Simplify-sin-6-

Question Number 11751 by tawa last updated on 31/Mar/17

Simplify : \sin (6 θ)

Answered by mrW1 last updated on 31/Mar/17

\sin (6 θ) = \sin (4 θ + 2 θ) = \sin (4 θ) \cos (2 θ) + \sin (2 θ) \cos (4 θ)

= 2 \sin (2 θ) \cos^{2} (2 θ) + \sin (2 θ) [1 - {2 \sin}^{2} (2 θ)]

= 2 \sin (2 θ) [1 - \sin^{2} (2 θ)] + \sin (2 θ) - {2 \sin}^{3} (2 θ)

= 2 \sin (2 θ) - {2 \sin}^{3} (2 θ) + \sin (2 θ) - {2 \sin}^{3} (2 θ)

= 3 \sin (2 θ) - {4 \sin}^{3} (2 θ)

= 3 \times 2 \sin θ \cos θ - 4 \times 8 \times \sin^{3} θ \cos^{3} θ

= 2 \sin θ \cos θ (3 - {16 \sin}^{2} θ \cos^{2} θ)

Commented by tawa last updated on 31/Mar/17

God bless you sir .

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