Question

What is the distance between the following polar coordinates?: `(6,(19pi)/12), (3,(15pi)/8)`

Answer 1

`4.804642396`

Explanation:

In general, the distance `d` between the points `(r_1, \theta_1)` & `(r_2, \theta_2)` is given as follows

`d=\sqrt{(r_1\cos\theta_1-r_2\cos\theta_2)^2+(r_1\sin\theta_1-r_2\sin\theta_2)^2}`

`=\sqrt{r_1^2\cos^2\theta_1+r_2^2\cos^2\theta_2-2r_1r_2\cos\theta_1\cos\theta_2+r_1^2\sin^2\theta_1+r_2^2\sin^2\theta_2-2r_1r_2\sin\theta_1\sin\theta_2}`

`=\sqrt{r_1^2(\cos^2\theta_1+\sin^2\theta_1)+r_2^2(\cos^2\theta_2+\sin^2\theta_2)-2r_1r_2(\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2)}`

`=\sqrt{r_1^2+r_2^2-2r_1r_2\cos(\theta_1-\theta_2)}`

Hence, the distance `d` between the given points `(r_1, \theta_1)\equiv(6, {19\pi}/12)` & `(r_2, \theta_2)\equiv(3, {15\pi}/8)` is given by substituting the corresponding values in above general formula as follows

`d=\sqrt{6^2+3^2-2\cdot 6\cdot 3\cos({19\pi}/12-{15\pi}/8)}`

`=\sqrt{45-36\cos({7\pi}/24)}`

`=4.804642396`