Find the distance between the points with polar coordinates (3, 120°) and (0.5, 49°)? Is the answer 2.88?
(3 , 120)
x = r cos T = 3 (-.5) = - 1.5
y = r sin T = 3 ( .866) = 2.598
(. 5 , 49)
x = .5 (cos 49) = .3280
y = .5 (sin 49) = .3774
change in x = 1.828
change in y = 2.221
d^2 = 1.828^2 + 2.221^2
d = 2.88
To find the distance between the two points with polar coordinates, you can use the formula:
d = √(r₁² + r₂² - 2r₁r₂cos(θ₂ - θ₁))
Let's use this formula to calculate the distance between the given points.
First, let's denote the polar coordinates of the first point as (r₁, θ₁) and the polar coordinates of the second point as (r₂, θ₂).
Given:
First point: (3, 120°)
Second point: (0.5, 49°)
Now, let's plug in the values into the distance formula and calculate:
d = √(3² + 0.5² - 2 * 3 * 0.5 * cos(49° - 120°))
= √(9 + 0.25 - 3 * 0.5 * cos(-71°))
= √(9 + 0.25 - 1.5 * cos(-71°))
= √(9 + 0.25 - 1.5 * cos(109°))
≈ √(9 + 0.25 - 1.5 * (-0.42262)) [converting the cosine value from degrees to radians using a calculator]
≈ √(9 + 0.25 + 1.26793)
≈ √(10.25 + 1.26793)
≈ √(11.51793)
≈ 3.39594
Therefore, the distance between the points with polar coordinates (3, 120°) and (0.5, 49°) is approximately 3.40, not 2.88.