Laplace Transform of cos(at)

The Laplace transform of cos(at) is s/(s² + a²). See the table rule behind it, then edit the function to transform anything else.

L{cos(at)} = s/(s² + a²). L{cos(at)} = s/(s² + a²) — the numerator is s for cosine.

f(t) — the function to transform

Use t as the variable. Examples: t^3, e^(-2t), sin(3t), t*cos(2t), 4t^2+3. Write δ(t) as delta(t).

Try an example

The result

L{cos(at)} = s/(s² + a²)

Worked example with numbers: L{cos(3t)} = s/(s^2 + 9)

L{cos(3t)} = s/(s^2 + 9) Table rule: L{e^(at)cos(bt)} = (s − a)/((s − a)² + b²)

L{cos(at)} = s/(s² + a²) — the numerator is s for cosine.

More trigonometric transforms

L{sin(at)} L{sin(2t)} L{cos(2t)}

Other transforms

L{1} L{t} L{t²} L{t³} L{t⁴} L{e^(at)}

Frequently asked questions

What is the Laplace transform of cos(at)? ▾

L{cos(at)} = s/(s² + a²). L{cos(at)} = s/(s² + a²) — the numerator is s for cosine.

Which rule gives the transform of cos(at)? ▾

L{e^(at)cos(bt)} = (s − a)/((s − a)² + b²) applied to cos(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.

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