IIT JEE , AIEEE Forum - Mathematics , Trignometry

vandna_iit

Find the value of sin A * (cos A) + sin 2A * (cos A)^2 + sin 3A * (cos A)^3 +.......................................sin nA (cos A)^n

sprinkle

Vandana!

let z is a complex number as z = cos A + i sin A

=> z^n = (cos A + i sin A)^n = cos nA + i sin nA (De-Moivre's Theorem)

Now consider the series:

S = ( z * cos A) + ( z * cos A)^2 + ( z * cos A)^3 + ......+ ( z * cos A)^n ........ (i)

=> S = (cos A + i sin A)* cosA + (cos 2A + i sin 2A) * (cos A)^2 + ..................
.......................(cos nA + i sin nA)* (cos A)^n

Now imaginary part of S is,
Im (S) = sin A * (cos A) + sin 2A * (cos A)^2 + sin 3A * (cos A)^3 +.......................................sin nA (cos A)^n = given expression

but (i) is a GP and so S = [(z cos A) ((z cosA)^n - 1) ] / (z cos A - 1)

thus:
sin A * (cos A) + sin 2A * (cos A)^2 + sin 3A * (cos A)^3 +.................sin nA (cos A)^n = Im ([(z cos A) ((z cosA)^n - 1) ] / (z cos A - 1)) Ans

RHS can be found easily.

konichiwa2x

Very nice question.....You are required to find the value of

let

note that if

then

suppose

see that:

After a little bit simplifying you'll get,

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