Question

Find Laplace transforms of sin 2t cos 3t

Answer 1

Answer:

see the image

Step-by-step explanation:

use trigo identity of 2SinA CosB to solve

Answer 2

Concept Introduction: Trigonometry is used in finding the lengths of the side of a triangle and it's angles.

Given:

We have been Given:

$\sin(2t) \cos(3t)$

To Find:

We have to Find: Solve the Laplace Transformation equation.

Solution:

According to the problem, we first trigonometrically solve the equation,

$2 \sin(a) \cos(b) = \sin(a + b) + \sin(a - b)$

therefore, according to the formula,

$\sin(2t) \cos(3t) = \frac{1}{2} \times ( \sin(2t + 3t) + \sin(2t - 3t) ) = \frac{1}{2} \times ( \sin(5t) - \sin(t)$

According to the Laplace Transformation equation,

$f(t) = \sin(at) = \frac{ {a}^{2} }{ {s}^{2} + {a}^{2} }$

therefore,

$= \frac{1}{2} \times ( \frac{ {5}^{2} }{ {s}^{2} + {5}^{2} } + \frac{ {1}^{2} }{ {s}^{2} + {1}^{2} } ) \\ = \frac{1}{2 } \times ( \frac{25}{ {s}^{2} + 25 } + \frac{1}{ {s}^{2} + 1} )$

Final Answer: The Laplace Transformation of the question is

$\frac{1}{2} \times ( \frac{25}{ {s}^{2} + 25 } + \frac{1}{ {s}^{2} + 1} )$

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