prove that : (cos a + i sin a)(to power n ) = cos na + i sin na

Asked by shuchi.saumya | 25th Sep, 2010, 12:00: AM

Expert Answer:

cosa+isinan=cosna+isinnaProve the result by mathematical induction.For n = 1 , cosa+isina1=cosa+isinait is true.Now assume for n=k,cosa+isinak=coska+isinka  -------(1)T.P.T the result is true for n =k+1i.e T.P.T cosa+isinak+1=cos(k+1)a+isin(k+1)aL.H.S = cosa+isinak+1           =cosa+isinakcosa+isina1          =(coska+isinka)(cosa+isina) Now multiply the 2 brackets and use the formula cos(A+B) and sin(A+B)           = cos(k+1)a+isin(k+1)a = R.H.SHence proved by induction.">

Answered by  | 25th Sep, 2010, 09:47: AM

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