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How do we derive the expansion for sin 3x?

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justaguide eNotes educator | Certified Educator

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I think by expansion you mean sin 3x in terms of sin x.

We start with the relations sin (x + y) = sin x* cos y + cos x*sin y and cos (x + y) = cos x* cos y – sin x*sin y

sin 3x = sin (2x + x) = sin 2x* cos x + cos...

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giorgiana1976 | Student

sin(2x+x) = sin2x*cosx + sin x*cos 2x

sin 2x = 2sinx*cos x

cos 2x = 1 - 2(sin x)^2

sin 3x = 2sin x*cos x*sin x + sin x*[1-2(sin x)^2]

sin 3x = 2(cos x)^2*sinx + sin x - 2 (sin x)^3

From the fundamental formula of trigonometry, we'll have:

(cos x)^2 = 1 - (sin x)^2

We'll try to express sin 3x in terms of sin x:

sin 3x = 2[1 - (sin x)^2]*sin x + sin x - 2(sin x)^3

sin 3x = 2sin x - 2(sin x)^3 + sin x - 2(sin x)^3

We'll combine like terms:

sin 3x = 3sin x - 4(sin x)^3

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