# 三角函数的图像与性质题型归纳总结1

昨天给我的一个学生讲课，由于学校已经上完三角函数的图像和性质这一章的内容，但是在做题的时候，仍然对三角函数的基础题型掌握得不够全面，本系列文章将对三角函数的图像与性质的题型进行系统的归纳总结，希望能帮助到有类似困惑的高中生们。

本文总结的题型：已知函数解析式确定函数性质

【思路提示】一般所给函数为*y*＝*A *sin(*ω x*＋*φ*)或*y*＝*A *cos(*ω x*＋*φ*)，*A>0,ω>0, *要根据* **y*＝sin *x**，y*＝cos* x*的整体性质求解。

**一、函数的奇偶性**

**例1*** f*(*x*)＝ sin(*x*＋*φ*)（0≤*φ*<π）是**R**上的偶函数，则 *φ *等于（ ）

A.0 B．\(\frac{\pi }{4}\) C．\(\frac{\pi }{2}\) D．π

【评注】由*y*＝sin*x *是奇函数，*y*＝cos*x *是偶函数可拓展得到关于三角函数奇偶性的重要结论：

（1）若*y*＝*A *sin(*ω x*＋*φ*)是奇函数，则\(\phi=k\pi(k\epsilon Z)\)

（2）若*y*＝*A *sin(*ω x*＋*φ*)是偶函数，则\(\phi=k\pi+\frac{\pi}{2}(k\epsilon Z)\)

（3）若*y*＝*A *cos(*ω x*＋*φ*)是奇函数，则\(\phi=k\pi+\frac{\pi}{2}(k\epsilon Z)\)

（4）若*y*＝*A cos*(*ω x*＋*φ*)是偶函数，则\(\phi=k\pi(k\epsilon Z)\)

（5）若*y*＝*A tan*(*ω x*＋*φ*)是偶函数，则\(\phi=\frac{k\pi }{2} (k\epsilon Z) \)

本质：只需要将*φ*等于\(\frac{\pi }{2}\)的倍数，利用诱导公式化简，按照题意消去*φ *即可

**二、函数的周期性**

**例2*** *函数\(y=\sin(2x + \frac{\pi }{6} ) + \cos(2x + \frac{\pi }{6} ) \) 的最小正周期为

【评注】关于三角函数周期的几个重要结论：

*y*＝*A *sin(*ω x*＋*φ*)+b ，*y*＝*A *cos(*ω x*＋*φ*)+b，*y*＝*A *tan(*ω x*＋*φ*)+b 的周期分别为：

\(\frac{2\pi }{\left | \varpi \right |}\)，\(\frac{2\pi }{\left | \varpi \right |}\)，\(\frac{\pi }{\left | \varpi \right |}\)

*y*＝|*A *sin(*ω x*＋*φ*)+b| ，*y*＝|*A *cos(*ω x*＋*φ*)+b|，*y*＝|*A *tan(*ω x*＋*φ*)+b| 的周期分别为：

\(\frac{\pi }{\left | \varpi \right |}\)，\(\frac{\pi }{\left | \varpi \right |}\)，\(\frac{\pi }{\left | \varpi \right |}\)

**三、函数的单调性**

**例3*** * 求函数\(y=\sin 2x+\frac{\pi }{6}\) 的单调增区间

【评注】求三角函数*y*＝*A *sin(*ω x*＋*φ*) (A>0, *ω>0*)的单调区间：

直接将*ω x*＋*φ *放入标准的* y＝ sin x *的增减区间（可见掌握基础的重要性），解不等式即可

对于此题，如果求增区间，则直接解下列不等式：

\(2k\pi -\frac{\pi }{2}\leqslant 2x+\frac{\pi }{6}\leqslant 2k\pi +\frac{\pi }{2} \)

对于*y*＝*A *cos(*ω x*＋*φ*)，*y*＝*A *tan(*ω x*＋*φ*) 有类似的做法，这里不再赘述

**注意: **如果将题目改一下，*ω*<0，对于函数*y*＝*A *sin(*ω x*＋*φ*) (A>0, *ω<0*)

可以使用诱导公式进行恒等变形，将函数变形为*y*＝-*A *sin(-*ω x – **φ*)

则函数 *y*＝*A *sin(-*ω x – **φ*) 的增区间为原函数的减区间，减区间为原函数的增区间

**四、函数的对称性（对称轴、对称中心）**

**例4*** *求函数 \(y=sin(2x+\frac{\pi }{3})\) 图像的对称轴方程

【评注】关于三角函数 *y*＝*A *sin(*ω x*＋*φ*) 的对称轴或者对称中心的求法，类似求单调区间的做法，将*ω x*＋*φ *等于其相应的标准三角函数的对称轴或者对称中心，求解方程即可

对于此题而言，只需将：

\(2x+\frac{\pi }{3} = k\pi+\frac{\pi}{2}\)

解方程即为此题的答案

**三角函数性质的综合**

【思路提示】三角函数的性质（奇偶性、周期性、单调性、对称性）中，对称性尤为重要；

（1）对称性=》奇偶性

若函数f(x)的图像关于y轴对称，则函数f(x)为偶函数；若函数f(x)的图像关于原点对称，则函数f(x)为奇函数

（2）对称性=》周期性

相邻两条对称轴之间的距离为T/2，相邻两个对称点之间的距离为T/2，相邻的对称中心与对称轴之间的距离为T/4

（3）对称性=》单调性

相邻两条对称轴之间，函数f(x)单调

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daw

2017年2月28日 下午11:49

我是一名高一学生，最近正在学习三角函数这一章，吴老师的这篇文章解答了我心中的疑惑，感谢吴老师

吴老师

2017年2月28日 下午11:50

客气啦，欢迎继续关注，后续将有三角函数其他部分的专题总结