Question

Решить уравнение 2-5cosx-cos2x=0

Answer 1

2 - 5Cosx - Cos2x = 0
2 - 5Cosx - (2Cos²x - 1) = 0
2 - 5Cosx - 2Cos²x + 1 = 0
2Cos²x + 5Cosx - 3 = 0
Cosx = 1/2                     Cosx = - 3 - решений нет, так как |- 3| > 1
x = + - π/3 + 2πn, n ∈ z

Answer 2

$2-5cosx-cos2x=0 \\ 2-5cosx-cos^2x+sin^2x=0 \\ 2-5cosx-cos^2x+1-cos^2x=0 \\ 2cos^2x+5cosx-3=0 \\ \\ cosx=t; t \in [-1;1]\\ \\ 2t^2+5t-3=0 \\ D=25+24=7^2 \\ t_1= \dfrac{-5+7}{4}=0,5 \\ t_2= \dfrac{-5-7}{4}=-3 \notin ODZ \\ \\ cosx=0,5 \\ x=б \dfrac{ \pi }{3}+2 \pi k;k \in Z$