Bounds for sine and cosine via eigenvalue estimation

Abstract

De ne n × n tridiagonal matrices T and S as follows: All entries of the main diagonal of T are zero and those of the rst super- and subdiagonal are one. The entries of the main diagonal of S are two except the (n, n) entry one, and those of the rst super- and subdiagonal are minus one. Then, denoting by (·) the largest eigenvalue, (T) = 2 cos n + 1 , (S−1) = 1 4 cos2 n 2n+1 . Using certain lower bounds for the largest eigenvalue, we provide lower bounds for these expressions and, further, lower bounds for sin x and cos x on certain intervals. Also upper bounds can be obtained in this way.