The Laplace transform of cosh(at) is s/(s^{2}-a^{2}). Here we focus on how to find the Laplace of cosh at, the hyperbolic cosine function.

The Laplace transform of cosh at is denoted by L{cosh at} and its formula is given by

L{cosh at} = s/(s^{2}-a^{2}).

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## Find Laplace Transform of cosh(at)

Answer: The Laplace transform of cosh at is s/(s^{2}-a^{2}). |

**Proof:**

By the definition of cosine hyperbolic functions, we have that

cosh(at) = $\dfrac{e^{at}+e^{-at}}{2}$

Now, we take Laplace transforms on both sides. By doing so, we get that

L{cosh (at)} = $L\big( \dfrac{e^{at}+e^{-at}}{2} \big)$

= $\dfrac{1}{2} L(e^{at}+e^{-at})$

= $\dfrac{1}{2} \big( L( e^{at}) + L(e^{-at}) \big)$ by the linearity property of Laplace transforms

= $\dfrac{1}{2} \big( \dfrac{1}{s-a} + \dfrac{1}{s+a} \big)$ by the Laplace formula of exponential functions which is L{e^{at}} =1/(s-a).

= $\dfrac{1}{2} \dfrac{s+a+s-a}{(s-a)(s+a)}$

= $\dfrac{1}{2} \dfrac{2s}{s^2-a^2}$

= $\dfrac{s}{s^2-a^2}$

So the Laplace transform of cosh(at) is equal to s/(s^{2}-a^{2}).

## Laplace Transform of cosh t

From above, L{cosh at} = s/(s^{2}-a^{2}).

Put a=1 in order to get the Laplace Transform of cosh t. Therefore,

L{cosh t} = s/(s^{2}-1).

Hence, the Laplace of cosh t is equal to s/(s^{2}-1).

More Laplace Transforms:

## FAQs

**Q1: What is the Laplace transform of cosh(at)?**

**Answer:** The Laplace transform of cosh(at) is s/(s^{2}-a^{2})

**Q2: What is the Laplace transform of cosh t?**

**Answer:** The Laplace transform of cosh t is equal to s/(s^{2}-1)

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.