Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 759

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\[1)\ y = 1 - 2\sin^{2}x\]

\[- 1 \leq \sin x \leq 1\]

\[0 \leq \sin^{2}x \leq 1\]

\[- 2 \leq - 2\sin^{2}x \leq 0\]

\[- 1 \leq 1 - 2\sin^{2}x \leq 1.\]

\[Ответ:\ \ E(y) = \lbrack - 1;\ 1\rbrack.\]

\[2)\ y = 2\cos^{2}x - 1\]

\[- 1 \leq \cos x \leq 1\]

\[0 \leq \cos^{2}x \leq 1\]

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

\[- 1 \leq 2\cos^{2}x - 1 \leq 1.\]

\[Ответ:\ \ E(y) = \lbrack - 1;\ 1\rbrack.\]

\[3)\ y = 3 - 2\sin^{2}x\]

\[- 1 \leq \sin x \leq 1\]

\[0 \leq \sin^{2}x \leq 1\]

\[- 2 \leq - 2\sin^{2}x \leq 0\]

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

\[Ответ:\ \ E(y) = \lbrack 1;\ 3\rbrack.\]

\[4)\ y = 2\cos^{2}x + 5\]

\[- 1 \leq \cos x \leq 1\]

\[0 \leq \cos^{2}x \leq 1\]

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

\[5 \leq 2\cos^{2}x + 5 \leq 7.\]

\[Ответ:\ \ E(y) = \lbrack 5;\ 7\rbrack.\]

\[5)\ y = \cos{3x} \bullet \sin x - \sin{3x} \bullet \cos x + 4 =\]

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

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

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

\[3 \leq 4 - \sin{2x} \leq 5.\]

\[Ответ:\ \ E(y) = \lbrack 3;\ 5\rbrack.\]

\[6)\ y = \cos{2x} \bullet \cos x + \sin{2x} \bullet \sin x - 3 =\]

\[= \cos(2x - x) - 3 = \cos x - 3;\]

\[- 1 \leq \cos x \leq 1\]

\[- 4 \leq \cos x - 3 \leq - 2.\]

\[Ответ:\ \ E(y) = \lbrack - 4;\ - 2\rbrack.\]