In this paper, we show that the asymptotic estimate for the expected number of K-level crossings of a random hyperbolic polynomial a1sinhx+a2sinh2x++ansinhnx, where aj(j=1,2,,n) are independent normally distributed random variables with mean zero and variance one, is (1/π)logn. This result is true for all K independent of x, provided KKn=O(n). It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomial a1coshx+a2cosh2x++ancoshnx, with aj(j=1,2,,n) as before, is also (1/π)logn.