(Solved): 1. Using the definition of Laplace transforms, prove that a. \( \mathcal{L}\left\{\sin \left(\frac{ ...

1. Using the definition of Laplace transforms, prove that a. \( \mathcal{L}\left\{\sin \left(\frac{2}{3} t\right)=\frac{6}{9 s^{2}+4}\right\} \) 2. Using standard Laplace transforms and trigonometric identities, find Laplace transforms of a. \( \cos ^{2} t \) b. \( 5 \sin (\omega t-\alpha) \) where \( \omega \) and \( \alpha \) are constants c. \( \sinh ^{2} t \) 3. Determine the Laplace transforms of \( 3 e^{5 t}(4 \cos 3 t-3 \sin 3 t) \)