If aj(j=1,2,,n) are independent, normally distributed random variables with mean 0 and variance 1, if p is one half of any odd positive integer except one, and if vnp is the mean number of zeros on (0,2π) of the trigonometric polynomial a1cosx+2pa2cos2x++npancosnx, then vnp=μp{(2n+1)+D1p+(2n+1)1D2p+(2n+1)2D3p}+O{(2n+1)3}, in which μp={(2p+1)/(2p+3)}½, and D1p, D2p and D3p are explicitly stated constants.