Question

Aratati ca tg20 + 4cos70 = radical din 3

Answer 1

$\frac{\sin{20}}{\cos{20}}+4\cos{70}=\sqrt{3}\Rightarrow \sin{20}+4\cos{20}\cos{70}=\sqrt{3}\cos{20}\Rightarrow 4\cos{20}\cos{70}=\sqrt{3}\cos{20}-\sin{20}$ Impartim relatia prin 2$2\cos{20}\cos{70}=\frac{\sqrt{3}}{2}\cos{20}-\sin{20}\frac{1}{2}=\sin{60}\cos{20}-\sin{20}\cos{60}=\sin{(60-20)}=\sin{40}$Mai stim ca$\cos{a}+\cos{b}=2\cos{\frac{a+b}{2}}\cos{\frac{a-b}{2}}$Atunci
a+b=2*70=140a-b=2*20=40
Adunam cele 2 relatii2a=180 adica a=90
Si avem atunci
90+b=140 adica b=140-90=50
Atunci
$2\cos{20}\cos{70}=\cos{90}+\cos{50}=0+\cos{50}$
Dar mai stim ca
$\cos{50}=\cos{90-40}=\sin{40}$
Deci ajungem la relatia
$2\cos{20}\cos{70}=<span>\cos{50}=</span>\sin{40}$ exact relatia pe care trebuia sa o demonstrezi