Задача 65691 ...
Условие
Решение
Все решения
По формулам приведения:
sin (π - a) = sin a
Подставляем:
sin 2x + sin 8x = sqrt(2)*cos 3x
По формуле суммы синусов:
[m]sin a + sin b = 2sin \frac{a+b}{2}cos \frac{a-b}{2}[/m]
Подставляем:
[m]sin(8x) + sin(2x) = 2sin \frac{8x+2x}{2}cos \frac{8x-2x}{2} = 2sin(5x)cos(3x)[/m]
2sin 5x*cos 3x = sqrt(2)*cos 3x
2sin 5x*cos 3x - sqrt(2)*cos 3x = 0
2cos 3x*(sin 5x - sqrt(2)/2) = 0
1) cos 3x = 0
3x = π/2 + π*k
x = π/6 + π/3*k, k ∈ Z
На промежутке [-π/2; 2π/3] будут корни:
[b]x1 = π/6 - 2π/3 = π/6 - 4π/6 = -3π/6 = -π/2
x2 = π/6 - π/3 = π/6 - 2π/6 = -π/6
x3 = π/6
x4 = π/6 + π/3 = π/6 + 2π/6 = 3π/6 = π/2[/b]
2) sin 5x - sqrt(2)/2 = 0
sin 5x = sqrt(2)/2
2a) 5x = π/4 + 2π*n
x = π/20 + π/10*n
На промежутке [-π/2; 2π/3] = [-10π/20; 40π/60] будут корни:
[b]x5 = π/20 - 5π/10 = π/20 - 10π/20 = -9π/20
x6 = π/20 - 4π/10 = π/20 - 8π/20 = -7π/20
x7 = π/20 - 3π/10 = π/20 - 6π/20 = -5π/20 = -π/4
x8 = π/20 - 2π/10 = π/20 - 4π/20 = -3π/20
x9 = π/20 - π/10 = π/20 - 2π/20 = -π/20
x10 = π/20
x11 = π/20 + π/10 = π/20 + 2π/20 = 3π/20
x12 = π/20 + 2π/10 = π/20 + 4π/20 = 5π/20 = π/4
x13 = π/20 + 3π/10 = π/20 + 6π/20 = 7π/20
x14 = π/20 + 4π/10 = π/20 + 8π/20 = 9π/20 = 27π/60
x15 = π/20 + 5π/10 = π/20 + 10π/20 = 11π/20 = 33π/60
x16 = π/20 + 6π/10 = π/20 + 12π/20 = 13π/20 = 39π/60[/b]
2b) 5x = 3π/4 + 2π*n
x = 3π/20 + π/10*n
На промежутке [-π/2; 2π/3] = [-10π/20; 40π/60] будут корни:
x17 = 3π/20 - 6π/10 = 3π/20 - 12π/20 = -9π/20 = x5
x18 = 3π/20 - 5π/10 = 3π/20 - 10π/20 = -7π/20 = x6
x19 = 3π/20 - 4π/10 = 3π/20 - 8π/20 = -5π/20 = x7
x20 = 3π/20 - 3π/10 = 3π/20 - 6π/20 = -3π/20 = x8
x21 = 3π/20 - 2π/10 = 3π/20 - 4π/20 = -π/20 = x9
x22 = 3π/20 - π/10 = 3π/20 - 2π/20 = π/20 = x10
x23 = 3π/20 = x11
x24 = 3π/20 + π/10 = 3π/20 + 2π/20 = 5π/20 = x12
x25 = 3π/20 + 2π/10 = 3π/20 + 4π/20 = 7π/20 = x13
x26 = 3π/20 + 3π/10 = 3π/20 + 6π/20 = 9π/20 = x14
x27 = 3π/20 + 4π/10 = 3π/20 + 8π/20 = 11π/20 = x15
x28 = 3π/20 + 5π/10 = 3π/20 + 10π/20 = 13π/20 = x16
Здесь все корни повторяются, поэтому их можно не писать.
[b]Ответ: на промежутке [-π/2; 2π/3] есть 16 различных корней.[/b]