Упр.22.11 ГДЗ Мордкович 10-11 класс (Алгебра)

a) sin 5x + 2sin 6x + sin 7x;
6) 2cos x + cos 2x + cos 4x.

а)

$$ \sin 5x + 2\sin 6x + \sin 7x = (\sin 5x + \sin 7x) + 2\sin 6x $$
$$ = 2\sin \frac{5x+7x}{2}\cos \frac{5x-7x}{2} + 2\sin 6x = 2\sin 6x\cos(-x) + 2\sin 6x $$
$$ = 2\sin 6x\cos x + 2\sin 6x = 2\sin 6x(\cos x+1) $$
$$ = 2\sin 6x \cdot 2\cos^2 \frac{x}{2} = 4\sin 6x\cos^2 \frac{x}{2}. $$

б)

$$ 2\cos x + \cos 2x + \cos 4x = 2\cos x + (\cos 2x + \cos 4x) $$
$$ = 2\cos x + 2\cos \frac{2x+4x}{2}\cos \frac{2x-4x}{2} = 2\cos x + 2\cos 3x\cos(-x) $$
$$ = 2\cos x + 2\cos 3x\cos x = 2\cos x(1+\cos 3x) $$
$$ = 2\cos x \cdot 2\cos^2 \frac{3x}{2} = 4\cos x\cos^2 \frac{3x}{2}. $$

Ответ

а) $$4\sin 6x\cos^2 \frac{x}{2}$$;
б) $$4\cos x\cos^2 \frac{3x}{2}$$.