# What is the distance between the following polar coordinates?:  (2,(23pi)/12), (3,(7pi)/8)

Nov 24, 2016

The distance is $\approx 4.9897$

#### Explanation:

Let's begin by rotating both points an additional $\frac{\pi}{12}$, because that will make the first point go around a full rotation and its angle will be zero:

$\left(2 , 0\right) , \left(3 , \frac{23 \pi}{24}\right)$

These points form a triangle with the origin.

side $a = 2$

side $b = 3$

The $\angle C \text{ between the sides} = \frac{23 \pi}{24}$

We can use the Law of Cosines to find the length of side c, which is, also, the distance between the two points:

c = sqrt(a^2 + b^2 - 2(a)(b)cos(C)

c = sqrt(2^2 + 3^2 - 2(2)(3)cos((23pi)/24)

$c \approx 4.9897$