Question

Arătați ca: sin20×sin40×sin60×sin80=3/16

Answer 1

[tex]sin20\cdot sin40\cdot sin60 \cdot sin80=\frac{3}{16}\\ \frac{\sqrt3}{2}\cdot sin20( sin40\cdot sin80)=\frac{3}{16}\\ Folosim\ formula:sinA\cdot sinB=\frac{1}{2}[cos(A-B)-cos(A+B)]\\ \frac{\sqrt{3}}{2}\cdot sin20 \cdot \frac{1}{2}(cos40-cos120)=\frac{3}{16}\\ Dar: -cos120=cos 60\\ \frac{\sqrt3}{2}\cdot sin20 \cdot \frac{1}{2}(cos40+cos60)=\frac{3}{16}\\ \frac{\sqrt3}{4}\cdot sin20 (cos40+\frac{1}{2})=\frac{3}{16}\\ \frac{\sqrt{3}}{4}\cdot sin20\cdot cos40+\frac{\sqrt3}{8}\cdot sin20=\frac{3}{16}\\[/tex]
[tex]Folosim\ formula:sinA\cdot cosB=\frac{1}{2}[sin(A+B)+sin(A-B)]\\ \frac{\sqrt3}{4}\cdot \frac{1}{2}[sin60+sin(-20)]+\frac{\sqrt3}{8}\cdot sin20=\frac{3}{16}\\ \frac{\sqrt{3}}{8}(\frac{\sqrt3}{2}-sin20)+\frac{\sqrt3}{8}\cdot sin20=\frac{3}{16}\\ \frac{3}{16}-\frac{\sqrt3}{8}\cdot sin20+\frac{\sqrt3}{8}\cdot sin20=\frac{3}{16}\\ \frac{3}{16}=\frac{3}{16} (A)[/tex]