Find the Laplace transforms of the given functions. displaystyle g{{left({t}right)}}={4} cos{{left({4}{t}right)}}-{9} sin{{left({4}{t}right)}}+{2} cos{{left({10}{t}right)}}

Question

Find the Laplace transforms of the given functions.

g (t) = 4 \cos (4 t) - 9 \sin (4 t) + 2 \cos (10 t)

Answer 1

Step 1 : Introduction
Given function is,

g (t) = 4 \cos (4 t) - 9 \sin (4 t) + 2 \cos (10 t)

We have to evaluate the Laplace transform of given function g(t)
step 2

g (t) = 4 \cos (4 t) - 9 \sin (4 t) + 2 \cos (10 t)

using Laplace transform properties

L {\sin a t} = \frac{a}{s^{2} + a^{2}}

L {\cos a t} = \frac{s}{s^{2} + a^{2}}

Applying Laplace transform on g(t) function

∴ L {g (t)} = L {4 \cos (4 t) - 9 \sin (4 t) + 2 \cos (10 t)}

L {g (t)} = 4 L {\cos (4 t)} - 9 L {\sin (4 t)} + 2 L {\cos (10 t)}

= 4 (\frac{s}{s^{2} + 4^{2}}) - 9 (\frac{4}{s^{2} + 4^{2}}) + 2 (\frac{s}{s^{2} + 10^{2}})

= \frac{4 s}{s^{2} + 16} - \frac{36}{s^{2} + 16} + \frac{2 s}{s^{2} + 100}

Thus , Laplace transform of given function g(t) is

L {g (t)} = \frac{4 s}{s^{2} + 16} - \frac{36}{s^{2} + 16} + \frac{2 s}{s^{2} + 100}

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