simplify e^{(4-i)t}

Solution

simplify

e^{(4 - i) t}

e^{4 t} cos (t) - i e^{4 t} sin (t)

Solution steps

e^{(4 - i) t}

Apply imaginary number rule:

e^{a + ib} = e^{a} (cos (b) + i sin (b))

= e^{4 t} (cos (- t) + i sin (- t))

cos (- t) = cos (t)

sin (- t) i = - i sin (t)

= e^{4 t} (cos (t) - i sin (t))

Rewrite

e^{4 t} (cos (t) - i sin (t))

in standard complex form:

e^{4 t} cos (t) - e^{4 t} sin (t) i

= e^{4 t} cos (t) - e^{4 t} sin (t) i

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