What is the distance between (4 , pi/12 ) and (5, pi/12 )?

1 Answer
Mar 10, 2016

$1$

Explanation:

$\left(r , \theta\right)$ in polar coordinates is $\left(r \cos \theta , r \sin \theta\right)$ in rectangular coordinates.

Hence, $\left(4 , \frac{\pi}{12}\right)$ in rectangular coordinates is

$\left(4 \cos \left(\frac{\pi}{12}\right) , 4 \sin \left(\frac{\pi}{12}\right)\right)$ or

$\left(4 \times 0.9659 , 4 \times 0.2588\right)$ or $\left(3.8636 , 1.0352\right)$

And, $\left(5 , \frac{\pi}{12}\right)$ in rectangular coordinates is

$\left(5 \cos \left(\frac{\pi}{12}\right) , 5 \sin \left(\frac{\pi}{12}\right)\right)$ or

$\left(5 \times 0.9659 , 5 \times 0.2588\right)$ or $\left(4.8295 , 1.294\right)$

The distance between the two points is $\sqrt{{\left(4.8295 - 3.8636\right)}^{2} + {\left(1.294 - 1.0352\right)}^{2}}$ or $\sqrt{{\left(0.9659\right)}^{2} + {\left(0.2588\right)}^{2}}$ or $\sqrt{0.93296281 + 0.06697744}$ or $\sqrt{0.99994025} = 1$

Actually the distance will be exactly $1$ as both points have same $\theta$ coordinate, bur difference between $r$ coordinate is exactly $1$. Minor difference has arisen because of rounding of numbers.