Question

Se considera a, b reale astfel incat a+b=pi/3. Aratati ca 2cos(b)=cos(a)+radical(3)*sin(a)

Answer 1

In general avem formula
$\cos{(x-y)}=\cos{x}\cos{y}+\sin{x}\sin{y}$
In cazul nostru avem
$a+b=\frac{\pi}{3}\Rightarrow b=\frac{\pi}{3}-a\Rightarrow \cos{b}=\cos{(\frac{\pi}{3}-a)}=\cos{\frac{\pi}{3}}\cos{a}+\sin{\frac{\pi}{3}}\sin{a}=\frac{1}{2}\cos{a}+\frac{\sqrt{3}}{2}\sin{a}\Rightarrow 2\cos{b}=\cos{a}+\sqrt{3}\sin{a}$