What is the distance between the following polar coordinates?: $(6,pi/3), (0,pi/2)$

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Answer 1 · 2018-01-18T20:08:42

6.

In cartesian coordinates $(x,y)$ , the distance $(d)$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by:

$d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$ .

Now let us transform that to polar coordinates. In this new system, we know that $x = rhocostheta$ and $y = rhosintheta$ . Then:

$d^2 = (rho_2costheta_2 - rho_1costheta_1)^2 + (rho_2sintheta_2 -rho_1sintheta_1)^2$ .

With a few algebric steps, we can rewrite the above expression as:

$d^2 = rho_2^2(cos^2theta_2 + sin^2theta_2) + rho_1^2(cos^2theta_1 + sin^2theta_1) - 2rho_1rho_2(costheta_1costheta_2 + sintheta_1sintheta_2)$ .

Now, by using the trigonometric relations, we obtain the following expression for $d$ :

$d = sqrt(rho_1^2 + rho_2^2 - 2rho_1rho_2cos(theta_2 - theta_1)$ .

Then, applying our points $(rho_1,theta_1) = (6,pi/3)$ and $(rho_2,theta_2) = (0,pi/2)$ :

$d = sqrt(36 + 0 - 0) = 6$ .

What is the distance between the following polar coordinates?: (6,pi/3), (0,pi/2) (6,pi/3), (0,pi/2)