The author gives some characterizations of the resolvent set \(\rho(A)\) of the generator \(A\) of a strongly continuous cosine function \(C(t)\) with the aid of the equation \[ u''(t)=Au(t)+f(t). \] One of these theorems shows that \(1\in\rho(C(1))\) if and only if, for every \(1\)-periodic function \(f\in C([0,1],X)\) (\(X\) is a Banach space), the above equation has a unique \(1\)-periodic mild solution of class \(C^1\). In case of a Hilbert space \(X\), the above equation has a unique \(1\)-periodic mild solution for any \(1\)-periodic function \(f\in C([0,1],X)\), if and only if, the associated sine function \(S(t)\) has the property that \(S(1)\) is invertible.