# What is the distance between the following polar coordinates?:  (4,(7pi)/4), (3,(3pi)/8)

Jan 3, 2016

$5.8508$ 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 $\left(4 , \frac{7 \pi}{4}\right)$ and $\left({r}_{2} , {\theta}_{2}\right)$ represent $\left(3 , \frac{3 \pi}{8}\right)$.
implies d=sqrt(4^2+3^2-2*4*3Cos((7pi)/4-(3pi)/8)
implies d=sqrt(16+9-24Cos((11pi)/8)
$\implies d = \sqrt{25 - 24 \cdot \left(- 0.3847\right)} = \sqrt{25 + 9.2328} = \sqrt{34.2328} = 5.8508$ units
$\implies d = 5.8508$ units (approx)
Hence the distance between the given points is $5.8508$.