# Needle probability distribution problem

## Homework Statement

part a(complete)
The needle on a broken car speedometer is free to swing, and bounces
perfectly off the pins at either end, so that if you give it a flick it is equally likely to
come to rest at any angle between 0 and $$\pi$$.
part b(attempting)
We consider the same device as the previous problem, but this time
we are interested in the x-coordinate of the needle point--that is, the "shadow", or
"projection", of the needle on the horizontal line.
1. what is $$\rho$$(x)?

## Homework Equations

x=rcos($$\theta$$)

## The Attempt at a Solution

assuming that my solution for the probability function for $$\theta$$ is correct, (1/$$\theta$$)
(i'm going to cheat and use a unit needle for a second)
I took x=cos($$\theta$$)
and found arccos(x)=$$\theta$$
and d$$\theta$$=-dx/($$\sqrt{1-x^{2}}$$
but I am unsure of the usefulness of this relationship

so I took dx=-sin($$\theta$$)d$$\theta$$

and got p(x)dx=(-dx/$$\pi$$)(sin(arccos(x)))^-1