Using the definition of the laplace transform not tables go

Question

Question 1:

Sketch the function given below.

f(t) = e -20tu(t).

Question 2:

Using the definition of the Laplace transform (not tables), go through all the calculus and algebra to calculate the transform of the above equation.

Question 3:

In class it was stated that

sin (θ) = e jθ - e -jθ/2j.

Using the odd and even properties of the cos and the sin function, and Euler equation, prove this statement.