三角函数的图像与性质题型归纳总结1

作者: 吴老师 分类: 三角函数 发布时间: 2017-02-27 14:55 浏览: 1,105 次

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yA sin(ω xφ)或yA cos(ω xφ),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(xφ)(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是奇函数,y=cos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yA sin(ω xφ)是奇函数,则\(\phi=k\pi(k\epsilon Z)\)

(2)若yA sin(ω xφ)是偶函数,则\(\phi=k\pi+\frac{\pi}{2}(k\epsilon Z)\)

(3)若yA cos(ω xφ)是奇函数,则\(\phi=k\pi+\frac{\pi}{2}(k\epsilon Z)\)

(4)若yA cos(ω xφ)是偶函数,则\(\phi=k\pi(k\epsilon Z)\)

(5)若yA tan(ω xφ)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:

yA sin(ω xφ)+b ,yA cos(ω xφ)+b,yA tan(ω xφ)+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 xφ)+b| ,y=|A cos(ω xφ)+b|,y=|A tan(ω xφ)+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yA sin(ω xφ) (A>0, ω>0)的单调区间:

直接将ω xφ 放入标准的 y= sinx 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yA cos(ω xφ),yA tan(ω xφ) 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yA sin(ω xφ) (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 yA 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 yA sin(ω xφ) 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 xφ 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

    1. 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

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