Find the distance between the points with polar coordinates (2, 120°) and (1, 45°)
there is a formula that says:
distance between two points (r1,Ø1) and (r2,Ø2) , where both are in polar form is
√ [ r1^2 + r2^2 - 2(r1)(r2)cos(Ø2-Ø1) }
= √ [ 4 + 1 - 2(2)(1)cos(120-45) ]
= √ [ 5 - 4cos75]
= √[ 5 - (√6 - √2)/4] ----> exact value, or
= appr. 1.99
or
you could convert them to cartesian form
(2, 120°) = (-1,√3)
(1,45°) = (√2/2 , √2/2)
distance = √[ (√2/2 + 1)^2 + (√2/2 - √3)^2 ]
= 1.99
notice that the formula I gave is just a variation of the Cosine Law formula.
To find the distance between two points with polar coordinates (r1, θ1) and (r2, θ2), you can use the formula for the distance between two points in a polar coordinate system:
distance = √((r2^2 + r1^2) - 2(r2)(r1)(cos(θ2 - θ1)))
Let's apply this formula to find the distance between the points (2, 120°) and (1, 45°).
Step 1: Convert polar coordinates to rectangular coordinates.
We can use the following formulas to convert polar coordinates to rectangular coordinates:
x = r * cos(θ)
y = r * sin(θ)
For the first point: (2, 120°)
x1 = 2 * cos(120°) = -1
y1 = 2 * sin(120°) = √3
For the second point: (1, 45°)
x2 = 1 * cos(45°) = √2/2
y2 = 1 * sin(45°) = √2/2
Step 2: Calculate the distance.
Now, substitute these values into the distance formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
= √((√2/2 - (-1))^2 + (√2/2 - √3)^2)
= √((√2/2 + 1)^2 + (√2/2 - √3)^2)
Simplify the expression:
distance = √((√2/2 + 1)^2 + (√2/2 - √3)^2)
distance = √((1 + (√2/2)^2 + 2(1)(√2/2) + 1) + (1 + (√2/2)^2 - 2(√2/2)(√3) + 3))
distance = √(1 + 1/2 + √2/2 + √2 + 1 + 1/2 - √6 + 3)
distance = √(7 + √2 + √2/2 - √6)
Therefore, the distance between the points with polar coordinates (2, 120°) and (1, 45°) is √(7 + √2 + √2/2 - √6).