sin-pi-7-sin-2pi-7-sin-3pi-7-

Question Number 206362 by MATHEMATICSAM last updated on 12/Apr/24

\sin \frac{π}{7} \times \sin \frac{2 π}{7} \times \sin \frac{3 π}{7} = ?

Answered by Skabetix last updated on 12/Apr/24

= \frac{\sqrt{7}}{8}

Answered by Berbere last updated on 13/Apr/24

s i n (\frac{π}{7}) = s i n (\frac{6 π}{7}); s i n (\frac{2 π}{7}) = s i n (\frac{5 π}{7}); s i n (\frac{3 π}{7}) = s i n (\frac{4 π}{7})

Z^{7} - 1 = \underset{k = 0}{\prod^{6}} (Z - e^{\frac{2 i k π}{7}})

\frac{Z^{7} - 1}{Z - 1} = \underset{k = 1}{\prod^{6}} (Z - e^{\frac{2 i k π}{7}}) = Σ z^{k}

z = 1, z - e^{\frac{2 i k π}{7}} = - e^{\frac{i k π}{7}} . 2 i s i n (\frac{k π}{7})

\Rightarrow {(- 2 i)}^{6} . e^{3 i π} Π s i n (\frac{k π}{7}) = 6 \Rightarrow \underset{k = 1}{\prod^{6}} s i n (\frac{k π}{7}) = \frac{7}{64}

s i n (\frac{π}{7}) s i n (\frac{3 π}{7}) s i n (\frac{5 π}{7}) = \sqrt{\frac{7}{64}} = \frac{\sqrt{7}}{8}

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