Polar coordinates are written as (r, theta)(r,θ), where rr is the distance from the origin and thetaθ is the angle with respect to the positive x-axis and the origin.
If we plot (3, {-15pi}/12)(3,−15π12) and (5, {-3pi}/8)(5,−3π8) on a graph, we notice we can form a triangle with the origin, (0,0)(0,0), as such:
![enter image source here]()
We know the length of two of the sides of the triangle, since the rr value of the coordinates tell us. We can also calculate the angle alphaα, since we have the angles of the two points.
|theta_1 - theta_2| = |{-15pi}/12 - {-3pi}/8| = {7pi}/8|θ1−θ2|=∣∣∣−15π12−−3π8∣∣∣=7π8
If the result was larger than piπ, we would simply subtract it from 2pi2π to get the angle inside of the triangle.
Finally, we know from trigonometry that if we know two sides and the angle in between, we can calculate the length of the other side using the Law of Cosines:
a^2 = b^2 + c^2 - 2bc cos(alpha)a2=b2+c2−2bccos(α)
we can use this law to arrive at an equation for the distance between two polar points, (r_1, theta_1)(r1,θ1) and (r_2, theta_2)(r2,θ2)
x = sqrt(r_1^2 + r_2^2 - 2r_1r_2cos(theta_1 - theta_2)x=√r21+r22−2r1r2cos(θ1−θ2)
or in this case,
x = sqrt(3^2 + 5^2 - 2*3*5*cos({7pi}/8)) ~~ 7.86 square
p.s. note the lack of absolute value signs in the final equation. That's because cos(x) = cos(-x)
