Find Lf(t) where f(t) = cos t, 0 < t < π sin t, sin t ≥ π Using the Heaviside unit step function.

$\displaystyle f(t)\ =\cos t\ [H(t)\ -H(t-\pi{})]\ +\ sint\ [H(t-\pi{})] \\[2ex] \displaystyle f(t)\ =\cos t\ [H(t)]\ +\ (sint\ -\cos t)\ [H(t-\pi{})] \\[5ex] \begin{align*} \displaystyle L\ \left[f(t)\right]\ &=\ L[\cos t\ H(t)]\ +\ L[(sint\ -\cos t)\ H(t-\pi{})] \\[2ex] \displaystyle &=\ L(\cos t)\ +\ e^{-\pi{}s}\ L[sin⁡(t+\pi{})\ –\cos ⁡(t+\pi{})] \\[2ex] \displaystyle &=\ L(\cos t)\ +\ e^{-\pi{}s}\ L[-\sin t\ +\cos t] \\[2ex] \displaystyle &=\ \frac{s}{s^2+\ 1}+\ e^{-\pi{}s}\ [\frac{-1}{s^2+\ 1}\ +\ \frac{s}{s^2+\ 1}] \\[2ex] \displaystyle &=\frac{s}{s^2+\ 1}+\ e^{-\pi{}s}\ [\frac{s-1}{s^2+\ 1}] \\[2ex] \end{align*}$