Rozwiązanie:
[tex]2\cos x + 5\sin x + 1=0[/tex]
Stosujemy podstawienie Weierstrassa:
[tex]$u=\tan \Big(\frac{x}{2}\Big)[/tex]
Przy czym:
[tex]$\sin x=\frac{2u}{1+u^{2}}[/tex]
[tex]$\cos x=\frac{1-u^{2}}{1+u^{2}}[/tex]
Mamy:
[tex]$2 \cdot \frac{1-u^{2}}{1+u^{2}} +5 \cdot \frac{2u}{1+u^{2}} +1=0[/tex]
Mnożymy obustronnie przez mianownik:
[tex]2(1-u^{2})+10u+1+u^{2}=0[/tex]
[tex]-u^{2}+10u+3=0[/tex]
[tex]$\Delta=100 -4 \cdot (-1) \cdot 3=112[/tex]
[tex]$u_{1}=\frac{-10+4\sqrt{7}}{-2}=5-2\sqrt{7}[/tex]
[tex]$u_{2}=\frac{-10-4\sqrt{7}}{-2} =5+2\sqrt{7}[/tex]
Stąd mamy:
[tex]$\tan \Big(\frac{x}{2}\Big)=5-2\sqrt{7} \vee \tan \Big(\frac{x}{2}\Big)=5+2\sqrt{7}[/tex]
[tex]x=2\arctan (5-2\sqrt{7})+2k\pi \vee x=2\arctan(5+2\sqrt{7})+2k\pi[/tex]
[tex]k \in \mathbb{Z}[/tex]
