Question

determine as transformadas de Laplace das funções definidas pelas seguintes expressões analíticas.

f(t) = cos^2 at

Answer 1

Vamos usar a seguinte relação trigonométrica:

$\cos(2\theta)=2\cos^2(\theta)-1\Longrightarrow 2\cos^2(\theta)=\cos(2\theta)+1\\ \Longrightarrow \cos^2(\theta)=\dfrac{1}{2}\cos(2\theta)+\dfrac{1}{2}$

Para $\theta=at\Longrightarrow \cos^2(at)=\dfrac{1}{2}\cos(2at)+\dfrac{1}{2}$

Assim, aplicando a Transformada de Laplace na função pedida:

$\mathcal{L}\{\cos^2(at)\}=\mathcal{L}\{\dfrac{1}{2}\cos(2at)+\dfrac{1}{2}\}\\\\ \mathcal{L}\{\cos^2(at)\}=\mathcal{L}\{\dfrac{1}{2}\cos(2at)\}+\mathcal{L}\{\dfrac{1}{2}\}\\\\ \mathcal{L}\{\cos^2(at)\}=\dfrac{1}{2}\mathcal{L}\{\cos(2at)\}+\dfrac{1}{2}\mathcal{L}\{1\}\\\\ \mathcal{L}\{\cos^2(at)\}=\dfrac{1}{2}\cdot\dfrac{s}{s^2+(2a)^2}+\dfrac{1}{2}\cdot\dfrac{1}{s}\\\\ \boxed{\mathcal{L}\{\cos^2(at)\}=\dfrac{s}{2s^2+8a^2}+\dfrac{1}{2s}}$