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# What is the expansion of,sin x + sin 3x +sin 5x +sin7x+.....+sin 29x ?Please explain..

Haroldo Martins
18 Points
9 years ago

sin x + sin 3x +sin 5x +sin7x+.....+sin 29x

façamos S=sin x + sin 3x +sin 5x +sin7x+.....+sin 29x

note que x, 3x, 5x formam nesta ordem uma PA de razão 2x, então multipliquemos ambos os membros da igualdade por 2.sen x então

(2.sen x). S=(2.sen x).(sin x + sin 3x +sin 5x +sin7x+.....+sin 29x )

(2.sen x). S=2.sin x.sen x +2. sin 3x.sin x +2.sin 5x.sin x +2.sin7x.sin x+.....+2.sin 29x.sin x

2.sena.senb=cos(a-b)-cos(a+b), assim

2.sin x.sen x=cos(x-x)-cos(x+x)=cos(0)-cos(2x)=1-cos(2x)

2. sin 3x.sin x=cos(3x-x)-cos(3x+x)=cos(2x)-cos(4x)

2.sin 5x.sin x=cos(5x-x)-cos(5x+x)=cos(4x)-cos(6x)

.............................................................................

2.sin 29x.sin x=cos(29x-x)-cos(29x+x)=cos(28x)-cos(30x)

S=1-cos(2x)+cos(2x)-cos(4x)+cos(4x)-cos(6x)+...+cos(28x)-cos(30x)

(2.sen x).S=1-cos(30x)

(2.sen x).S=1-[cos²(15x)-sen²(15x)]

(2.sen x).S=1-cos²(15x)+sen²(15x)

(2.sen x).S=sen²(15x)+sen²(15x)

(2.sen x).S=2.sen²(15x)

S=sen²(15x)/sen(x)