СИСТЕМА

[tehribo]

|cos(2y)+0.5=(cos(y)-0.5)(1+2sin(2x))
|sin(y)[tg³(x)+cotg³(x)]=3cotg(y)


Отговор:
{(n/4+k₁n ; -n/3+k₂n) ; (n/4+k₁n ; n/3+k₃n) }

[Реклама]

[Tony_89]

Аз получавам друг отговор.

ДС:

|sin(y) ≠ 0
|sin(x) ≠ 0
|cos(x) ≠ 0

|y ≠ k*п
|x ≠ k*п/2

cos(2*y) + 1/2 = 2*cos(y)² - 1 + 1/2 = 2*cos(y)² - 1/2 =

= 2*(cos(y)² - 1/4) = 2*(cos(y) - 1/2)*(cos(y) + 1/2)

2*(cos(y) - 1/2)*(cos(y) + 1/2) - (cos(y) - 1/2)*(1 + 2*sin(2*x)) = 0

(cos(y) - 1/2)*(2*(cos(y) + 1/2) - 1 - 2*sin(2*x)) = 0

(cos(y) - 1/2)*(2*cos(y) + 1 - 1 - 2*sin(2*x)) = 0

(cos(y) - 1/2)*(2*cos(y) - 2*sin(2*x)) = 0 /*(1/2)

(cos(y) - 1/2)*(cos(y) - sin(2*x)) = 0

tg(x)³ + cotg(x)³ = (tg(x) + cotg(x))*(tg(x)² - tg(x)*cotg(x) + cotg(x)²) =

= (sin(x)/cos(x) + cos(x)/sin(x))*(sin(x)²/cos(x)² + cos(x)²/sin(x)² - 1) =

= ((sin(x)² + cos(x)²)/(sin(x)*cos(x)))*((sin(x)⁴ + cos(x)⁴ - sin(x)²*cos(x)²)/(sin(x)²*cos(x)²)) =

= (1/(sin(x)*cos(x)))*((sin(x)² + cos(x)²)² - 3*sin(x)²*cos(x)²)/(sin(x)²*cos(x)²)) =

= (1 - 3*sin(x)²*cos(x)²)/(sin(x)³*cos(x)³) =

= (1 - (3/4)*4*sin(x)²*cos(x)²)/(( 1/8 )*8*sin(x)³*cos(x)³) =

= (1 - (3/4)*sin(2*x)²)/(( 1/8 )*sin(2*x)³) =

= (8 - 6*sin(2*x)²)/sin(2*x)³

sin(y)*((8 - 6*sin(2*x)²)/sin(2*x)³) = 3*cotg(y)

(8 - 6*sin(2*x)²)/sin(2*x)³ = 3*cos(y)/sin(y)²

|(cos(y) - 1/2)*(cos(y) - sin(2*x)) = 0
|(8 - 6*sin(2*x)²)/sin(2*x)³ = 3*cos(y)/sin(y)²

I случай: cos(y) = 1/2

y = ±п/3 + 2*k*п

sin(y)² = 1 - cos(y)² = 3/4

(8 - 6*sin(2*x)²)/sin(2*x)³ = 3*(1/2)/(3/4) = 2

Нека sin(2*x) = u E [-1 ; 1]

(8 - 6*u²)/u³ = 2

8 - 6*u² = 2*u³

2*u³ + 6*u² - 8 = 0

2*u³ - 2*u² + 8*u² - 8 = 0

2*u²*(u - 1) + 8*(u² - 1) = 0

2*u²*(u - 1) + 8*(u - 1)*(u + 1) = 0

(u - 1)*( 2*u² + 8*u + 8 ) = 0 /*(1/2)

(u - 1)*(u² + 4*u + 4) = 0

(u - 1)*(u + 2)² = 0

u - 1 = 0 или (u + 2)² = 0

u - 1 = 0

u = 1

sin(2*x) = 1

2*x = п/2 + 2*l*п

x = п/4 + l*п

(u + 2)² = 0

u + 2 = 0

u = -2, което не е решение

II случай: cos(y) = sin(2*x)

sin(y)² = 1 - cos(y)² = 1 - sin(2*x)²

(8 - 6*sin(2*x)²)/sin(2*x)³ = 3*sin(2*x)/(1 - sin(2*x)²)

Нека sin(2*x) = u E [-1 ; 1]

(8 - 6*u²)/u³ = 3*u/(1 - u²)

(8 - 6*u²)*(1 - u²) = 3*u⁴

8 - 8*u² - 6*u² + 6*u⁴ = 3*u⁴

3*u⁴ - 14*u² + 8 = 0

(u² - 4)*(u² - 2/3) = 0

(u - 2)*(u + 2)*(u - SQRT(2/3))*(u + SQRT(2/3)) = 0

u - 2 = 0 или u + 2 = 0 или u - SQRT(2/3) = 0 или u + SQRT(2/3) = 0

u = 2, което не е решение

u = -2, което не е решение

u = SQRT(2/3)

u = -SQRT(2/3)

sin(2*x) = ±SQRT(2/3)

cos(y) = ±SQRT(2/3)

(п/4 + l*п ; ±п/3 + 2*k*п)

U

|sin(2*x) = ±SQRT(2/3)
|cos(y) = ±SQRT(2/3)