Determine the Laplace transform ofg(t)=4sin(8(t−7))u7(t)

The Laplace transform of a function g(t) is given by the integral from 0 to infinity of e^(-st)g(t) dt.

The Laplace transform of a function g(t) multiplied by a unit step function u_c(t) (which is 0 for t < c and 1 for t >= c) is given by the Laplace transform of g(t-c) multiplied by e^(-cs), where s is the complex frequency variable.

The Laplace transform of sin(at) is given by a/(s^2 + a^2).

So, to find the Laplace transform of g(t) = 4sin(8(t−7))u_7(t), we first find the Laplace transform of 4sin(8t), which is 4*8/(s^2 + 64) = 32/(s^2 + 64).

Then, we multiply this by e^(-7s) to account for the unit step function u_7(t), giving us the final answer of 32e^(-7s)/(s^2 + 64).