Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 1213

\[\boxed{\mathbf{1213}\mathbf{.}}\]

\[1)\ 4\sin{3x} + \sin{5x} -\]

\[- 2\sin x \bullet \cos{2x} = 0\]

\[4\sin{3x} + \sin{5x} + \sin{2x} \bullet\]

\[\cdot \cos x - 2\sin x \bullet \cos{2x} -\]

\[- \sin{2x} \bullet \cos x = 0\]

\[4\sin{3x} + \sin{5x} + \sin(2x - x) -\]

\[- \sin(2x + x) = 0\]

\[4\sin{3x} + \sin{5x} + \sin x -\]

\[- \sin{3x} = 0\]

\[3\sin{3x} + 2 \bullet \sin\frac{5x + x}{2} \bullet\]

\[\cdot \cos\frac{5x - x}{2} = 0\]

\[3\sin{3x} + 2 \bullet \sin{3x} \bullet \cos{2x} = 0\]

\[\sin{3x} \bullet \left( 3 + 2\cos{2x} \right) = 0\]

\[Первое\ уравнение:\]

\[\sin{3x} = 0\]

\[3x = \arcsin 0 + \pi n = \pi n\]

\[x = \frac{1}{3} \bullet \pi n = \frac{\text{πn}}{3}.\]

\[Второе\ уравнение:\]

\[3 + 2\cos{2x} = 0\]

\[2\cos{2x} = - 3\]

\[\cos{2x} = - \frac{3}{2} - корней\ нет.\]

\[Ответ:\ \ \frac{\text{πn}}{3}.\]

\[2)\ 6\cos{2x} \bullet \sin x + 7\sin{2x} = 0\]

\[6\cos{2x} \bullet \sin x +\]

\[+ 14\sin x \bullet \cos x = 0\]

\[2\sin x \bullet \left( 3\cos{2x} + 7\cos x \right) = 0\]

\[3\cos{2x} + 7\cos x = 0\]

\[3\cos^{2}x - 3\sin^{2}x + 7\cos x = 0\]

\[3\cos^{2}x - 3\left( 1 - \cos^{2}x \right) +\]

\[+ 7\cos x = 0\]

\[3\cos^{2}x - 3 + 3\cos^{2}x +\]

\[+ 7\cos x = 0\]

\[6\cos^{2}x + 7\cos x - 3 = 0\]

\[Пусть\ y = \cos x:\]

\[6y^{2} + 7y - 3 = 0\]

\[D = 7^{2} + 4 \bullet 6 \bullet 3 = 49 + 72 =\]

\[= 121 = 11^{2}\]

\[y_{1} = \frac{- 7 - 11}{2 \bullet 6} = - \frac{18}{12} = - \frac{3}{2}\text{\ \ }и\ \]

\[\ y_{2} = \frac{- 7 + 11}{2 \bullet 6} = \frac{4}{12} = \frac{1}{3}.\]

\[Первое\ уравнение:\]

\[2\sin x = 0\]

\[\sin x = 0\]

\[x = \arcsin 0 + \pi n = \pi n.\]

\[Второе\ уравнение:\]

\[\cos x = - \frac{3}{2} - корней\ нет.\]

\[Третье\ уравнение:\]

\[\cos x = \frac{1}{3}\]

\[x = \pm \arccos\frac{1}{3} + 2\pi n.\]

\[Ответ:\ \ \pi n;\ \ \pm \arccos\frac{1}{3} + 2\pi n.\]

\[3)\sin x\sin{5x} = 1\]

\[16\sin^{2}x\cos^{4}x -\]

\[- 12\sin^{2}x\cos^{2}x + \sin^{2}x -\]

\[- {sin²}x - \cos^{2}x = 0\]

\[- 16\cos^{6}x + 28\cos^{4}x -\]

\[- 13\cos^{2}x = 0\]

\[- \cos^{2}x \cdot\]

\[\cdot \left( 16\cos^{4}x - 28\cos^{2}x + 13 \right) =\]

\[= 0\]

\[\cos^{2}x = 0\]

\[x = \frac{\pi}{2} + \pi n.\]

\[16\cos^{4}x - 28\cos^{2}x + 13 = 0\]

\[Пусть\ \cos^{2}x = y:\]

\[16y^{2} - 28y + 13 = 0\]

\[D_{1} = 196 - 208 < 0 - нет\ \]

\[корней.\]

\[Ответ:x = \frac{\pi}{2} + \pi n.\]

\[4)\sin x\cos{4x} = - 1\]

\[\frac{1}{2}\left( \sin{5x} + \sin{3x} \right) = 1\]

\[- \frac{1}{2} \leq \sin{5x} \leq \frac{1}{2}\]

\[- \frac{1}{2} \leq \sin{3x} \leq \frac{1}{2}\]

\[Левая\ часть\ равна\ ( - 1),\ при\]

\[\sin{5x} = - 1\ и\sin{3x} = - 1:\]

\[x = - \frac{\pi}{2} + 2\pi n.\]

\[Ответ:x = - \frac{\pi}{2} + \pi n.\]