ГДЗ по алгебре и начала математического анализа 11 класс Колягин Задание 1035

\[1)\ y = 0,5 + \sin\left( x - \frac{\pi}{4} \right)\]

\[- 1 \leq \sin\left( x - \frac{\pi}{4} \right) \leq 1\]

\[- 0,5 \leq 0,5 + \sin\left( x - \frac{\pi}{4} \right) \leq 1,5.\]

\[Ответ:\ \ E(y) = \lbrack - 0,5;\ 1,5\rbrack.\]

\[2)\ y = 0,5\cos x + \sin x\]

\[C = \sqrt{{0,5}^{2} + 1^{2}} = \sqrt{0,25 + 1} =\]

\[= \sqrt{1,25};\]

\[y =\]

\[= \sqrt{1,25}\left( \frac{0,5}{\sqrt{1,25}} \bullet \cos x + \frac{1}{\sqrt{1,25}} \bullet \sin x \right) =\]

\[= \sqrt{1,25}\cos(x - a);\]

\[a = \arcsin\frac{1}{\sqrt{1,25}}.\]

\[- 1 \leq \cos a \leq 1\]

\[- \sqrt{1,25} \leq \sqrt{1,25}\cos a \leq \sqrt{1,25}.\]

\[Ответ:\ \ E(y) = \left\lbrack - \sqrt{1,25};\ \sqrt{1,25} \right\rbrack.\]