Derive the transform of f(t)=sin kt by using the identity sin kt =frac{1}{2}u (e^{i kt}-e^{-i kt})

Carol Gates 2021-01-27 Answered
Derive the transform of f(t)=sinkt by using the identity sinkt=12u(eikteikt)
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Expert Answer

unessodopunsep
Answered 2021-01-28 Author has 105 answers
Step 1
Consider the given identity,
sinkt=1(2i)(eikteikt
Apply the Laplace transform on both sides,
L{sinkt}=L{1(2i)(eikteikt)}
Step 2
Apply the linearity of Laplace transform,
L{sinkt}=1(2i)[L{eikt}L{eikt}]
=1(2i)[1sik1s(ik)](L{eat}=1(sa))
=1(2i)[1(sik)1(s+ik)]
=1(2i)[(s+ik)(sik)(sik)(s+ik)]
=1(2i)[(2ik)(s2i2k2)]
=k(s2+k2)
Hence, L{sinkt}=k(s2+k2)
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