Find the Laplace Transform of sin^4t

The Laplace transform of sin^4t is denoted by L{sin⁴t}, and it is equal to 3/(8s) – s/[2(s²+4)] + s/[8(s²+16)]. So the Laplace formula of sin⁴t is equal to

L{sin⁴t} = $\dfrac{3}{8s} – \dfrac{s}{2(s^2+4)} + \dfrac{s}{8(s^2+16)}$.

Let us now prove the above Laplace transform formula of the fourth power of sint.

Laplace of sin⁴t

Question: What is the Laplace of sin⁴t?

Answer: L{sin4t} = 3/(8s) – s/[2(s2+4)] + s/[8(s2+16)].

Proof:

To find the Laplace transform of sin⁴t, we will follow the below steps.

Step 1:

First, we simplify the function sin⁴t using the trigonometric identity: 2sin²t = 1-cos2t. Therefore,

sin⁴t = sin²t × sin²t

⇒ sin⁴t = $\big( \dfrac{1-\cos 2t}{2}\big)^2$

⇒ sin⁴t = $\dfrac{1-2\cos 2t+\cos^2 2t}{4}$

⇒ sin⁴t = $\dfrac{1-2\cos 2t+ \frac{1+\cos 4t}{2}}{4}$ by the identity 2cos²θ = 1+cos2θ.

⇒ sin⁴t = $\dfrac{2-4\cos 2t+ 1+\cos 4t}{8}$

⇒ sin⁴t = $\dfrac{3-4\cos 2t+\cos 4t}{8}$.

Step 2:

So the Laplace of sin⁴t will be equal to

L{sin⁴t} = L$\big\{ \dfrac{3-4\cos 2t+\cos 4t}{8} \big\}$

= $\dfrac{1}{8} \big[ 3L\{1\} – 4L\{\cos 2t\}+ L\{\cos 4t\} \big]$, by the property of Laplace transforms.

= $\dfrac{1}{8} \big[ \dfrac{3}{s} – 4 \dfrac{s}{s^2+2^2}+ \dfrac{s}{s^2+4^2} \big]$ as we know that L{1} = 1/s and L{cosat} = s/(s²+a²).

= $\dfrac{3}{8s} – \dfrac{s}{2(s^2+4)}+ \dfrac{s}{8(s^2+16)}$.

Thus, the Laplace transform of sin⁴t is equal to L{sin⁴t} = 3/(8s) – s/[2(s²+4)] + s/[8(s²+16)].

Also Read:

Laplace transform of sinat/t

Laplace transform of u(t-2)

Laplace transform of t²u(t-1)

Laplace transform of (1-e^t)/t

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