# What is the distance between (-5 ,( 5 pi)/12 ) and (-2 , ( pi )/2 )?

Jun 13, 2017

The distance between the dots is $\frac{1296 + {\pi}^{2}}{144}$

#### Explanation:

The distance in $x$ is $3$, since $| - 5 - \left(- 2\right) | = | - 3 | = 3$.
The distance in $y$ is $\frac{\pi}{12}$, since $\frac{\pi}{2} - \frac{5 \pi}{12} = \frac{6 \pi}{12} - \frac{5 \pi}{12} = \frac{\pi}{12}$.

With these information, we have the difference in both $x$ and $y$ axis. Now, we can apply Pythagoras Theorem:

${\left(D x\right)}^{2} + {\left(D y\right)}^{2} = {D}^{2}$, being
$D x$ the distance in $x$ and $D y$ the distance in $y$. So, we have:
${3}^{2} + {\left(\frac{\pi}{12}\right)}^{2} = 9 + \left({\pi}^{2} / 144\right) = \frac{1296 + {\pi}^{2}}{144}$.

Jun 13, 2017

$= \sqrt{29 - 20 \cos \left(- \frac{\pi}{12}\right)} \approx 3.11$

#### Explanation:

The distance is sqrt(r_1^2+r_2^2-2r_1r_2cos(theta_1-theta_2) if we are given ${P}_{1} = \left({r}_{1} , {\theta}_{1}\right)$ and ${P}_{2} = \left({r}_{2} , {\theta}_{2}\right)$.

This is an application of the cosine law. Taking the difference between ${\theta}_{1}$ and ${\theta}_{2}$ gives us the angle between side ${r}_{1}$ and side ${r}_{2}$. And the cosine law gives us the length of the ${3}^{r d}$ side.

So, for the two points given,

Distance: $\sqrt{{\left(- 5\right)}^{2} + {\left(- 2\right)}^{2} - 2 \left(- 5\right) \left(- 2\right) \cos \left(\frac{5 \pi}{12} - \frac{\pi}{2}\right)}$

$= \sqrt{25 + 4 - 20 \cos \left(- \frac{\pi}{12}\right)}$

$= \sqrt{29 - 20 \cos \left(- \frac{\pi}{12}\right)} \approx 3.11$