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| 7.3.2 正弦型函数的性质与图像题目答案及解析如下,仅供参考!
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第七章 三角函数
7.3 三角函数的性质与图像
7.3.2 正弦型函数的性质与图像
已知函数$f(x)=A\sin(\omega x+\phi )(A\gt 0,\omega \gt 0,|\phi |\lt \dfrac{\pi }{2})$,且$f\left(x\right)$图象的相邻两条对称轴之间的距离为$\dfrac{\pi }{2}$,请从条件①、条件②、条件③中任意选择两个作为已知条件作答.
条件①:$f\left(x\right)$的最小值为$-2$;
条件②:$f\left(x\right)$的图象的一个对称中心为$(\dfrac{5\pi }{12},0)$;
条件③:$f\left(x\right)$的图象经过点$(\dfrac{5\pi }{6},-1)$.
$(1)$求$f\left(x\right)$的解析式;
$(2)$在锐角$\triangle ABC$中,内角$A$,$B$,$C$所对的边分别为$a$,$b$,$c$,$A=\dfrac{\pi }{3}$,$b=f\left(A\right)$,求$\triangle ABC$面积的取值范围.
$(1)$$\\ f(x)=2\\sin(2x+\\dfrac{\\pi }{6})$;
$(2)$($\\dfrac{\\sqrt{3}}{8},\\dfrac{\\sqrt{3}}{2})$
"]]$(1)\because f\left(x\right)$图象的相邻两条对称轴之间的距离为$\dfrac{\pi }{2}$,$\therefore \dfrac{T}{2}=\dfrac{\pi }{2}$,即$T=\pi $,
$\therefore \omega =\dfrac{2\pi }{T}=\dfrac{2\pi }{\pi }=2$,$\therefore f\left(x\right)=A\sin \left(2x+\varphi \right)$.
若选①②,则$A=2$,$2×\dfrac{5\pi }{12}+\phi =k\pi $,$k\in \bf Z$,即$\phi =k\pi -\dfrac{5\pi }{6}$,$k\in\bf Z$,
$\because |\phi |\lt \dfrac{\pi }{2}$,$\therefore \phi =\dfrac{\pi }{6}$,$\therefore f(x)=2\sin(2x+\dfrac{\pi }{6})$;
若选①③,则$A=2$,$2\sin(2×\dfrac{5\pi }{6}+\phi )=-1$,即$\sin(\dfrac{5\pi }{3}+\phi )=-\dfrac{1}{2}$,
$\because |\phi |\lt \dfrac{\pi }{2}$,$\therefore \dfrac{7\pi }{6}\lt \dfrac{5\pi }{3}+\phi \lt \dfrac{13\pi }{6}$,$\therefore \dfrac{5\pi }{3}+\phi =\dfrac{11\pi }{6}$,得$\phi =\dfrac{\pi }{6}$,
$\therefore f(x)=2\sin(2x+\dfrac{\pi }{6})$;
若选②③,$2×\dfrac{5\pi }{12}+\phi =k\pi $,$k\in \bf Z$,即$\phi =k\pi -\dfrac{5\pi }{6}$,$k\in\bf Z$,
$\because |\phi |\lt \dfrac{\pi }{2}$,$\therefore \phi =\dfrac{\pi }{6}$,此时$f(x)=A\sin(2x+\dfrac{\pi }{6})$,
$\because f(\dfrac{5\pi }{6})=-1$,$A\sin(2×\dfrac{5\pi }{6}+\dfrac{\pi }{6})=-1$,$Asin\dfrac{11\pi }{6}=-1$,$A=2$,
$\therefore f(x)=2\sin(2x+\dfrac{\pi }{6})$.
$(2)$由$(1)$知,$f(x)=2\sin(2x+\dfrac{\pi }{6})$,$\because A=\dfrac{\pi }{3}$,
$\therefore b=f(\dfrac{\pi }{3})=2\sin(\dfrac{2\pi }{3}+\dfrac{\pi }{6})=1$,
$\because \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$,$\therefore c=\dfrac{b\sin C}{\sin B}=\dfrac{\sin(A+B)}{\sin B}=\dfrac{\sqrt{3}}{2\tan B}+\dfrac{1}{2}$,
$\therefore S=\dfrac{1}{2}bc\sin A=\dfrac{\sqrt{3}}{4}c=\dfrac{3}{8\tan B}+\dfrac{\sqrt{3}}{8}$,
$\because A+B \gt \dfrac{\pi }{2}$,$0 \lt B \lt \dfrac{\pi }{2}$,$\therefore \dfrac{\pi }{6} \lt B \lt \dfrac{\pi }{2}$,$\therefore \tan B \gt \dfrac{\sqrt{3}}{3}$,
$\therefore \dfrac{\sqrt{3}}{8}\lt {S}_{△ABC}\lt \dfrac{\sqrt{3}}{2}$,即$\triangle ABC$面积的取值范围是($\dfrac{\sqrt{3}}{8},\dfrac{\sqrt{3}}{2})$.
| 7.3.2 正弦型函数的性质与图像题目答案及解析(完整版)