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МатематикаРешебник по алгебре 11 класс Никольский Параграф 1. Функции и их графики Задание 35
Задание 35
\[\boxed{\mathbf{35.}}\]
\[\textbf{а)}\ y = \left\lbrack \sin x \right\rbrack;\ \ X = R;\]
\[\left\lbrack \sin(x + T) \right\rbrack = \left\lbrack \sin(x + 2\pi) \right\rbrack =\]
\[= \left\lbrack \sin x \right\rbrack\]
\[Периодическая,\ T = 2\pi.\]
\[\textbf{б)}\ y = \left\{ \sin x \right\};\ \ X = R;\]
\[\left\{ \sin(x + T) \right\} = \left\{ \sin(x + 2\pi) \right\} =\]
\[= \left\{ \sin x \right\}\]
\[Периодическая,\ T = 2\text{π.}\]
\[\textbf{в)}\ y = \left\lbrack \cos x \right\rbrack;\ \ \]
\[y = \left\lbrack \cos x \right\rbrack = \left\lbrack \sin\left( x + \frac{\pi}{2} \right) \right\rbrack;\]
\[\ \ X = R;\]
\[\left\lbrack \sin\left( x + \frac{\pi}{2} + T \right) \right\rbrack =\]
\[= \left\lbrack \sin\left( x + \frac{\pi}{2} + 2\pi \right) \right\rbrack =\]
\[= \left\lbrack \sin\left( x + \frac{\pi}{2} \right) \right\rbrack\]
\[Периодическая,\ T = 2\text{π.}\]
\[\textbf{г)}\ y = \left\{ \cos x \right\} = \left\{ \sin\left( x + \frac{\pi}{2} \right) \right\};\ \]
\[\ X = R;\]
\[\left\{ \sin\left( x + \frac{\pi}{2} + T \right) \right\} =\]
\[= \left\{ \sin\left( x + \frac{\pi}{2} + 2\pi \right) \right\} =\]
\[= \left\{ \sin\left( x + \frac{\pi}{2} \right) \right\}\]
\[Периодическая,\ T = 2\text{π.}\]
\[\textbf{д)}\ y = \left| \left\lbrack \sin x \right\rbrack \right|;\ \ X = R;\]
\[\left| \left\lbrack \sin(x + T) \right\rbrack \right| = \left| \left\lbrack \sin(x + 2\pi) \right\rbrack \right| =\]
\[= \left| \left\lbrack \sin x \right\rbrack \right|\]
\[Периодическая,\ T = 2\text{π.}\]
\[\textbf{е)}\ y = \left| \left\{ \sin x \right\} \right|;\ \ X = R;\]
\[\left| \left\{ \sin(x + T) \right\} \right| =\]
\[= \left| \left\{ \sin(x + 2\pi) \right\} \right| = \left| \left\{ \sin x \right\} \right|\]
\[Периодическая,\ T = 2\text{π.}\]
\[\textbf{ж)}\ y = \left| \left\lbrack \cos x \right\rbrack \right| = \left| \left\lbrack \sin\left( x + \frac{\pi}{2} \right) \right\rbrack \right|;\ \]
\[\ X = R;\]
\[\left| \left\lbrack \sin\left( x + \frac{\pi}{2} + T \right) \right\rbrack \right| =\]
\[= \left| \left\lbrack \sin\left( x + \frac{\pi}{2} + 2\pi \right) \right\rbrack \right| =\]
\[= \left| \left\lbrack \sin\left( x + \frac{\pi}{2} \right) \right\rbrack \right|\]
\[Периодическая,\ T = 2\text{π.}\]
\[\textbf{з)}\ y = \left\{ \cos x \right\} = \left| \left\{ \sin\left( x + \frac{\pi}{2} \right) \right\} \right|;\ \ \]
\[X = R;\]
\[\left| \left\{ \sin\left( x + \frac{\pi}{2} + T \right) \right\} \right| =\]
\[= \left| \left\{ \sin\left( x + \frac{\pi}{2} + 2\pi \right) \right\} \right| =\]
\[= \left| \left\{ \sin\left( x + \frac{\pi}{2} \right) \right\} \right|\]
\[Периодическая,\ T = 2\text{π.\ }\]
