Question

What is the distance between the following polar coordinates?: `(3,(12pi)/8), (9,(5pi)/8)`

Answer 1

`d approx 9.97`

Explanation:

We can convert each to Cartesian:

`(3, (3pi)/2) rightarrow 3e^(i(3pi)/2) = 3(-i) = -3i`
`(9, (5pi)/8) rightarrow 9e^(i(5pi)/8) = -(9sqrt(2-sqrt(2)))/2 + i (9sqrt(2+sqrt(2)))/2`

In order to do the trig functions for `(5pi)/8`, I used the half angle formulae and the angle addition formulae (just to be careful about signs):

`(5pi)/8 = pi/2 + pi/8`

`cos(pi/8) = cos(1/2 (pi/4)) = sqrt( (1+cos(pi/4)) /2) = sqrt( 2 + sqrt(2))/2`

`sin(pi/8) = sin(1/2 (pi/4)) = sqrt( (1-cos(pi/4)) /2) = sqrt( 2 - sqrt(2))/2`

`cos(pi/2 + pi/8) = - sin(pi/8) = -sqrt( 2- sqrt(2))/2`
`sin(pi/2 + pi/8) = cos(pi/8) = sqrt(2 + sqrt(2))/2`

Now, we can just apply Pythagorean theorem:

`d^2 = Delta x^2 + Delta y^2`
`= 81/4 * (2 - sqrt(2)) + ((9sqrt(2+sqrt2))/2 + 3)^2`
`= 81/4 * (2 - sqrt(2)) + 81/4 * (2+sqrt2) + 9 + 2 * 3 * 9/2 *sqrt(2+sqrt2)`
`= 81/2 + 18/2 + 27sqrt(2+sqrt2)`
`d^2= 99/2 + 27sqrt(2+sqrt2)`

Which means that the distance is actually

`d = sqrt(99/2 + 27sqrt(2+sqrt2)) approx 9.97`

Answer 2

Distance=`18` units.

Explanation:

Polar coordinates are written as `[r,theta]`. From Pythagoras, `x^2+y^2=r^2`....and from the same triangle, `x=rcostheta`, `y=rsintheta`

So `[3, [12pi]/[8]]`= means `x=3cos12pi/[8]` and `y=3sin12pi/[8]`

So coordinates of this point is`[0,1]`