Step by step
The question
Question: find the Laplace transform of sin(3t).
ℒ{sin(3t)} =
1 The idea
The Laplace transform turns a function of t into a function of s.
It is an integral, but we almost never work it out: we use a table made from it.
ℒ{f(t)} = ∫ from 0 to ∞ of e^(−st) f(t) dt
The table rows we need.
sin(bt) → bs² + b²
2 Use the table
Each part by its row: sin(3t) gives 3/(s² + 9).
ℒ{sin(3t)} = 3s2 + 9
Careful!
sin has b on top: 3, not 1.
Careful: not ℒ{sin(3t)} = 1s² + 9 ✗
3 Check
Let's check: read the table backwards.
3/(s² + 9) comes from sin(3t): the question.
Right!
backwards: 3s2 + 9 → sin(3t)
Let's look back.
First: the idea.
Then: use the table.
Last: check.
Answer
Careful: not ℒ{sin(3t)} = 1s² + 9 ✗