Question

Cum se rezolva sin x+7cos x=5?

Answer 1

[tex]\sin x+7\cos x=5\\ 7\cos x=5-\sin x|()^2\\ 49\cos^2x=25-10\sin x+\sin^2 x\\ 49(1-\sin^2 x)=25-10\sin x+\sin^2 x\\ 49-49\sin^2 x=25-10\sin x+\sin^2 x\\ 50\sin^2 x-10\sin x-24=0|:2\\ 25\sin^2 x-5\sin x-12=0\\ \Delta=25+4\cdot 12\cdot 25=1225\Rightarrow \sqrt{\Delta}=35\\ i)\sin x=\dfrac{5+35}{50}=\dfrac{4}{5}\\ S_1=(-1)^k\cdot \arcsin \dfrac{4}{5}+k\cdot \pi,k\in \mathbb{Z}[/tex]
[tex]ii)\sin x=\dfrac{5-35}{50}=\dfrac{-3}{5}\\ S_2=(-1)^{k+1}\cdot \arcsin \dfrac{3}{5}+k\cdot \pi,k\in \mathbb{Z}\\ S=S_1\cup S_2[/tex]