Sin5x + sin3x = ....

TriGonoMetRi

sin a + sin b = 2 sin (a+b)/2 cos (a-b)/2

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Misal :
a = 5x
b = 3x

sin 5x + sin 3x
= 2 sin (5x + 3x)/2 cos (5x - 3x)/2
= 2 sin 4x cos x

= 2. sin 4x cos x
= 2 (2 sin 2x cos 2x) cos x
= 4 (2 sin x cos x)(cos^2 x - sin^2 x) cos x
= 8 sin x cos x cos x (cos^2 x - sin^2 x)
= 8 sin x cos^2 x (cos^2 x - sin^2 x)
= 8 (sin x cos^4 x - sin^3 x cos^2 x)