The fraction sin(t)/t is a function with numerator sin(t) and denominator t. The Laplace transform of sin(t)/t is tan^{-1}(1/s). In this article, we will learn how to find the Laplace transform of sin(t)/t.

Table of Contents

## Laplace Transform of sint/t Formula

sint/t Laplace formula: The Laplace transform formula of sin(t)/t is given below:

L{sin(t)/t} = tan^{-1}(1/s).

## What is the Laplace Transform of sint/t?

**Answer:** The Laplace transform of sin(t)/t is tan^{-1}(1/s).

*Proof:*

We will use the division by t Laplace transform formula here. The formula is given below.

$L\{\frac{f(t)}{t} \} =\int_s^\infty F(s) ds$, where $L\{f(t)\}=F(s)$ **…(I)**

**Step 1:** Put f(t)=sin t.

∴ F(s) = L{f(t)} = L{sin t} = 1/(s^{2}+1)

**Step 2:** So from (I), we get the Laplace transform of sin(t)/t which is

L{sin(t)/t} = $\int_s^\infty \dfrac{1}{s^2+1} ds$

= $\Big[ \tan^{-1} s\Big]_s^\infty$

= tan^{-1} ∞ – tan^{-1} s

= π/2 – tan^{-1} s

= cot^{-1} s

= tan^{-1} (1/s).

So the Laplace transform of sin(t)/t is tan^{-1}(1/s).

Find the Laplace transform of sin(t)/t.Summary:L{sin(t)/t} = tan ^{-1}(1/s). |

**Also Read:**

Laplace transform of t: | 1/s^{2} |

Laplace transform of sin at: | a/(s^{2}+a^{2}) |

Laplace transform of cos at: | s/(s^{2}+a^{2}) |

Laplace transform of e^{-t}: | 1/(s+1) |

Laplace transform of 1: | 1/s |

## FAQs

**Q1: What is the Laplace transform of sin(at)/t?**

Answer: The Laplace transform of sin(at)/t is tan^{-1}(a/s).

**Q2: Find the Laplace transform of sin(2t)/t.**

Answer: The Laplace transform of sin(2t)/t is tan^{-1}(2/s).

**Q3: What is the Laplace transform of sin(3t)/t?**

Answer: The Laplace transform of sin(3t)/t is tan^{-1}(3/s).

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.