人教A版高中數(shù)學(xué)（必修第一冊(cè)）同步講義 5.4三角函數(shù)圖象（原卷版）

第第頁(yè)第05講5.4.1正弦函數(shù)、余弦函數(shù)的圖象課程標(biāo)準(zhǔn)學(xué)習(xí)目標(biāo)①理解并掌握用單位圓作正弦函數(shù)以及作余弦函數(shù)的圖象的方法.掌握數(shù)形結(jié)合的優(yōu)勢(shì)。②通過(guò)兩類(lèi)函數(shù)圖象認(rèn)識(shí)函數(shù)圖象的特點(diǎn)，并能通過(guò)兩類(lèi)圖象的形狀掌握兩類(lèi)函數(shù)的性質(zhì)。會(huì)作正弦函數(shù)、余弦函數(shù)的圖象的同時(shí)，能認(rèn)識(shí)圖象與三角函數(shù)的密切關(guān)系，并能解決與圖象有關(guān)的三角函數(shù)問(wèn)題知識(shí)點(diǎn)01：正弦函數(shù)的圖象正弦函數(shù)SKIPIF1<0,SKIPIF1<0的圖象叫做正弦曲線.知識(shí)點(diǎn)02：正弦函數(shù)圖象的畫(huà)法(1)幾何法:①在單位圓上,將點(diǎn)SKIPIF1<0繞著點(diǎn)SKIPIF1<0旋轉(zhuǎn)SKIPIF1<0弧度至點(diǎn)SKIPIF1<0,根據(jù)正弦函數(shù)的定義,點(diǎn)SKIPIF1<0的縱坐標(biāo)SKIPIF1<0.由此,以SKIPIF1<0為橫坐標(biāo),SKIPIF1<0為縱坐標(biāo)畫(huà)點(diǎn),即得到函數(shù)圖象上的點(diǎn)SKIPIF1<0.②將函數(shù)SKIPIF1<0,SKIPIF1<0的圖象不斷向左、向右平行移動(dòng)(每次移動(dòng)SKIPIF1<0個(gè)單位長(zhǎng)度).(2)“五點(diǎn)法”:在函數(shù)SKIPIF1<0,SKIPIF1<0的圖象上,以下五個(gè)點(diǎn):SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0在確定圖象形狀時(shí)起關(guān)鍵作用.描出這五個(gè)點(diǎn),函數(shù)SKIPIF1<0,SKIPIF1<0的圖象形狀就基本確定了.因此,在精確度要求不高時(shí),常先找出這五個(gè)關(guān)鍵點(diǎn),再用光滑的曲線將它們連接起來(lái),得到正弦函數(shù)的簡(jiǎn)圖.【即學(xué)即練1】用五點(diǎn)法作出函數(shù)SKIPIF1<0的大致圖象．知識(shí)點(diǎn)03：余弦函數(shù)的圖象余弦函數(shù)SKIPIF1<0,SKIPIF1<0的圖象叫做余弦曲線.知識(shí)點(diǎn)04：余弦函數(shù)圖象的畫(huà)法(1)要得到SKIPIF1<0,SKIPIF1<0的圖象,只需把SKIPIF1<0,SKIPIF1<0的圖象向左平移SKIPIF1<0個(gè)單位長(zhǎng)度即可,這是因?yàn)镾KIPIF1<0.(2)用“五點(diǎn)法”:畫(huà)余弦函數(shù)SKIPIF1<0在SKIPIF1<0上的圖象時(shí),所取的五個(gè)關(guān)鍵點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0再用光滑的曲線連接起來(lái).【即學(xué)即練1】作出函數(shù)SKIPIF1<0的圖象題型01用“五點(diǎn)法”作三角函數(shù)的圖象【典例1】用“五點(diǎn)法”在給定的坐標(biāo)系中，畫(huà)出函數(shù)SKIPIF1<0在SKIPIF1<0上的大致圖像.

【典例2】（1）作出函數(shù)SKIPIF1<0的簡(jiǎn)圖；（2）作出函數(shù)SKIPIF1<0的簡(jiǎn)圖．【變式1】用五點(diǎn)法畫(huà)出函數(shù)SKIPIF1<0一個(gè)周期的圖象.【變式2】已知函數(shù)SKIPIF1<0，SKIPIF1<0．在用“五點(diǎn)法”作函數(shù)SKIPIF1<0的圖象時(shí)，列表如下：SKIPIF1<0xSKIPIF1<0完成上述表格，并在坐標(biāo)系中畫(huà)出函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的圖象；

【變式3】用五點(diǎn)法分別畫(huà)下列函數(shù)在SKIPIF1<0上的圖象:(1)SKIPIF1<0;(2)SKIPIF1<0.題型02利用圖象解三角不等式【典例1】不等式SKIPIF1<0的解集為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】不等式SKIPIF1<0的解集為．【變式1】不等式SKIPIF1<0SKIPIF1<0的解集是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式2】在(0，2π)內(nèi)使sinx>|cosx|的x的取值范圍是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式3】不等式SKIPIF1<0的解集為.題型03利用圖象求方程的解或函數(shù)零點(diǎn)的個(gè)數(shù)問(wèn)題【典例1】函數(shù)SKIPIF1<0的零點(diǎn)個(gè)數(shù)為.【典例2】已知函數(shù)SKIPIF1<0在SKIPIF1<0上恰有2個(gè)零點(diǎn)，則SKIPIF1<0的取值范圍為．【典例3】函數(shù)SKIPIF1<0的零點(diǎn)個(gè)數(shù)為．【典例4】已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上有且僅有10個(gè)零點(diǎn)，則ω的取值范圍是.【典例5】方程SKIPIF1<0的解的個(gè)數(shù)是．【變式1】已知函數(shù)SKIPIF1<0，則SKIPIF1<0在SKIPIF1<0上的零點(diǎn)個(gè)數(shù)為．【變式2】已知函數(shù)SKIPIF1<0），若方程SKIPIF1<0在SKIPIF1<0上恰有5個(gè)實(shí)數(shù)解，則實(shí)數(shù)SKIPIF1<0的取值范圍為．【變式3】已知函數(shù)SKIPIF1<0，若存在實(shí)數(shù)SKIPIF1<0滿足SKIPIF1<0SKIPIF1<0互不相等SKIPIF1<0，則SKIPIF1<0的取值范圍是.【變式4】已知關(guān)于x的方程SKIPIF1<0在SKIPIF1<0上有兩個(gè)不同的實(shí)數(shù)根，則m的取值范圍是．【變式5】已知函數(shù)SKIPIF1<0，令SKIPIF1<0在區(qū)間SKIPIF1<0上恰有2個(gè)零點(diǎn)SKIPIF1<0，則SKIPIF1<0，SKIPIF1<0.A夯實(shí)基礎(chǔ)1．函數(shù)SKIPIF1<0的圖象中與y軸最近的最高點(diǎn)的坐標(biāo)為（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<02．用“五點(diǎn)法”作SKIPIF1<0的圖象，首先描出的五個(gè)點(diǎn)的橫坐標(biāo)是（）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<03．三角函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的圖像為（）A． B．C． D．4．從函數(shù)SKIPIF1<0的圖象來(lái)看，當(dāng)SKIPIF1<0時(shí)，對(duì)于SKIPIF1<0的x有（

）A．0個(gè) B．1個(gè) C．2個(gè) D．3個(gè)5．函數(shù)SKIPIF1<0與SKIPIF1<0圖像交點(diǎn)的個(gè)數(shù)為（

）A．0 B．1 C．2 D．36．函數(shù)SKIPIF1<0的簡(jiǎn)圖是（

）A．B．C． D．7．函數(shù)SKIPIF1<0，SKIPIF1<0的圖象在區(qū)間SKIPIF1<0的交點(diǎn)個(gè)數(shù)為（

）A．3 B．4 C．5 D．68．華羅庚說(shuō)：“數(shù)缺形時(shí)少直觀，形少數(shù)時(shí)難入微，數(shù)形結(jié)合百般好，隔離分家萬(wàn)事休.”所以研究函數(shù)時(shí)往往要作圖，那么函數(shù)SKIPIF1<0的部分圖像可能是（

）A．

B．

C．

D．

二、多選題9．（多選）函數(shù)SKIPIF1<0與SKIPIF1<0有一個(gè)交點(diǎn)，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．0C．1 D．SKIPIF1<010．函數(shù)SKIPIF1<0，SKIPIF1<0的圖像與直線SKIPIF1<0（t為常數(shù)，SKIPIF1<0）的交點(diǎn)可能有（

）A．0個(gè) B．1個(gè) C．2個(gè) D．3個(gè)三、填空題11．若函數(shù)SKIPIF1<0在SKIPIF1<0上有且僅有3個(gè)零點(diǎn)，則SKIPIF1<0的最小值為.12．不等式SKIPIF1<0的解集為.四、解答題13．用“五點(diǎn)法”在給定的坐標(biāo)系中，畫(huà)出函數(shù)SKIPIF1<0在SKIPIF1<0上的大致圖像.

14．函數(shù)SKIPIF1<0，用五點(diǎn)作圖法畫(huà)出函數(shù)SKIPIF1<0在SKIPIF1<0上的圖象；（先列表，再畫(huà)圖）15．作函數(shù)SKIPIF1<0的圖象.16．已知函數(shù)SKIPIF1<0（1）作出該函數(shù)的圖象；（2）若SKIPIF1<0，求SKIPIF1<0的值；（3）若SKIPIF1<0，討論方程SKIPIF1<0的解的個(gè)數(shù)．B能力提升1．設(shè)函數(shù)SKIPIF1<0的定義域?yàn)镽，SKIPIF1<0，SKIPIF1<0，當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0，則函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上零點(diǎn)的個(gè)數(shù)為（

）A．4 B．5 C．6 D．72．若SKIPIF1<0(SKIPIF1<0)在SKIPIF1<0上有且只有兩個(gè)零點(diǎn)，則SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．函數(shù)SKIPIF1<0的圖象與直線SKIPIF1<0有且僅有兩個(gè)不同的交點(diǎn)，求實(shí)數(shù)SKIPIF1<0的取值范圍．

第06講5.4.2正弦函數(shù)、余弦函數(shù)的性質(zhì)課程標(biāo)準(zhǔn)學(xué)習(xí)目標(biāo)①結(jié)合正弦函數(shù)、余弦函數(shù)的圖象掌握正、余弦函數(shù)的性質(zhì)。②會(huì)求正、余弦函數(shù)的周期，單調(diào)區(qū)間、對(duì)稱點(diǎn)、對(duì)稱軸及最值，及結(jié)合函數(shù)的圖象會(huì)求函數(shù)的解析式，并能求出相關(guān)的基本量。會(huì)求正、余弦函數(shù)的最小正周期，單調(diào)區(qū)間，對(duì)稱點(diǎn)，對(duì)稱區(qū)間，會(huì)求兩類(lèi)函數(shù)的最值.知識(shí)點(diǎn)01：函數(shù)的周期性1.周期函數(shù)的定義一般地,設(shè)函數(shù)SKIPIF1<0的定義域?yàn)镾KIPIF1<0,如果存在一個(gè)非零常數(shù)SKIPIF1<0,使得對(duì)每一個(gè)SKIPIF1<0,都有SKIPIF1<0,且SKIPIF1<0,那么函數(shù)SKIPIF1<0就叫做周期函數(shù).非零常數(shù)SKIPIF1<0叫做這個(gè)函數(shù)的周期.2.最小正周期的定義如果在周期函數(shù)f(x)的所有周期中存在一個(gè)最小的正數(shù),那么這個(gè)最小正數(shù)就叫做SKIPIF1<0的最小正周期.

【即學(xué)即練1】函數(shù)SKIPIF1<0的最小正周期為SKIPIF1<0，則SKIPIF1<0.知識(shí)點(diǎn)02：正弦函數(shù)、余弦函數(shù)的周期性和奇偶性函數(shù)奇偶性SKIPIF1<0奇函數(shù)SKIPIF1<0偶函數(shù)SKIPIF1<0當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0為奇函數(shù)；當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0為偶函數(shù)；SKIPIF1<0當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0為奇函數(shù)；當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0為偶函數(shù)；【即學(xué)即練2】已知函數(shù)SKIPIF1<0SKIPIF1<0的最小正周期為SKIPIF1<0，則SKIPIF1<0.知識(shí)點(diǎn)03：正弦、余弦型函數(shù)的常用周期函數(shù)最小正周期SKIPIF1<0或SKIPIF1<0（SKIPIF1<0）SKIPIF1<0SKIPIF1<0或SKIPIF1<0SKIPIF1<0SKIPIF1<0或SKIPIF1<0（SKIPIF1<0）SKIPIF1<0SKIPIF1<0無(wú)周期SKIPIF1<0SKIPIF1<0【即學(xué)即練3】函數(shù)SKIPIF1<0的最小正周期為.知識(shí)點(diǎn)04：正弦函數(shù)、余弦函數(shù)的圖象和性質(zhì)函數(shù)SKIPIF1<0SKIPIF1<0圖象定義域定義域SKIPIF1<0SKIPIF1<0值域SKIPIF1<0SKIPIF1<0周期性SKIPIF1<0SKIPIF1<0奇偶性奇函數(shù)偶函數(shù)單調(diào)性在每一個(gè)閉區(qū)間SKIPIF1<0(SKIPIF1<0)上都單調(diào)遞增;在每一個(gè)閉區(qū)間SKIPIF1<0(SKIPIF1<0上都單調(diào)遞減在每一個(gè)閉區(qū)間SKIPIF1<0(SKIPIF1<0)上都單調(diào)遞增;在每一個(gè)閉區(qū)間SKIPIF1<0(SKIPIF1<0)上都單調(diào)遞減最值當(dāng)SKIPIF1<0(SKIPIF1<0)時(shí)，SKIPIF1<0;當(dāng)SKIPIF1<0(SKIPIF1<0)時(shí)，SKIPIF1<0;當(dāng)SKIPIF1<0(SKIPIF1<0)時(shí)，SKIPIF1<0;當(dāng)SKIPIF1<0(SKIPIF1<0)時(shí)，SKIPIF1<0;圖象的對(duì)稱性對(duì)稱中心為SKIPIF1<0(SKIPIF1<0),對(duì)稱軸為直線SKIPIF1<0(SKIPIF1<0)對(duì)稱中心為SKIPIF1<0(SKIPIF1<0),對(duì)稱軸為直線SKIPIF1<0(SKIPIF1<0)【即學(xué)即練4】y＝cosSKIPIF1<0的單調(diào)遞減區(qū)間為.題型01三角函數(shù)的周期問(wèn)題及簡(jiǎn)單應(yīng)用【典例1】下列函數(shù)，最小正周期為SKIPIF1<0的是（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例2】（多選）下列函數(shù)中是奇函數(shù)，且最小正周期是SKIPIF1<0的函數(shù)是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例3】求下列函數(shù)的最小正周期．(1)SKIPIF1<0；(2)SKIPIF1<0.【變式1】下列函數(shù)中，最小正周期為π的函數(shù)是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式2】（多選）下列函數(shù)中，是周期函數(shù)的是（）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式3】求下列函數(shù)的最小正周期．(1)SKIPIF1<0；(2)SKIPIF1<0．題型02三角函數(shù)的奇偶性及其應(yīng)用【典例1】已知函數(shù)SKIPIF1<0，則SKIPIF1<0是SKIPIF1<0為奇函數(shù)的（

）A．充要條件 B．充分不必要條件C．必要不充分條件 D．既不充分也不必要條件【典例2】（多選）已知函數(shù)SKIPIF1<0為偶函數(shù)，則SKIPIF1<0的取值可以為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例3】已知函數(shù)SKIPIF1<0，SKIPIF1<0是奇函數(shù)，則SKIPIF1<0的值為．【典例4】若函數(shù)SKIPIF1<0為偶函數(shù)，則SKIPIF1<0的最小正值為.【變式1】使函數(shù)SKIPIF1<0為奇函數(shù)，則SKIPIF1<0的一個(gè)值可以是（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】（多選）若函數(shù)SKIPIF1<0是偶函數(shù)，則SKIPIF1<0的值不可能為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式3】設(shè)函數(shù)SKIPIF1<0的最大值為SKIPIF1<0，最小值為SKIPIF1<0，則SKIPIF1<0.【變式4】若函數(shù)SKIPIF1<0為奇函數(shù)，則SKIPIF1<0的最小值為.題型03函數(shù)奇偶性與周期性、單調(diào)性，對(duì)稱性的綜合問(wèn)題【典例1】已知SKIPIF1<0是SKIPIF1<0上的奇函數(shù)，且SKIPIF1<0在區(qū)間SKIPIF1<0上是單調(diào)函數(shù)，則SKIPIF1<0的最大值為（

）A．3 B．4 C．5 D．6【典例2】已知SKIPIF1<0，且SKIPIF1<0的最小正周期為2.若存在SKIPIF1<0，使得對(duì)于任意SKIPIF1<0，都有SKIPIF1<0，則SKIPIF1<0為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例3】寫(xiě)出一個(gè)同時(shí)滿足下列三個(gè)條件的函數(shù)SKIPIF1<0的解析式.①SKIPIF1<0；②SKIPIF1<0；③SKIPIF1<0在SKIPIF1<0上單調(diào)遞增.【變式1】已知SKIPIF1<0是SKIPIF1<0上的奇函數(shù)，若SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱，且SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)是單調(diào)函數(shù)，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】已知函數(shù)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0（

）A．3 B．3或7 C．5 D．7【變式3】（多選）已知定義域?yàn)镽的函數(shù)SKIPIF1<0滿足SKIPIF1<0，函數(shù)SKIPIF1<0，若函數(shù)SKIPIF1<0為奇函數(shù)，則SKIPIF1<0的值可以為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式4】某函數(shù)SKIPIF1<0滿足以下三個(gè)條件：①SKIPIF1<0是偶函數(shù)；②SKIPIF1<0；③SKIPIF1<0的最大值為4．請(qǐng)寫(xiě)出一個(gè)滿足上述條件的函數(shù)SKIPIF1<0的解析式．題型04求三角函數(shù)的單調(diào)區(qū)間【典例1】已知函數(shù)SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的單調(diào)遞增區(qū)間是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0，SKIPIF1<0 D．SKIPIF1<0，SKIPIF1<0【典例2】函數(shù)SKIPIF1<0單調(diào)減區(qū)間為【典例3】用“五點(diǎn)法”作出函數(shù)SKIPIF1<0的圖象，并指出它的最小正周期、最值及單調(diào)區(qū)間.【變式1】已知函數(shù)SKIPIF1<0，則SKIPIF1<0在SKIPIF1<0上的單調(diào)遞增區(qū)間為（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【變式2】y＝cosSKIPIF1<0的單調(diào)遞減區(qū)間為.【變式3】求函數(shù)SKIPIF1<0的單調(diào)遞增區(qū)間．題型05利用單調(diào)性比較三角函數(shù)值的大小【典例1】已知SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】下列各式中正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例3】比較下列各組數(shù)的大小.(1)SKIPIF1<0與SKIPIF1<0；(2)cos1與sin2.【典例4】不求值，分別比較下列各組中兩個(gè)三角函數(shù)值的大?。海?）SKIPIF1<0與SKIPIF1<0；

（2）SKIPIF1<0與SKIPIF1<0.【變式1】令SKIPIF1<0，SKIPIF1<0，判斷a與b的大小關(guān)系是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．無(wú)法判斷【變式2】利用三角函數(shù)的單調(diào)性，比較下列各組數(shù)的大?。?1)SKIPIF1<0，SKIPIF1<0；(2)SKIPIF1<0，SKIPIF1<0．【變式3】比較下列各組數(shù)的大?。?1)SKIPIF1<0與SKIPIF1<0；(2)SKIPIF1<0與SKIPIF1<0；(3)SKIPIF1<0與SKIPIF1<0．題型06已知三角函數(shù)的單調(diào)情況求參數(shù)問(wèn)題【典例1】已知函數(shù)SKIPIF1<0在SKIPIF1<0時(shí)有最大值，且SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞增，則SKIPIF1<0的最大值是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】已知函數(shù)SKIPIF1<0，對(duì)于SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0在區(qū)SKIPIF1<0上單調(diào)遞增，則SKIPIF1<0的最大值是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例3】已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)遞減，則實(shí)數(shù)SKIPIF1<0的取值范圍為.【變式1】已知函數(shù)SKIPIF1<0的周期為SKIPIF1<0，且滿足SKIPIF1<0，若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0不單調(diào)，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式2】若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0單調(diào)遞減，且最小值為負(fù)值，則SKIPIF1<0的值可以是（

）A．1 B．SKIPIF1<0 C．2 D．SKIPIF1<0【變式3】（多選）已知SKIPIF1<0，函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞減，則實(shí)數(shù)SKIPIF1<0的取值可以是（

）A．1 B．SKIPIF1<0 C．SKIPIF1<0 D．2【變式4】已知函數(shù)SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱，且SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)單調(diào)，則SKIPIF1<0的最大值為．題型07三角函數(shù)的對(duì)稱性【典例1】設(shè)函數(shù)SKIPIF1<0的最小正周期為SKIPIF1<0，則它的一條對(duì)稱軸方程為（）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0【典例2】已知函數(shù)SKIPIF1<0在SKIPIF1<0內(nèi)單調(diào)遞減，SKIPIF1<0是函數(shù)SKIPIF1<0的一條對(duì)稱軸，且函數(shù)SKIPIF1<0為奇函數(shù)，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例3】已知函數(shù)SKIPIF1<0的最小正周期為SKIPIF1<0，其圖象關(guān)于直線SKIPIF1<0對(duì)稱，則SKIPIF1<0.【典例4】設(shè)函數(shù)SKIPIF1<0，若SKIPIF1<0的圖象關(guān)于點(diǎn)SKIPIF1<0對(duì)稱，則SKIPIF1<0的值可以是．（寫(xiě)出一個(gè)滿足條件的值即可）【變式1】函數(shù)SKIPIF1<0的一條對(duì)稱軸為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】已知常數(shù)SKIPIF1<0，如果函數(shù)SKIPIF1<0的圖像關(guān)于點(diǎn)SKIPIF1<0中心對(duì)稱，那么SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式3】已知函數(shù)SKIPIF1<0滿足SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0（

）A．3 B．3或7 C．5 D．7【變式4】已知函數(shù)SKIPIF1<0的圖象關(guān)于點(diǎn)SKIPIF1<0對(duì)稱，那么SKIPIF1<0的最小值為．題型08利用三角函數(shù)的有界性和單調(diào)性求值域或最大(小)值【典例1】函數(shù)SKIPIF1<0的值域?yàn)椋镜淅?】已知函數(shù)SKIPIF1<0．(1)求SKIPIF1<0的單調(diào)區(qū)間；(2)求SKIPIF1<0在SKIPIF1<0上的值域．【典例3】已知函數(shù)SKIPIF1<0.(1)求函數(shù)SKIPIF1<0的對(duì)稱中心和單調(diào)遞減區(qū)間；(2)若SKIPIF1<0，求函數(shù)SKIPIF1<0的值域.【變式1】函數(shù)y=2cos(2x+SKIPIF1<0)，xSKIPIF1<0[-SKIPIF1<0，SKIPIF1<0]的值域是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】已知函數(shù)SKIPIF1<0.(1)求SKIPIF1<0的最小正周期；(2)當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0的最小值和最大值.【變式3】已知函數(shù)SKIPIF1<0.(1)求函數(shù)SKIPIF1<0的最小正周期；(2)求函數(shù)SKIPIF1<0的單調(diào)遞增區(qū)間.(3)求函數(shù)SKIPIF1<0在SKIPIF1<0上的最大值.題型09換元法求值域或最大(小)值(可化為一元二次函數(shù)型）【典例1】函數(shù)SKIPIF1<0的最大值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】求函數(shù)SKIPIF1<0，SKIPIF1<0的最大值．【典例3】已知函數(shù)SKIPIF1<0（SKIPIF1<0）的定義域?yàn)镾KIPIF1<0，且函數(shù)SKIPIF1<0的最大值為3，最小值為1，求a，b的值．【變式1】函數(shù)SKIPIF1<0，SKIPIF1<0的值域?yàn)椋咀兪?】函數(shù)SKIPIF1<0的最小值是．【變式3】已知SKIPIF1<0，求函數(shù)SKIPIF1<0的值域．題型10分式型求值域或最大(小)值【典例1】函數(shù)SKIPIF1<0的定義域是，值域是．【典例2】求函數(shù)SKIPIF1<0的值域．【變式1】函數(shù)SKIPIF1<0的值域?yàn)?【變式2】函數(shù)SKIPIF1<0的值域?yàn)椋瓵夯實(shí)基礎(chǔ)1．函數(shù)SKIPIF1<0的一個(gè)單調(diào)減區(qū)間是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．函數(shù)SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．函數(shù)SKIPIF1<0是（

）A．奇函數(shù) B．偶函數(shù) C．非奇非偶函數(shù) D．以上都不對(duì)4．函數(shù)SKIPIF1<0的最小正周期為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．已知函數(shù)SKIPIF1<0，若SKIPIF1<0，在區(qū)間SKIPIF1<0上沒(méi)有零點(diǎn)，則SKIPIF1<0的取值共有（

）A．4個(gè) B．5個(gè) C．6個(gè) D．7個(gè)6．使函數(shù)SKIPIF1<0為奇函數(shù)，則SKIPIF1<0的一個(gè)值可以是（）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．已知函數(shù)SKIPIF1<0的值域是SKIPIF1<0，則實(shí)數(shù)SKIPIF1<0的值等于（

）A．2 B．-2 C．SKIPIF1<0 D．SKIPIF1<08．已知函數(shù)SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．1二、多選題9．下列函數(shù)中是奇函數(shù)，且最小正周期是SKIPIF1<0的函數(shù)是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<010．設(shè)函數(shù)SKIPIF1<0，則下列結(jié)論正確的是（

）A．SKIPIF1<0是奇函數(shù)B．SKIPIF1<0的周期是SKIPIF1<0C．SKIPIF1<0的圖象關(guān)于點(diǎn)SKIPIF1<0對(duì)稱D．SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱三、填空題11．已知SKIPIF1<0，設(shè)函數(shù)SKIPIF1<0，則SKIPIF1<0的單調(diào)遞減區(qū)間是．12．寫(xiě)出一個(gè)同時(shí)滿足下列條件的函數(shù)SKIPIF1<0解析式.①SKIPIF1<0；②SKIPIF1<0.四、解答題13．已知函數(shù)SKIPIF1<0圖象相鄰兩對(duì)稱軸之間的距離為SKIPIF1<0且SKIPIF1<0.(1)求函數(shù)SKIPIF1<0的解析式；(2)求函數(shù)SKIPIF1<0的單調(diào)區(qū)間.14．已知函數(shù)SKIPIF1<0，(1)求不等式SKIPIF1<0的解集(2)若SKIPIF1<0求函數(shù)SKIPIF1<0的值域15．已知函數(shù)SKIPIF1<0(1)用“五點(diǎn)法”畫(huà)出函數(shù)SKIPIF1<0在一個(gè)周期內(nèi)的圖象；(2)直接寫(xiě)出函數(shù)SKIPIF1<0的值域和最小正周期．列表：SKIPIF1<0SKIPIF1<0SKIPIF1<0作圖：

B能力提升1．若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上恰有兩個(gè)零點(diǎn)，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．已知函數(shù)SKIPIF1<0（SKIPIF1<0），若SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)有且僅有3個(gè)零點(diǎn)和3條對(duì)稱軸，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．已知函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上單調(diào)，且SKIPIF1<0，則（

）A．SKIPIF1<0，SKIPIF1<0B．SKIPIF1<0，SKIPIF1<0C．SKIPIF1<0，SKIPIF1<0D．SKIPIF1<0，SKIPIF1<04．已知函數(shù)SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0在SKIPIF1<0上恰有三個(gè)零點(diǎn)，則φ的取值范圍是．5．設(shè)函數(shù)SKIPIF1<0的最大值為SKIPIF1<0，最小值為SKIPIF1<0，則SKIPIF1<0.C綜合素養(yǎng)1．已知函數(shù)SKIPIF1<0的圖象經(jīng)過(guò)點(diǎn)SKIPIF1<0，且SKIPIF1<0圖象鄰的兩條對(duì)稱軸之間的距離是SKIPIF1<0.(1)求SKIPIF1<0的單調(diào)遞增區(qū)間；(2)若對(duì)任意的SKIPIF1<0，不等式SKIPIF1<0恒成立，求m的取值范圍.2．已知函數(shù)SKIPIF1<0，(1)若SKIPIF1<0，則SKIPIF1<0的最小值為SKIPIF1<0，求SKIPIF1<0的解析式．(2)在（1）的條件下，若SKIPIF1<0在SKIPIF1<0上的值域是SKIPIF1<0，求實(shí)數(shù)SKIPIF1<0的取值范圍；3．已知函數(shù)SKIPIF1<0，SKIPIF1<0(1)求函數(shù)SKIPIF1<0在SKIPIF1<0上的單調(diào)遞增區(qū)間；(2)求不等式SKIPIF1<0的解集；(3)若方程SKIPIF1<0在SKIPIF1<0上有兩個(gè)不同的實(shí)數(shù)解，求實(shí)數(shù)SKIPIF1<0的取值范圍．

第07講5.4.3正切函數(shù)的性質(zhì)與圖象課程標(biāo)準(zhǔn)學(xué)習(xí)目標(biāo)①理解與掌握正切函數(shù)的性質(zhì)，并能運(yùn)用正切函數(shù)的性質(zhì)解決與正切函數(shù)相關(guān)的周期性、奇偶性，定義域、值域、單調(diào)性等問(wèn)題。②掌握正切函數(shù)的圖象的畫(huà)法，會(huì)運(yùn)用正切函數(shù)的圖象研究正切函數(shù)的性質(zhì)，并能解決與正切函數(shù)有關(guān)的相關(guān)量問(wèn)題。會(huì)運(yùn)用正切函數(shù)的圖象與性質(zhì)解決與正切函數(shù)有關(guān)的周期、奇偶性、單調(diào)性及值域等問(wèn)題.知識(shí)點(diǎn)01：正切函數(shù)的圖象【即學(xué)即練1】函數(shù)SKIPIF1<0在一個(gè)周期內(nèi)的大致圖象是（

）A．

B．

C．

D．

知識(shí)點(diǎn)02：正切（型）函數(shù)的性質(zhì)正切函數(shù)SKIPIF1<0正切型函數(shù)SKIPIF1<0定義域SKIPIF1<0由SKIPIF1<0SKIPIF1<0SKIPIF1<0值域SKIPIF1<0SKIPIF1<0周期性SKIPIF1<0SKIPIF1<0奇偶性奇函數(shù)當(dāng)SKIPIF1<0時(shí)SKIPIF1<0是奇函數(shù)單調(diào)性在SKIPIF1<0，SKIPIF1<0上單調(diào)遞增當(dāng)SKIPIF1<0,SKIPIF1<0時(shí)，由SKIPIF1<0，解出單調(diào)增區(qū)間對(duì)稱性對(duì)稱中心：SKIPIF1<0SKIPIF1<0；無(wú)對(duì)稱軸令：SKIPIF1<0SKIPIF1<0SKIPIF1<0，對(duì)稱中心為：SKIPIF1<0，無(wú)對(duì)稱軸【即學(xué)即練2】（多選）下列說(shuō)法中正確的是（

）A．對(duì)于定義在實(shí)數(shù)SKIPIF1<0上的函數(shù)SKIPIF1<0中滿足SKIPIF1<0，則函數(shù)SKIPIF1<0是以2為周期的函數(shù)B．函數(shù)SKIPIF1<0的單調(diào)遞增區(qū)間為SKIPIF1<0，SKIPIF1<0C．函數(shù)SKIPIF1<0為奇函數(shù)D．角SKIPIF1<0的終邊上一點(diǎn)坐標(biāo)為SKIPIF1<0，則SKIPIF1<0題型01正切函數(shù)的定義域【典例1】函數(shù)SKIPIF1<0的定義域?yàn)椋镜淅?】函數(shù)SKIPIF1<0的定義域?yàn)?【變式1】函數(shù)SKIPIF1<0的定義域?yàn)椋咀兪?】函數(shù)SKIPIF1<0的定義域是．題型02正切函數(shù)的值域【典例1】函數(shù)SKIPIF1<0，SKIPIF1<0的值域?yàn)椋?/p>

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】函數(shù)SKIPIF1<0，SKIPIF1<0的值域?yàn)椋镜淅?】函數(shù)SKIPIF1<0的值域?yàn)?【變式1】函數(shù)SKIPIF1<0的值域?yàn)椋咀兪?】函數(shù)SKIPIF1<0的值域?yàn)?【變式3】函數(shù)SKIPIF1<0的值域?yàn)轭}型03求正切函數(shù)的單調(diào)區(qū)間【典例1】函數(shù)SKIPIF1<0的單調(diào)區(qū)間是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】SKIPIF1<0的單調(diào)遞減區(qū)間為.【變式1】若函數(shù)SKIPIF1<0在SKIPIF1<0上為嚴(yán)格減函數(shù)，則實(shí)數(shù)SKIPIF1<0的取值范圍是．【變式2】函數(shù)SKIPIF1<0的單調(diào)遞減區(qū)間為．題型04正切函數(shù)單調(diào)性的應(yīng)用【典例1】已知函數(shù)SKIPIF1<0，若SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)單調(diào)遞減，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】已知函數(shù)SKIPIF1<0在SKIPIF1<0上是嚴(yán)格減函數(shù)，則實(shí)數(shù)SKIPIF1<0的取值范圍是．【變式1】已知函數(shù)SKIPIF1<0在SKIPIF1<0內(nèi)是減函數(shù)，則SKIPIF1<0的取值范圍為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上是增函數(shù)，則實(shí)數(shù)a的取值范圍是．題型05正切函數(shù)的周期性與奇偶性【典例1】函數(shù)SKIPIF1<0的最小正周期為．【典例2】已知SKIPIF1<0（其中SKIPIF1<0為常數(shù)且SKIPIF1<0），如果SKIPIF1<0，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．3 C．SKIPIF1<0 D．5【變式1】已知函數(shù)SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．1 D．4【變式2】函數(shù)SKIPIF1<0的最小正周期是（）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．π【變式3】已知SKIPIF1<0，SKIPIF1<0.題型06正切函數(shù)圖象的對(duì)稱性【典例1】以點(diǎn)SKIPIF1<0為對(duì)稱中心的函數(shù)是（

）．A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】已知函數(shù)SKIPIF1<0的最小正周期為SKIPIF1<0，其圖像的一個(gè)對(duì)稱中心的坐標(biāo)為SKIPIF1<0，則曲線SKIPIF1<0的對(duì)稱中心坐標(biāo)為（

）A．SKIPIF1<0，SKIPIF1<0 B．SKIPIF1<0，SKIPIF1<0C．SKIPIF1<0，SKIPIF1<0 D．SKIPIF1<0，SKIPIF1<0【典例3】函數(shù)SKIPIF1<0的圖象的對(duì)稱中心為.【變式1】函數(shù)SKIPIF1<0圖象的對(duì)稱中心可能是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】（多選）下列坐標(biāo)所表示的點(diǎn)是函數(shù)SKIPIF1<0的圖像的對(duì)稱中心的是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式3】（多選）已知函數(shù)SKIPIF1<0，則下列敘述中，正確的是（

）A．函數(shù)SKIPIF1<0的圖象關(guān)于點(diǎn)SKIPIF1<0對(duì)稱 B．函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞增C．函數(shù)SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對(duì)稱 D．函數(shù)SKIPIF1<0是偶函數(shù)題型07與正切（型）函數(shù)有關(guān)的值域（最值）問(wèn)題【典例1】函數(shù)SKIPIF1<0的值域?yàn)椋镜淅?】若函數(shù)SKIPIF1<0，SKIPIF1<0的圖象都在SKIPIF1<0軸上方，則實(shí)數(shù)SKIPIF1<0的取值范圍為.【變式1】函數(shù)SKIPIF1<0的值域是【變式2】若函數(shù)SKIPIF1<0在SKIPIF1<0上單調(diào)遞減，且在SKIPIF1<0上的最大值為SKIPIF1<0，則SKIPIF1<0.題型08正切函數(shù)圖象與性質(zhì)的綜合應(yīng)用【典例1】畫(huà)出函數(shù)SKIPIF1<0的圖象．(1)根據(jù)圖象判斷其定義域、值域、單調(diào)區(qū)間、奇偶性、周期性；(2)求不等式SKIPIF1<0的解集．【典例2】已知函數(shù)SKIPIF1<0，其中SKIPIF1<0.(1)若SKIPIF1<0，求函數(shù)SKIPIF1<0的最小正周期以及函數(shù)圖象的對(duì)稱中心；(2)若SKIPIF1<0在閉區(qū)間SKIPIF1<0上是嚴(yán)格增函數(shù)，求正實(shí)數(shù)SKIPIF1<0的取值范圍.【變式1】設(shè)函數(shù)SKIPIF1<0(1)求函數(shù)SKIPIF1<0的定義域、最小正周期．(2)求不等式SKIPIF1<0的解集．【變式2】已知函數(shù)SKIPIF1<0，SKIPIF1<0．(1)若SKIPIF1<0，求SKIPIF1<0的最小正周期與函數(shù)圖像的對(duì)稱中心；(2)若SKIPIF1<0在SKIPIF1<0上是嚴(yán)格增函數(shù)，求SKIPIF1<0的取值范圍；(3)若方程SKIPIF1<0在SKIPIF1<0上至少存在2022個(gè)根，且b－a的最小值不小于2022，求SKIPIF1<0的取值范圍．【變式3】已知函數(shù)SKIPIF1<0．（1）當(dāng)SKIPIF1<0時(shí)，求SKIPIF1<0的最小正周期及單調(diào)區(qū)間；（2）若SKIPIF1<0