Ctg (a+pi) + 4 tg (3pi/2-a) + 2 tg (a+5pi/2) ctga = -0.25

[latex]ctga + 4ctga+2tg(a+3 \pi - \frac{ \pi }{2} )*ctga=-0.25[/latex] [latex]5ctga+2tg(a-\frac{ \pi }{2} )*ctga=-0.25[/latex] [latex]5ctga-2tg(\frac{ \pi }{2}-a)*ctga=-0.25[/latex] [latex]5ctga-2ctga*ctga=-0.25[/latex] [latex]-2ctg^{2}a+5ctga+0.25=0[/latex] [latex]2ctg^{2}a-5ctga-0.25=0[/latex] [latex]8ctg^{2}a-20ctga-1=0[/latex] Замена: [latex]ctga=t[/latex], t∈R [latex]8t^{2}-20t-1=0, D=20^{2}+4*8=400+32=432[/latex] [latex]t_{1}= \frac{20- \sqrt{432}}{16} = \frac{20-4 \sqrt{27}}{16}= \frac{5- \sqrt{27}}{4}[/latex] [latex]t_{2}= \frac{20+ \sqrt{432}}{16} = \frac{20+4 \sqrt{27}}{16}= \frac{5+ \sqrt{27}}{4}[/latex] Вернемся к замене: 1) [latex]ctga=\frac{5- \sqrt{27}}{4}[/latex] [latex]a=arcctg(\frac{5- \sqrt{27}}{4})+ \pi k[/latex] 2) [latex]ctga=(\frac{5+ \sqrt{27}}{4}[/latex] [latex]a=arcctg(\frac{5+ \sqrt{27}}{4})+ \pi k[/latex] k∈Z