geometric probablity?

Question

A point P is placed at random on the circumfrence of the unit circle x^2+y^2=1. A point Q is placed at random in the interior of the circle. If the rectangle R has as its diagonal the line segment PQ, what is the probablity that R is contained entirely in the unit circle?

Answer 1

Since P is on the unit circle, P has coordinates (cos t, sin t) for some angle t. For the rectangle R to be contained in the circle, we must have

|Qx| <= |cos t|, |Qy| <= |sin t|

where (Qx,Qy) are the coordinates of Q.

Therefore, Q must fall within a rectangle of area 4|cos t * sin t| = 2|sin 2*t|, but it can fall anywhere within the circle, which has an area of pi.

We conclude that the probability is 2|sin 2t| / pi.