Goniometrické rovnice
1.) 2sin^2(x) + cos^2(x) + sinxcosx=1
2.) 4cos^3(x)=cosx
3.)2sin^2(x/3) + 3cos(x/3)=0
2.) 4cos^3(x)=cosx
3.)2sin^2(x/3) + 3cos(x/3)=0
Odpovědi
Diskuze
1 + sin^2(x) + sin(x)cos(x) = 1
sin(x)(sin(x) + cos(x)) = 0
1. sin(x) = 0 => x_1 = kπ
2. sin(x) + cos(x) = 0 | :(cos(x))≠0 => x≠(2k+1)π/2
tg(x) = -1
x_2 = 3π/4 + kπ
cos(x)(4cos^2(x)-1) = 0
1) cos(x) = 0 => x_1 = (2k+1)π/2
2) 4cos^2(x) = 1
cos^2(x)= 1/4 => cos(x) = ±1/2 => x_2 = π/3 + 2kπ, x_3 = 2π/3 + 2kπ, x_4 = 4π/3 + 2kπ, x_5 = 5π/3 + 2kπ
2(1-cos^2(x/3)) + 3cos(x/3) = 0
s: cos(x/3) = a
2a^2 - 3a - 2 = 0
a_1,2 = (3±√(9+16))/4 = (3±5)/4
a_1 = 2 => cos(x) = 2 => ∅
a_2 = -1/2 => cos(x) = -1/2 => x_1 = 2π/3 + 2kπ, x_2 = 4π/3 + 2kπ