * cos(π/8))

Calculate the cosine value:

cos(π/8) ≈ 0.9239

Substitute this value into the distance formula:

distance = √(9 + 4 - 12 * 0.9239)
distance ≈ √(13 - 11.0868)
distance ≈ √1.9132
distance ≈ 1.383

Therefore, the distance between the given polar coordinates is approximately 1.383 units.

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:

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

Given $ (r_1, \theta_1) = (3, \frac{\pi}{4}) $ and $ (r_2, \theta_2) = (2, \frac{7\pi}{8}) $, substitute these values into the formula and calculate the distance.