# What is the distance between (1 ,( pi)/4 ) and (4 , ( 3 pi )/2 )?

Jan 3, 2016

The first coordinate is from the unit circle ($\frac{\pi}{4} = {45}^{o}$) and the second coordinate lies directly on the y-axis ($\frac{3 \pi}{2} = {270}^{o}$)

#### Explanation:

$\left(1 , \frac{\pi}{4}\right)$ from the unit circle is $\left(\frac{\sqrt{2}}{2} , \frac{\sqrt{2}}{2}\right)$

$\left(4 , \frac{3 \pi}{2}\right)$ on the y-axis $\left(- 4 , 0\right)$

Using the distance formula ...

distance =sqrt[(-4-sqrt2/2)^2+((0-sqrt2/2)^2]

$\approx 4.7599$

hope that helped

Jan 5, 2016

$4.761$ units

#### Explanation:

The distance formula for polar coordinates is

d=sqrt(r_1^2+r_2^2-2r_1r_2Cos(theta_1-theta_2)
Where $d$ is the distance between the two points, ${r}_{1}$, and ${\theta}_{1}$ are the polar coordinates of one point and ${r}_{2}$ and ${\theta}_{2}$ are the polar coordinates of another point.
Let $\left({r}_{1} , {\theta}_{1}\right)$ represent $\left(1 , \frac{\pi}{4}\right)$ and $\left({r}_{2} , {\theta}_{2}\right)$ represent $\left(4 , \frac{3 \pi}{2}\right)$.
implies d=sqrt(1^2+4^2-2*1*4Cos((pi)/4-(3pi)/2)
implies d=sqrt(1+16-8Cos((-5pi)/4)
$\implies d = \sqrt{17 - 8 \cdot \left(- 0.7085\right)} = \sqrt{17 + 5.668} = \sqrt{22.668} = 4.761$ units
$\implies d = 4.761$ units (approx)
Hence the distance between the given points is $4.761$ units.