Question

Demonstrați ca sin20°+sin50°+sin80°= sin100°+sin130°+sin160°

Answer 1

Răspuns:

Explicație pas cu pas:

$\sin 20\textdegree+\sin50\textdegree+\sin80\textdegree=\sin100\textdegree+\sin130\textdegree+\sin160\textdegree\\(\sin 100\textdegree-\sin80\textdegree)+(\sin130\textdegree-\sin50\textdegree)+(\sin160\textdegree-\sin20\textdegree)=0\\2\cdot\sin \dfrac{100\textdegree-80\textdegree}{2}\cdot\cos\dfrac{100\textdegree+80\textdegree}{2}+2\cdot\sin\dfrac{130\textdegree-50\textdegree}{2}\cdot\cos\dfrac{130\textdegree+50\textdegree}{2}+$

$+2\cdot\sin\dfrac{160\textdegree-20\textdegree}{2}\cdot\cos\dfrac{160\textdegree+20\textdegree}{2}=0\\2\cdot\sin10\textdegree\cdot\cos90\textdegree+2\cdot\sin40\textdegree\cdot\cos 90\textdegree +2\cdot\sin70\textdegree\cdot\cos90\textdegree=0\\2\cdot\sin10\textdegree\cdot 0+2\cdot\sin40\textdegree\cdot0+2\cdot\sin70\textdegree\cdot0=0\\0+0+0=0\\\boxed{0=0},\texttt{ adevarat.}$