Задача 80541 ...
Условие
Решение
Формула суммы синусов:
[m]\sin a + \sin b = 2\sin \frac{a+b}{2} \cos \frac{a-b}{2}[/m]
Подставляем:
[m]2\sin \frac{5x+x}{2} \cos \frac{5x-x}{2} - 2\cos 2x = 0[/m]
[m]2\sin 3x \cos 2x - 2\cos 2x = 0[/m]
[m]2\cos 2x \cdot (\sin 3x - 1) = 0[/m]
а) cos 2x = 0; 2x = π/2 + π*k; [b]x1 = π/4 + π/2*k, k ∈ Z[/b]
б) sin 3x = 1; 3x = π/2 + 2π*n; [b]x2 = π/6 + 2π/3*n, n ∈ Z[/b]
2) cos 5x + cos x + 2cos 3x = 0
Формула суммы косинусов:
[m]cos a + cos b = 2\cos \frac{a+b}{2} \cos \frac{a-b}{2}[/m]
Подставляем:
[m]2\cos \frac{5x+x}{2} \cos \frac{5x-x}{2} + 2\cos 3x = 0[/m]
[m]2\cos 3x \cos 2x + 2\cos 3x = 0[/m]
[m]2\cos 3x \cdot (\cos 2x + 1) = 0[/m]
а) cos 3x = 0; 3x = π/2 + π*k; [b]x1 = π/6 + π/3*k, k ∈ Z[/b]
б) cos 2x = -1; 2x = π + 2π*n; [b]x2 = π/2 + π*n, n ∈ Z[/b]
3) sin x - sqrt(2)*sin 3x = -sin 5x
sin 5x + sin x - sqrt(2)*sin 3x = 0
Как в 1) номере:
[m]2\sin \frac{5x+x}{2} \cos \frac{5x-x}{2} - \sqrt{2}\sin 3x = 0[/m]
[m]2\sin 3x \cos 2x - \sqrt{2}\sin 3x = 0[/m]
[m]\sin 3x \cdot (2\cos 2x - \sqrt{2}) = 0[/m]
а) sin 3x = 0; 3x = π*k; [b]x1 = π/3*k, k ∈ Z[/b]
б) cos 2x = sqrt(2)/2; 2x = ± π/4 + 2π*n; [b]x2 = ± π/8 + π*n, n ∈ Z[/b]
4) cos x - cos 3x - 2sin 2x = 0
Формула разности косинусов:
[m]cos a - cos b = -2\sin \frac{a+b}{2} \sin \frac{a-b}{2}[/m]
Подставляем:
[m]-2\sin \frac{x+3x}{2} \sin \frac{x-3x}{2} - 2\sin 2x = 0[/m]
[m]2\sin 2x \sin x - 2\sin 2x = 0[/m]
[m]2\sin 2x \cdot (\sin x - 1) = 0[/m]
а) sin 2x = 0; 2x = π*k; [b]x1 = π/2*k, k ∈ Z[/b]
б) sin x = 1; [b]x2 = π/2 + 2π*n, n ∈ Z[/b]