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18学年高中数学三角函数章末小结与测评教学案北师大版41802032126


内部文件,版权追溯 第一章 三角函数 一、角的概念 1.角不仅有大小而且有正负,角的概念的推广重在“旋转”两字.其旋转方向决定了角的 正负,由此确定了角的分类. 2.象限角及非象限角,都是相对于坐标系而言的,应注意平面直角坐标系的建立方法,即 角的顶点与坐标原点重合,角的始边与 x 轴的正半轴重合,只有在这一前提下,才能讨论象限角 与非象限角. 3.终边相同的角有无数个,在所有与角 α 终边相同的角的集合可表示为 S= {β |β =k×360°+α ,k∈Z}.终边相同的角不一定相等,但相等的角终边一定相同. 二、角度制与弧度制 弧度制是以“弧度”为单位来度量角的单位制,而角度制是以“度”为单位来度量角的单位 制,两种单位不能混用,如 π +k×360°或 60°+2kπ ,k∈Z 的写法是不允许的,尤其是当角 6 是用字母表示时更要注意,如角是在弧度制下,就不能写成 k×360°+α ,k∈Z 等. 三、三角函数的定义 1.三角函数的定义有两种 (1)角 α 的终边上任取一点 P(x,y),|OP|=r,则 sin α = ,cos α = ;tan α = . (2)角 α 的终边与以原点为圆心,以单位长为半径的圆交于点 P(x,y),则 sin α =y,cos α =x,tan α = . 2.用三角函数线解基本的三角不等式的步骤为: (1)先作出取等号的角; (2)利用三角函数线的直观性,在单位圆中确定满足不等式的角范围. 3.诱导公式 π π 2kπ +α ,π ±α ,-α ,2π ±α , ±α 的诱导公式可归纳为:k× +α (k∈Z)的三角 2 2 函数值.当 k 为偶数时,得 α 的同名三角函数值;当 k 为奇数时,得 α 的余名三角函数值,然 后在前面加上一个把 α 看成锐角时原函数值的符号,概括为“奇变偶不变,符号看象限”,这 -1- y r x r y x y x 里的奇偶指整数 k 的奇偶. 四、三角函数的图像与性质 y=sin x y=cos x y=tan x 图像 定义域 (-∞,+∞) π {x|x∈R, x≠ +kπ , 2 k∈Z} 值域 周期性 奇偶性 奇函数 π ? 在?2kπ - , 2 ? π 2kπ + ? ? 2? (k∈Z)上增; 单调性 π ? 在?2kπ + 2 ? 3π 2kπ + ? 2 ? ? (k∈Z)上减 对称轴 对称 (kπ ,0)(k∈Z) 中心 , [-1,1] 周期 T=2π 偶函数 (-∞,+∞) 周期 T=π 奇函数 在[2kπ -π , π π 2kπ ](k∈Z)上增; 在(kπ - , kπ + ) 2 2 在[2kπ ,2kπ + π ](k∈Z)上减 (k∈Z)上增 x=kπ + (k∈Z) π 2 x=kπ (k∈Z) π (kπ + ,0)(k∈ 2 Z) __ ( kπ 2 ,0)(k∈Z) 五、函数 y=Asin(ω x+φ )的图像 1.由 y=sin x 的图像变换得到 y=Asin(ω x+φ )的图像 -2- (1)三角函数图像的变化规律和方法,由 y=sin x→y=sin(x+φ ),此步骤只是平移,而由 y=sin x→y=sin(ω x+φ )可由两条思路:①y=sin x→y=sin(x+φ )→y=sin(ω x+φ )即 先平移后伸缩;②y=sin x→y=sin ω x→y=sin(ω x+φ )即先伸缩再平移.不论哪一条路径, 每一次变换都是对字母 x 而言的. (2)“先平移后伸缩”和“先伸缩后平移”,两条路径平移的单位不同 ;“先平移后伸缩” ?φ ? 平 移 |φ | 个 单 位 , “ 先 伸 缩 后 平 移 ” 则 须 平 移 ? ? 个 单 位 . 主 要 程 序 如 下 : ①y = sin ?ω ? x 平移| ― → ― → y = sin(ω x + φ ) ― ― → y = Asin(ω x + φ ) ;②y = φ― | 个单位y = sin(x + φ ) ― 周期变换 平移变换 振幅变换 φ― ? sin x ― ― → y=sin ω x平移? ― → ― → y=Asin(ω x+φ ). 个单位y=sin(ω x+φ ) ― ? ? ?ω ? 2.由图像确定函数 y=Asin(ω x+φ )的解析式,主要从以下三个方面来考虑 (1)A 的确定:根据图像的“最高点、最低点”确定 A. 2π (2)ω 的确定:结合图像先求周期 T,然后由 T= (ω >0)确定 ω . ω φ (3)φ 的确定:根据函数 y=Asin(ω x+φ )最开始与 x 轴的交点(靠近原点)的横坐标为- ω 平移变换 周期变换 振幅变换 ?即令ω x+φ =0,x=-φ ?确定 φ . ? ω? ? ? 3.函数 y=Asin(ω x+φ )的性质 (1)求形如 y=Asin(ω x+φ )(其中 A≠0,ω >0)的函数的单调区间,可以通过列不等式的方 法求解,列不等式的原则是:把“ω x+φ ”视为一个“整体”;再根据 y=sin x 的增减区间列 不等式. π (2)对于函数 y=Asin(ω x+φ )(A≠0,ω ≠0),当 φ =kπ ,k∈Z 时,是奇函数;当 φ = 2 +kπ ,k∈Z 时,是偶函数. 2π (3)函数 y=Asin(ω x+φ )的周期 T= . |ω | [典例 1] 已知 f(α ) π ? ?3π ? ? sin?-α + ?cos? -α ?tan(α +5π ) 2? ? 2 ? ? = , tan(-α -π )sin(α -3π ) -3- 13π (1)化简 f(α );(2)若 α =- ,求 f(α )的值. 3 cos α (-sin α )tan α [解] (1)f(α )= =-cos α . (-tan α )(-sin α ) ?

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