Question

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

Answer 1

The distance between the dots is `(1296+pi^2)/144`

Explanation:

The distance in `x` is `3`, since `|-5-(-2)| = |-3| = 3`.
The distance in `y` is `pi/12`, since `pi/2 - (5pi)/12 = (6pi)/12 - (5pi)/12 = pi/12`.

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

`(Dx)^2 + (Dy)^2 = D^2`, being
`Dx` the distance in `x` and `Dy` the distance in `y`. So, we have:
`3^2 + (pi/12)^2 = 9 + (pi^2/144) = (1296+pi^2)/144`.

Answer 2

`=sqrt(29-20cos(-pi/12))~~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=(r_1, theta_1)` and `P_2=(r_2, theta_2)`.

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^(rd)` side.

So, for the two points given,

Distance: `sqrt((-5)^2+(-2)^2-2(-5)(-2)cos((5pi)/12-pi/2))`

`=sqrt(25+4-20cos(-pi/12))`

`=sqrt(29-20cos(-pi/12))~~3.11`