Write each function in terms of unit step functions. Find...

Write each function in terms of unit step functions. Find the Laplace transform of the given function.
$$f(t)=\left\{\begin{array}{lr}
0, & 0 \leq t<3 \pi / 2 \\
\sin t, & t \geq 3 \pi / 2
\end{array}\right.$$

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Key Concepts

Time Shifting Property

The time shifting property of Laplace transforms states that if a function is delayed in time, its Laplace transform is multiplied by an exponential decay factor. Specifically, if L{f(t)} = F(s), then for a delayed version represented as u(t-a)f(t-a), the Laplace transform is given by e^(–as)F(s). This shifting is particularly useful when dealing with functions that become active only after a certain time, allowing a straightforward computation of their transforms.

Laplace Transform

The Laplace transform is an integral transform that converts a time-domain function into a complex frequency-domain representation. Defined generally as L{f(t)} = ???? e^(–st)f(t) dt, it is a crucial tool in solving differential equations, analyzing linear systems, and handling initial value problems by simplifying operations such as differentiation into algebraic ones.

Unit Step Function

The unit step function, often denoted as u(t-a), is used to represent a function that 'turns on' at a specific time a. It is defined to be 0 when t is less than a and 1 when t is greater than or equal to a. This property makes it a useful tool for rewriting piecewise functions as a single expression that activates the intended behavior at the right moment.

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