Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 1244

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\[1)\ \ 1 + 2\sin x = \sin{2x} + 2\cos x\]

\[\cos^{2}x + \sin^{2}x + 2\sin x =\]

\[= 2\sin x \bullet \cos x + 2\cos x\]

\[\left( \cos^{2}x + \sin^{2}x - 2\sin x \bullet \cos x \right) +\]

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

\[\left( \cos x - \sin x \right)^{2} -\]

\[- 2\left( \cos x - \sin x \right) = 0\]

\[\left( \cos x - \sin x \right) \bullet\]

\[\bullet \left( \cos x - \sin x - 2 \right) = 0\]

\[Первое\ уравнение:\]

\[\cos x - \sin x = 0\ \ \ \ \ |\ :\cos x\]

\[1 - tg\ x = 0\]

\[tg\ x = 1\]

\[x = arctg\ 1 + \pi n = \frac{\pi}{4} + \pi n.\]

\[Второе\ уравнение:\]

\[\cos x - \sin x - 2 = 0\]

\[\cos x - \sin x = 2 - корней\ нет.\]

\[Ответ:\ \ \frac{\pi}{4} + \pi n.\]

\[2)\ 1 + 3\cos x = \sin{2x} + 3\sin x\]

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

\[= 2\sin x \bullet \cos x + 3\sin x\]

\[\left( \cos^{2}x + \sin^{2}x - 2\sin x \bullet \cos x \right) +\]

\[+ \left( 3\cos x - 3\sin x \right) = 0\]

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

\[+ 3\left( \cos x - \sin x \right) = 0\]

\[\left( \cos x - \sin x \right) \bullet\]

\[\bullet \left( \cos x - \sin x + 3 \right) = 0\]

\[Первое\ уравнение:\]

\[\cos x - \sin x = 0\ \ \ \ \ |\ :\cos x\]

\[1 - tg\ x = 0\]

\[tg\ x = 1\]

\[x = arctg\ 1 + \pi n = \frac{\pi}{4} + \pi n.\]

\[Второе\ уравнение:\]

\[\cos x - \sin x + 3 = 0\]

\[\cos x - \sin x = - 3 - корней\]

\[\ нет.\]

\[Ответ:\ \ \frac{\pi}{4} + \pi n.\]