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МатематикаРешебник по алгебре и начала математического анализа 10 класс Колягин Задание 1255
Задание 1255
\[\boxed{\mathbf{1255}\mathbf{.}}\]
\[1)\sin{2x} + \cos{2x} = 2\ tg\ x + 1\]
\[2\sin x \bullet \cos x +\]
\[+ \left( \cos^{2}x - \sin^{2}x \right) = 2\ tg\ x + 1\]
\[2\sin x \bullet \cos x + \left( 1 - \sin^{2}x \right) -\]
\[- \sin^{2}x - 2\ tg\ x - 1 = 0\]
\[2\sin x \bullet \cos x - 2\sin^{2}x -\]
\[- \frac{2\sin x}{\cos x} = 0\]
\[2\sin x \bullet\]
\[\bullet \left( \cos x - \sin x - \frac{1}{\cos x} \right) = 0\]
\[\cos x - \sin x - \frac{1}{\cos x} = 0\ \ \]
\[\ \ \ |\ :\cos x\]
\[1 - tg\ x - \frac{1}{\cos^{2}x} = 0\]
\[- tg\ x - \left( \frac{1}{\cos^{2}x} - 1 \right) = 0\]
\[- tg\ x - tg^{2}\ x = 0\]
\[Пусть\ y = tg\ x:\]
\[- y - y^{2} = 0\]
\[y^{2} + y = 0\]
\[y(y + 1) = 0\]
\[y_{1} = 0\ \ и\ \ y_{2} = - 1.\]
\[Первое\ уравнение:\]
\[\sin x = 0\]
\[x = \arcsin 0 + \pi n = \pi n.\]
\[Второе\ уравнение:\]
\[tg\ x = 0\]
\[x = arctg\ 0 + \pi n = \pi n.\]
\[Третье\ уравнение:\]
\[tg\ x = - 1\]
\[x = - arctg\ 1 + \pi n = - \frac{\pi}{4} + \pi n.\]
\[Ответ:\ \ \pi n;\ \ - \frac{\pi}{4} + \pi n.\]
\[2)\sin{2x} - \cos{2x} = tg\ x\]
\[2\sin x \bullet \cos x -\]
\[- \left( \cos^{2}x - \sin^{2}x \right) = tg\ x\]
\[2\sin x \bullet \cos x - \left( 1 - \sin^{2}x \right) +\]
\[+ \sin^{2}x - tg\ x = 0\]
\[2\sin x \bullet \cos x - 1 + 2\sin^{2}x -\]
\[- \frac{\sin x}{\cos x} = 0\ \ \ \ \ | \bullet \cos x\]
\[2\sin x \bullet \cos^{2}x - \cos x +\]
\[+ 2\sin^{2}x \bullet \cos x - \sin x = 0\]
\[2\sin x \bullet \cos x \bullet \left( \cos x + \sin x \right) -\]
\[- \left( \cos x + \sin x \right) = 0\]
\[\left( \cos x + \sin x \right) \bullet\]
\[\bullet \left( 2\sin x \bullet \cos x - 1 \right) = 0\]
\[\left( \cos x + \sin x \right)\left( \sin{2x} - 1 \right) = 0\]
\[Первое\ уравнение:\]
\[\cos x + \sin x = 0\ \ \ \ \ |\ :\cos x\]
\[1 + tg\ x = 0\]
\[tg\ x = - 1\]
\[x = - arctg\ 1 + \pi n = - \frac{\pi}{4} + \pi n.\]
\[Второе\ уравнение:\]
\[\sin{2x} - 1 = 0\]
\[\sin{2x} = 1\]
\[2x = \arcsin 1 + 2\pi n = \frac{\pi}{2} + 2\pi n\]
\[x = \frac{1}{2} \bullet \left( \frac{\pi}{2} + 2\pi n \right) = \frac{\pi}{4} + \pi n.\]
\[Ответ:\ \ \frac{\pi}{4} + \frac{\text{πn}}{2}.\]
