Chapter 10.3, Problem 72E

To determine

To prove: The distance between two points $\left(r_1,{\theta }_1\right)$ and $\left(r_2,{\theta }_2\right)$ is $d=\sqrt{r_1{}^2+r_2{}^2-2r_1{r}_2\cos \left({\theta }_1-{\theta }_2\right)}$ .

Answer to Problem 72E

It is proved that the distance between two points $\left(r_1,{\theta }_1\right)$ and $\left(r_2,{\theta }_2\right)$ in polar coordinates is $d=\sqrt{r_1{}^2+r_2{}^2-2r_1{r}_2\cos \left({\theta }_1-{\theta }_2\right)}$ .

Explanation of Solution

Given information: The given two points are $\left(r_1,{\theta }_1\right)$ and $\left(r_2,{\theta }_2\right)$ .

Formula used: The distance between two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ in rectangular coordinates is $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$ .

Calculation:

Consider the polar coordinates of a point be $\left(r,\theta \right)$ . Write the rectangular coordinates of the same point.

  $\left(x,y\right)=\left(r\cos \theta ,r\sin \theta \right)$

Find the rectangular coordinates of the point $\left(r_1,{\theta }_1\right)$ .

  $\left(x_1,y_1\right)=\left(r_1\cos {\theta }_1,r_1\sin {\theta }_1\right)$

Find the rectangular coordinates of the point $\left(r_2,{\theta }_2\right)$ .

  $\left(x_2,y_2\right)=\left(r_2\cos {\theta }_2,r_2\sin {\theta }_2\right)$

Apply the formula for the distance between the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ .

  $\begin{array}{c}d=\sqrt{{\left(r_2\cos {\theta }_2-r_1\cos {\theta }_1\right)}^2+{\left(r_2\sin {\theta }_2-r_1\sin {\theta }_1\right)}^2}\\ =\sqrt{\begin{array}{l}\left(r_2{}^2{\cos }^2{\theta }_2+r_1{}^2{\cos }^2{\theta }_1-2\cdot r_2\cos {\theta }_2\cdot r_1\cos {\theta }_1\right)\\ +\left(r_2{}^2{\sin }^2{\theta }_2+r_1{}^2{\sin }^2{\theta }_1-2\cdot r_2\sin {\theta }_2\cdot r_1\sin {\theta }_1\right)\end{array}}\\ =\sqrt{r_2{}^2+r_1{}^2-2r_1{r}_2\left(\cos {\theta }_2\cdot \cos {\theta }_1+\sin {\theta }_2\cdot \sin {\theta }_1\right)}\\ =\sqrt{r_2{}^2+r_1{}^2-2r_1{r}_2\cos \left({\theta }_1-{\theta }_2\right)}\end{array}$

Hence, it is proved that the distance between two points $\left(r_1,{\theta }_1\right)$ and $\left(r_2,{\theta }_2\right)$ in polar coordinates is $d=\sqrt{r_1{}^2+r_2{}^2-2r_1{r}_2\cos \left({\theta }_1-{\theta }_2\right)}$ .