Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 7

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

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

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

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

\[- 2 \leq - 2\cos{2x} \leq 2;\]

\[- 1 \leq 1 - 2\cos{2x} \leq 3;\]

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

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

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

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

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

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

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

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

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

\[y = \frac{1 + 4\left( 1 + \cos{2x} \right)}{4}\]

\[y = \frac{1}{4} + 1 + \cos{2x}\]

\[y = 1,25 + \cos{2x}.\]

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

\[0,25 \leq 1,25 + \cos{2x} \leq 2,25;\]

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

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

\[y = 10 - 4,5\left( 1 - \cos{6x} \right)\]

\[y = 10 - 4,5 + 4,5\cos{6x}\]

\[y = 5,5 + 4,5\cos{6x}.\]

\[- 1 \leq \cos{6x} \leq 1;\]

\[- 4,5 \leq 4,5\cos{6x} \leq 4,5;\]

\[1 \leq 5,5 + 4,5\cos{6x} \leq 10;\]

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

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

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

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

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

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

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

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

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

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

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

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

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