Laplace Transform of sin(at)
The Laplace transform of sin(at) is a/(s² + a²). See the table rule behind it, then edit the function to transform anything else.
L{sin(at)} = a/(s² + a²). L{sin(at)} = a/(s² + a²). The numerator is the frequency, not s.
Use t as the variable. Examples: t^3, e^(-2t), sin(3t), t*cos(2t), 4t^2+3. Write δ(t) as delta(t).
The result
L{sin(at)} = a/(s² + a²)
Worked example with numbers: L{sin(3t)} = 3/(s^2 + 9)
- L{sin(3t)} = 3/(s^2 + 9) Table rule: L{e^(at)sin(bt)} = b/((s − a)² + b²)
L{sin(at)} = a/(s² + a²). The numerator is the frequency, not s.
More trigonometric transforms
Other transforms
Frequently asked questions
What is the Laplace transform of sin(at)? ▾
L{sin(at)} = a/(s² + a²). L{sin(at)} = a/(s² + a²). The numerator is the frequency, not s.
Which rule gives the transform of sin(at)? ▾
L{e^(at)sin(bt)} = b/((s − a)² + b²) applied to sin(3t).
How do I get back from F(s) to f(t)? ▾
Use the inverse Laplace transform. Split F(s) into partial fractions and invert each piece with the same table read backwards — the inverse Laplace transform calculator does this step by step.
About this Calculator
The Laplace transform of sin(at) is a/(s² + a²). See the table rule behind it, then edit the function to transform anything else.