= Using the angular uniform in the Box-Muller radius breaks normality
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{title2=$Y=\sqrt{-2\log V}\sin(2\pi V)$}
If $V$ is uniform on $(0,1)$, the displayed random variable is not standard normal. For $V\leq1/2$ it is nonnegative; for $V>1/2$, its radius is at most $\sqrt{2\log2}$, so it is always bounded below by $-\sqrt{2\log2}$. A <standard normal distribution> has an unbounded lower tail.
The <Box-Muller transform> instead uses a common radius $\sqrt{-2\log U}$ independent of the angle $2\pi V$ in both Cartesian coordinates. Their polar density is $re^{-r^2/2}/(2\pi)$, and dividing by the coordinate Jacobian $r$ gives the product of two standard normal densities. Changing a logarithm's uniform input changes the distribution, even if the resulting expression superficially resembles that formula.
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