Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 8

\[1)\ y = \sin x - 5\cos x\]

\[y = \sqrt{1 + 25}\sin(x - \varphi)\]

\[y = \sqrt{26}\sin(x - \varphi).\]

\[- 1 \leq \sin(x - \varphi) \leq 1;\]

\[- \sqrt{26} \leq \sqrt{26}\sin(x - \varphi) \leq \sqrt{26};\]

\[E(y) = \left\lbrack - \sqrt{26};\ \sqrt{26} \right\rbrack.\]

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

\[\sin x_{0} = - \frac{- 2}{2 \bullet 1} = \frac{2}{2} = 1.\]

\[y(1) = 1 - 2 = - 1;\]

\[y( - 1) = 1 + 2 = 3;\]

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

\[= 6 + 4\cos{2x} - 3\sin{2x} =\]

\[= 6 + \sqrt{16 + 9}\sin(2x - \varphi) =\]

\[= 6 + 5\sin(2x - \varphi).\]

\[- 1 \leq \sin(2x - \varphi) \leq 1;\]

\[- 5 \leq 5\sin(2x - \varphi) \leq 5;\]

\[1 \leq 6 + 5\sin(2x - \varphi) \leq 11;\]

\[E(y) = \lbrack 1;\ 11\rbrack.\]

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

\[\cos x_{0} = - \frac{3}{2 \bullet 1} = 1,5.\]

\[y(1) = 1 + 3 = 4;\]

\[y( - 1) = 1 - 3 = - 2;\]

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