Why does my random trajectory systematic biais in angle? - r

Question

I have a tracjectory in 2D (list of x,y positions).
I am trying to measure the angles of the motion between consecutive points.
So I calculate the scalar product of the two consecutive vectors, divide by the vector norms, and this gives me the cosinus of the angles I am looking for.
However, when I generate totally random trajectories (by generating random x and random y), I always have a high number of cos results very close to -1, or 1. While I was expecting to have all cos between -1 and 1 equally likely.
Here's my code to generate the trajectories (after correction from the comments below), and calculate the cosinus:
cost = c()
t = seq(0,500,0.5)
x = 1*runif( length(t),-1,1 )
y = 1*runif( length(t),-1,1 )
x = cumsum(x)
y = cumsum(y)
step = 1
dstep = 2
for ( j in 1:((length(t)-dstep)))
{
x1 = x[j+step]-x[j]
y1 = y[j+step]-y[j]
x2 = x[j+dstep]-x[j+step]
y2 = y[j+dstep]-y[j+step]
n1 = sqrt( x1*x1 + y1*y1 )
n2 = sqrt( x2*x2 + y2*y2 )
if ( (n1*n2) > 0 )
{
scal = x1*x2 + y1*y2
cost = c( cost, scal/(n1*n2) )
#print( paste(n1, " ", n2, " ", n1*n2, " ", scal, " ", x1, " ", x2, " ", scal/(n1*n2), sep="") )
}
}
When i look at the histogram of the cost results, I always have a high number of cost very close to -1 and 1:
> hist(cost, plot=F)
$breaks
[1] -1.00 -0.95 -0.90 -0.85 -0.80 -0.75 -0.70 -0.65 -0.60 -0.55 -0.50 -0.45
[13] -0.40 -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15
[25] 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
[37] 0.80 0.85 0.90 0.95 1.00
$counts
[1] 108 43 32 20 22 21 19 20 19 17 16 19 8 19 23 17 15 10 18
[20] 22 15 19 14 15 18 16 21 11 18 20 16 35 23 24 24 20 23 33
[39] 37 107
Any idea where I'm wrong or why it should do that ?
Thanks for help

Answer 1

In case somebody else meet this problem, here's the summary of the solution from the comments:
Actually this distribution of the cos is what you get when angles are uniformly distributed! Consider hist(cos(runif(1000, min = 0, max = 2*pi))). So it's working as expected. cos just moves quickly over 0 and slowly over 1 and -1. See plot(cos, from = 0, to = 2*pi).
Which is indeed explained there: https://math.stackexchange.com/questions/1153339/distribution-of-cosine-of-uniformly-random-variables
The solution is thus that it is normal to have more values of cosinus close to 1 and -1 from a distribution of totally random angles.