Question

Arătați ca cos^2 10 grade + cos^2 50 grade + cos^2 70 grade = 3/2.
Cos^2 ( cosinus la puterea a doua).

Answer 1

Stim de urmatoarea formula
$\cos{2x}=2\cos^{2}{x}-1\Rightarrow \cos^{2}{x}=\frac{1+\cos{2x}}{2}$
Aplicam aceasta formula pentru cele cele 3 cosinusuri la patrat. Notam acea suma cu S
$S=\frac{1+\cos{2*10}}{2}+\frac{1+\cos{2*50}}{2}+\frac{1+\cos{2*70}}{2}=\frac{3}{2}+\frac{\cos{20}}{2}+\frac{\cos{100}}{2}+\frac{\cos{140}}{2}$
Cosinusul unui unghi intre 90 si 180 de grade, atunci
$\cos{180-x}=-\cos{x}$
Atunci, pentru x=100 si x=140 avem
$\cos{100}=\cos{180-80}=-\cos{80}$
$\cos{140}=\cos{180-40}=-\cos{40}<span>$
Atunci ajungem la suma
$S=\frac{3}{2}+\frac{\cos{20}-(\cos{80}+\cos{40})}{2}$
Formula pentru o suma de 2 cosinusi este
$\cos{a}+\cos{b}=2*\cos{\frac{a+b}{2}}*\cos{\frac{a-b}{2}}$
Atunci
$\cos{80}+\cos{40}=2*\cos{\frac{80+40}{2}}*\cos{\frac{80-40}{2}}=2*\cos{60}*\cos{20}=2*\frac{1}{2}*\cos{20}=\cos{20}<span>$
Atunci
$S=\frac{3}{2}+\frac{\cos{20}-\cos{20}}{2}=\frac{3}{2}$