人教A版高中數(shù)學(xué)（必修第一冊）同步講義第39講 5.4.3正切函數(shù)的性質(zhì)與圖象（原卷版）

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)性等問題。②掌握正切函數(shù)的圖象的畫法，會運用正切函數(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)量問題。會運用正切函數(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)性及值域等問題.知識點01：正切函數(shù)的圖象【即學(xué)即練1】（2023·全國·高二專題練習(xí)）函數(shù)SKIPIF1<0在一個周期內(nèi)的大致圖象是（

）A．

B．

C．

D．

【答案】A【詳解】由正切函數(shù)的圖象與性質(zhì)可知SKIPIF1<0在SKIPIF1<0上單調(diào)遞增，圖象為A，故選：A知識點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，由SKIPIF1<0，解出單調(diào)增區(qū)間對稱性對稱中心：SKIPIF1<0SKIPIF1<0；無對稱軸令：SKIPIF1<0SKIPIF1<0SKIPIF1<0，對稱中心為：SKIPIF1<0，無對稱軸【即學(xué)即練2】（多選）（2023秋·山東德州·高三德州市第一中學(xué)校考開學(xué)考試）下列說法中正確的是（

）A．對于定義在實數(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的終邊上一點坐標(biāo)為SKIPIF1<0，則SKIPIF1<0【答案】AB【詳解】A.若對SKIPIF1<0，滿足SKIPIF1<0，則函數(shù)SKIPIF1<0是以2為周期的函數(shù)，故A正確；B.令SKIPIF1<0，解得：SKIPIF1<0，SKIPIF1<0，所以函數(shù)的單調(diào)遞增區(qū)間為SKIPIF1<0，SKIPIF1<0，故B正確；C.SKIPIF1<0為偶函數(shù)，故C錯誤；D.角SKIPIF1<0的終邊上一點坐標(biāo)為SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0，故D錯誤.故選：AB題型01正切函數(shù)的定義域【典例1】（2023·全國·高一課堂例題）函數(shù)SKIPIF1<0的定義域為．【典例2】（2023秋·山西朔州·高一統(tǒng)考期末）函數(shù)SKIPIF1<0的定義域為.【變式1】（2023·全國·高一課堂例題）函數(shù)SKIPIF1<0的定義域為．【變式2】（2023春·上海奉賢·高一?？计谥校┖瘮?shù)SKIPIF1<0的定義域是．題型02正切函數(shù)的值域【典例1】（2023·全國·高一假期作業(yè)）函數(shù)SKIPIF1<0，SKIPIF1<0的值域為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023·全國·高一假期作業(yè)）函數(shù)SKIPIF1<0，SKIPIF1<0的值域為．【典例3】（2023·高一課時練習(xí)）函數(shù)SKIPIF1<0的值域為.【變式1】（2023·高一課時練習(xí)）函數(shù)SKIPIF1<0的值域為．【變式2】（2023·全國·高一假期作業(yè)）函數(shù)SKIPIF1<0的值域為.【變式3】（2023秋·高一課時練習(xí)）函數(shù)SKIPIF1<0的值域為題型03求正切函數(shù)的單調(diào)區(qū)間【典例1】（2023春·高一單元測試）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】（2023·全國·高三專題練習(xí)）SKIPIF1<0的單調(diào)遞減區(qū)間為.【變式1】（2023·全國·高一假期作業(yè)）若函數(shù)SKIPIF1<0在SKIPIF1<0上為嚴(yán)格減函數(shù)，則實數(shù)SKIPIF1<0的取值范圍是．【變式2】（2023·全國·高三專題練習(xí)）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】（2023·全國·高三專題練習(xí)）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】（2023·高一課時練習(xí)）已知函數(shù)SKIPIF1<0在SKIPIF1<0上是嚴(yán)格減函數(shù)，則實數(shù)SKIPIF1<0的取值范圍是．【變式1】（2023·高一課時練習(xí)）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】（2023·高一課時練習(xí)）若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上是增函數(shù)，則實數(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】（2023·全國·高一假期作業(yè)）函數(shù)SKIPIF1<0的最小正周期為．【典例2】（2023秋·廣西貴港·高三平南縣中學(xué)校考階段練習(xí)）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】（2023秋·陜西·高三校聯(lián)考階段練習(xí)）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】（2023秋·高一課時練習(xí)）函數(shù)SKIPIF1<0的最小正周期是（）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．π【變式3】（2023春·山東濰坊·高一校聯(lián)考期中）已知SKIPIF1<0，SKIPIF1<0.題型06正切函數(shù)圖象的對稱性【典例1】（2023·全國·高二專題練習(xí)）以點SKIPIF1<0為對稱中心的函數(shù)是（

）．A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023春·遼寧·高一遼寧實驗中學(xué)?？茧A段練習(xí)）已知函數(shù)SKIPIF1<0的最小正周期為SKIPIF1<0，其圖像的一個對稱中心的坐標(biāo)為SKIPIF1<0，則曲線SKIPIF1<0的對稱中心坐標(biāo)為（

）A．SKIPIF1<0，SKIPIF1<0 B．SKIPIF1<0，SKIPIF1<0C．SKIPIF1<0，SKIPIF1<0 D．SKIPIF1<0，SKIPIF1<0【典例3】（2023春·遼寧鐵嶺·高一昌圖縣第一高級中學(xué)?？茧A段練習(xí)）函數(shù)SKIPIF1<0的圖象的對稱中心為.【變式1】（2023春·河南平頂山·高一校聯(lián)考階段練習(xí)）函數(shù)SKIPIF1<0圖象的對稱中心可能是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】（多選）（2023春·安徽蕪湖·高一校聯(lián)考期中）下列坐標(biāo)所表示的點是函數(shù)SKIPIF1<0的圖像的對稱中心的是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式3】（多選）（2023·全國·高一專題練習(xí)）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SKIPIF1<0對稱 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）問題【典例1】（2023·高一課時練習(xí)）函數(shù)SKIPIF1<0的值域為．【典例2】（2023·全國·高一專題練習(xí)）若函數(shù)SKIPIF1<0，SKIPIF1<0的圖象都在SKIPIF1<0軸上方，則實數(shù)SKIPIF1<0的取值范圍為.【變式1】（2023·全國·高一假期作業(yè)）函數(shù)SKIPIF1<0的值域是【變式2】（2023·全國·高三專題練習(xí)）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】（2023秋·高一課時練習(xí)）畫出函數(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】（2023春·上海虹口·高一上外附中?？计谀┮阎瘮?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)，求正實數(shù)SKIPIF1<0的取值范圍.【變式1】（2023春·四川南充·高一校考階段練習(xí)）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】（2023秋·高一校考課時練習(xí)）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個根，且b－a的最小值不小于2022，求SKIPIF1<0的取值范圍．【變式3】（2023秋·高一課時練習(xí)）已知函數(shù)SKIPIF1<0．（1）當(dāng)SKIPIF1<0時，求SKIPIF1<0的最小正周期及單調(diào)區(qū)間；（2）若SKIPIF1<0在SKIPIF1<0上恒成立，求SKIPIF1<0的取值范圍．A夯實基礎(chǔ)B能力提升C綜合素養(yǎng)A夯實基礎(chǔ)一、單選題1．（2023秋·山西晉中·高三介休一中?？茧A段練習(xí)）函數(shù)SKIPIF1<0的最小正周期是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．2 D．42．（2023秋·陜西漢中·高二西鄉(xiāng)縣第一中學(xué)校考開學(xué)考試）函數(shù)SKIPIF1<0的定義域是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．（2023·全國·高一專題練習(xí)）函數(shù)SKIPIF1<0的單調(diào)區(qū)間是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0SKIPIF1<04．（2023·全國·高一假期作業(yè)）下列各式中正確的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<05．（2023春·北京·高一北京師大附中?？计谥校┖瘮?shù)SKIPIF1<0|在區(qū)間（SKIPIF1<0，SKIPIF1<0）內(nèi)的圖象是（

）A． B．C． D．6．（2023·河南·校聯(lián)考模擬預(yù)測）已知函數(shù)SKIPIF1<0對任意SKIPIF1<0都有SKIPIF1<0，且函數(shù)SKIPIF1<0的圖象關(guān)于SKIPIF1<0對稱，當(dāng)SKIPIF1<0時，SKIPIF1<0.則下列結(jié)論正確的是（

）A．函數(shù)SKIPIF1<0的圖象關(guān)于點SKIPIF1<0對稱B．函數(shù)SKIPIF1<0的圖象關(guān)于直線SKIPIF1<0對稱C．函數(shù)SKIPIF1<0的最小正周期為2D．當(dāng)SKIPIF1<0時，SKIPIF1<07．（2023秋·福建福州·高三福建省福州第一中學(xué)校考開學(xué)考試）函數(shù)SKIPIF1<0的圖像如圖所示，圖中陰影部分的面積為SKIPIF1<0，則SKIPIF1<0（

）

A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．（2023春·河北衡水·高一校考階段練習(xí)）函數(shù)SKIPIF1<0在SKIPIF1<0上的最大值為SKIPIF1<0，最小值為SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題9．（2023秋·吉林長春·高三長春外國語學(xué)校?？茧A段練習(xí)）已知函數(shù)SKIPIF1<0，則（

）A．SKIPIF1<0B．SKIPIF1<0的最小正周期為SKIPIF1<0C．把SKIPIF1<0向左平移SKIPIF1<0可以得到函數(shù)SKIPIF1<0D．SKIPIF1<0在SKIPIF1<0上單調(diào)遞增10．（2023·全國·高一專題練習(xí)）已知函數(shù)SKIPIF1<0，則下列描述中正確的是（

）．A．函數(shù)SKIPIF1<0的圖象關(guān)于點SKIPIF1<0成中心對稱B．函數(shù)SKIPIF1<0的最小正周期為2C．函數(shù)SKIPIF1<0的單調(diào)增區(qū)間為SKIPIF1<0，SKIPIF1<0D．函數(shù)SKIPIF1<0的圖象沒有對稱軸三、填空題11．（2023春·廣東陽江·高一廣東兩陽中學(xué)校考期末）已知SKIPIF1<0，請寫出一個滿足條件的角SKIPIF1<0．12．（2023春·遼寧錦州·高一?？计谥校㏒KIPIF1<0，若SKIPIF1<0，則SKIPIF1<0．四、解答題13．（2023·全國·高一課堂例題）畫出函數(shù)SKIPIF1<0在SKIPIF1<0上的簡圖．14．（2023春·遼寧撫順·高一校聯(lián)考期中）已知函數(shù)SKIPIF1<0的最小正周期為SKIPIF1<0，(1)求SKIPIF1<0圖象的對稱中心；(2)求不等式SKIPIF1<0在SKIPIF1<0上的解集．15．（2023秋·高一單元測試）已知SKIPIF1<0.（1）求SKIPIF1<0的最小正周期；（2）若SKIPIF1<0是奇函數(shù)，則SKIPIF1<0應(yīng)滿足什么條件？并求出滿足SKIPIF1<0的SKIPIF1<0值.B能力提升1．（2023·全國·高三專題練習(xí)）若SKIPIF1<0，（SKIPIF1<0），則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．0 D．SKIPIF1<02．（2023春·湖南·高二校聯(lián)考階段練習(xí)）若SKIPIF1<0，則SKIPIF1<0的最小值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2023·全國·高三專題練習(xí)）已知定義在SKIPIF1<0上的奇函數(shù)SKIPIF1<0滿足SKIPIF1<0，且當(dāng)SKIPIF1<0時，SKIPIF1<0