I was reading today in the book "The number $\pi$" (Eymard, Lafon) about the famed Buffon's needle problem. Assume that every strip (two adjacent rays) is of height $d$ and that the needle is of length $\ell <d$. Now, look at the angle $\vartheta$ between the center of the needle and the perpendicular line to the rays. Also, define $t$ as the distance between the center of the needle to the closest ray. I think that the formula $$t=\frac{\ell}{2}\cos \vartheta$$ is wrong.
Am I wrong?
Well Strictly speaking you are correct insofar as the formula $$ t = \frac{l}{2}\cos \theta $$ is wrong.
But the book does not claim that this is a formula that always holds.