Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 696

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

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

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

\[= 3\sin^{2}x - \left( 1 - \sin^{2}x \right) =\]

\[= 4\sin^{2}x - 1;\]

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

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

\[0 \leq \sin^{2}x \leq 4\]

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

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

\[2)\ y = 1 - 8\cos^{2}x \bullet \sin^{2}x =\]

\[= 1 - 2 \bullet 4\sin^{2}x \bullet \cos^{2}x =\]

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

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

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

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

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

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

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

\[3)\ y = \frac{1 + 8\cos^{2}x}{4}\]

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

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

\[0 \leq 8\cos^{2}x \leq 8\]

\[1 \leq 1 + 8\cos^{2}x \leq 9\]

\[\frac{1}{4} \leq \frac{1 + 8\cos^{2}x}{4} \leq \frac{9}{4}.\]

\[Ответ:\ \ E(y) = \lbrack 0,25;\ 2,25\rbrack.\]

\[4)\ y = 10 - 9\sin^{2}{3x}\]

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

\[0 \leq \sin^{2}{3x} \leq 1\]

\[- 1 \leq - \sin^{2}{3x} \leq 0\]

\[- 9 \leq - 9\sin^{2}{3x} \leq 0\]

\[1 \leq 10 - 9\sin^{2}{3x} \leq 10.\]

\[Ответ:\ \ E(y) = \lbrack 1;\ 10\rbrack.\]

\[5)\ y = 1 - 2\left| \cos x \right|\]

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

\[0 \leq \left| \cos x \right| \leq 1\]

\[- 1 \leq - \left| \cos x \right| \leq 0\]

\[- 2 \leq - 2\left| \cos x \right| \leq 0\]

\[- 1 \leq 1 - 2\left| \cos x \right| \leq 1.\]

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

\[6)\ y = \sin x + \sin\left( x + \frac{\pi}{3} \right) =\]

\[= 2 \bullet \sin\frac{x + x + \frac{\pi}{3}}{2} \bullet \cos\frac{x - x - \frac{\pi}{3}}{2} =\]

\[= 2 \bullet \sin\left( x + \frac{\pi}{6} \right) \bullet \cos\left( - \frac{\pi}{6} \right) =\]

\[= 2 \bullet \cos\frac{\pi}{6} \bullet \sin\left( x + \frac{\pi}{6} \right) =\]

\[= \sqrt{3}\sin\left( x + \frac{\pi}{6} \right);\]

\[- 1 \leq \sin\left( x + \frac{\pi}{6} \right) \leq 1\]

\[- \sqrt{3} \leq \sqrt{3}\sin\left( x + \frac{\pi}{6} \right) \leq \sqrt{3}.\]

\[Ответ:\ \ E(y) = \left\lbrack - \sqrt{3};\ \sqrt{3} \right\rbrack.\]