2 интересни задачки

[brood2]

sin⁸x + cos⁸x =1/8



sin8x - cos6x=√3(sin6x + cos8x)


п.с. -корен втори от 3 по скоба sin6x+cos8x!![/list]

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[Tony_89]

sin(x)⁸ + cos(x)⁸ = (sin(x)⁴ + cos(x)⁴)² - 2*sin(x)⁴*cos(x)⁴ =

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

= (1 - sin(2*x)²/2)² - sin(2*x)⁴/8 =

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

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

8 - 8*sin(2*x)² + sin(2*x)⁴ = 1

sin(2*x)⁴ - 8*sin(2*x)² + 7 = 0

Нека sin(2*x)² = u E [0 ; 1]

u² - 8*u + 7 = 0

(u - 7)*(u - 1) = 0

u - 7 = 0 или u - 1 = 0

u - 7 = 0

u = 7, което не е решение

u - 1 = 0

u = 1

sin(2*x)² = 1

sin(2*x) = ±1

2*x = п/2 + k*п

x = п/4 + k*п/2



sin(8*x) - cos(6*x) = √3(sin(6*x) + cos(8*x))

sin(8*x) - √3*cos(8*x) - cos(6*x) - √3*sin(6*x) = 0

2*((1/2)*sin(8*x) - (√3/2)*cos(8*x)) - 2*((1/2)*cos(6*x) + (√3/2)*sin(6*x)) = 0 /:2

cos(п/3)*sin(8*x) - sin(п/3)*cos(8*x) - (sin(п/6)*cos(6*x) + cos(п/6)*sin(6*x)) = 0

sin(8*x - п/3) - sin(6*x + п/6) = 0

sin(8*x - п/3) = sin(6*x + п/6)

8*x - п/3 = 6*x + п/6 + 2*k*п

2*x = п/2 + 2*k*п

x = п/4 + k*п