Question

sin (5п/8)*sin(7п/8)*sin (7п/8)
Упростить

Answer 1

$\sin\frac{5\pi}8\cdot\sin\frac{7\pi}8\cdot\sin\frac{7\pi}4=\\=\frac{\sin\left(\frac{5\pi}8+\frac{7\pi}8-\frac{7\pi}4\right)+\sin\left(\frac{7\pi}8+\frac{7\pi}4-\frac{5\pi}8\right)+\sin\left(\frac{5\pi}8-\frac{7\pi}8+\frac{7\pi}4\right)-\sin\left(\frac{5\pi}8+\frac{7\pi}8+\frac{7\pi}4\right)}4=\\=\frac14\left(\sin\left(-\frac\pi4\right)+\sin2\pi+\sin\frac{3\pi}2-\sin\frac{13\pi}4\right)=$
$=\frac14\left(-\frac1{\sqrt2}+0-1-\sin\left(2\pi+\frac{5\pi}4\right)\right)=-\frac14\left(-\frac1{\sqrt2}-1-\sin\frac{5\pi}4\right)=\\=-\frac14\left(-\frac1{\sqrt2}-1+\frac1{\sqrt2}\right)=-\frac14\cdot\left(-1\right)=\frac14$