新高考數(shù)學(xué)二輪復(fù)習(xí)對(duì)點(diǎn)題型第6講三角函數(shù)的圖象與性質(zhì)（學(xué)生版）

第六講三角函數(shù)的圖象與性質(zhì)真題展示2022新高考一卷第六題記函數(shù)SKIPIF1<0的最小正周期為SKIPIF1<0．若SKIPIF1<0，且SKIPIF1<0的圖像關(guān)于點(diǎn)SKIPIF1<0，SKIPIF1<0中心對(duì)稱(chēng)，則SKIPIF1<0SKIPIF1<0A．1 B．SKIPIF1<0 C．SKIPIF1<0 D．3試題亮點(diǎn)三角函數(shù)是一類(lèi)重要的函數(shù)，三角函數(shù)的周期性是其基本性質(zhì)，三角函數(shù)的周期性決定了該函數(shù)的很多其他性質(zhì).刻畫(huà)三角函數(shù)周期性的是頻率ao.理解頻率a對(duì)三角函數(shù)的各種幾何性質(zhì)和代數(shù)性質(zhì)的影響，是考查和評(píng)價(jià)考生的基本要求.試題亮點(diǎn)如下：（1）試題巧妙地設(shè)計(jì)了正弦型三角函數(shù)圖像的中心對(duì)稱(chēng)性，反過(guò)來(lái)要求考生經(jīng)過(guò)分析與綜合，判斷正弦型函數(shù)頻率的取值或最小正周期的取值，這是對(duì)考生全面掌握三角函數(shù)性質(zhì)及其研究方法的一次很好的檢驗(yàn).（2）在試題的求解過(guò)程中，要求考生熟練掌握基本三角函數(shù)（y=sinx）的性質(zhì)，及其與復(fù)合函數(shù)（y=sin（wx+q））的性質(zhì)之間的關(guān)系，有利于指導(dǎo)教師在高中數(shù)學(xué)教學(xué)中整體把握三角函數(shù)的教學(xué).（3）數(shù)學(xué)正向問(wèn)題的解決主要依靠形式邏輯推理思維，其解決路徑是清晰的、確定的；而數(shù)學(xué)反向問(wèn)題的解決需要建立在辯證邏輯思維的基礎(chǔ)上，其解決路經(jīng)需要分析與綜合判斷．辯證邏輯思維是考生未來(lái)進(jìn)入高等學(xué)校學(xué)習(xí)，進(jìn)一步開(kāi)展科學(xué)研究需要運(yùn)用的主要的思維方式.因此，試題有利于考查考生未來(lái)的學(xué)習(xí)潛能，有利于檢測(cè)考生的辯證邏輯思維能力，對(duì)高中數(shù)學(xué)教學(xué)具有引導(dǎo)作用.知識(shí)要點(diǎn)整理一、正弦函數(shù)、余弦函數(shù)的圖象函數(shù)y＝sinxy＝cosx圖象圖象畫(huà)法五點(diǎn)法五點(diǎn)法關(guān)鍵五點(diǎn)，eq\b\lc\(\rc\)(\a\vs4\al\co1(\f(π,2)，1))，(π，0)，eq\b\lc\(\rc\)(\a\vs4\al\co1(\f(3π,2)，－1))，(0,1)，eq\b\lc\(\rc\)(\a\vs4\al\co1(\f(π,2)，0))，(π，－1)，eq\b\lc\(\rc\)(\a\vs4\al\co1(\f(3π,2)，0))，(2π，1)正(余)弦曲線(xiàn)正(余)弦函數(shù)的叫做正(余)弦曲線(xiàn)二、正切函數(shù)的圖象與性質(zhì)解析式y(tǒng)＝tanx圖象定義域eq\b\lc\{\rc\}(\a\vs4\al\co1(x\b\lc\|\rc\(\a\vs4\al\co1(x≠\f(π,2)＋kπ，k∈Z))))值域R最小正周期π奇偶性奇函數(shù)單調(diào)性在每一個(gè)區(qū)間eq\b\lc\(\rc\)(\a\vs4\al\co1(－\f(π,2)＋kπ，\f(π,2)＋kπ))(k∈Z)上都單調(diào)遞增對(duì)稱(chēng)性對(duì)稱(chēng)中心eq\b\lc\(\rc\)(\a\vs4\al\co1(\f(kπ,2)，0))(k∈Z)三、函數(shù)的周期性1．函數(shù)的周期性一般地，設(shè)函數(shù)f(x)的定義域?yàn)镈，如果存在一個(gè)，使得對(duì)每一個(gè)x∈D都有x＋T∈D，且，那么函數(shù)f(x)就叫做周期函數(shù)．叫做這個(gè)函數(shù)的周期．2．最小正周期如果在周期函數(shù)f(x)的所有周期中存在一個(gè)，那么這個(gè)最小正數(shù)叫做f(x)的最小正周期．四、正弦函數(shù)、余弦函數(shù)的周期性和奇偶性函數(shù)y＝sinxy＝cosx圖象定義域RR周期2kπ(k∈Z且k≠0)2kπ(k∈Z且k≠0)最小正周期2π2π奇偶性奇函數(shù)偶函數(shù)正弦函數(shù)、余弦函數(shù)的單調(diào)性與最值正弦函數(shù)余弦函數(shù)圖象定義域RR值域[][]單調(diào)性在每一個(gè)閉區(qū)間eq\b\lc\[\rc\](\a\vs4\al\co1(2kπ－\f(π,2)，2kπ＋\f(π,2)))(k∈Z)上都單調(diào)遞增，在每一個(gè)閉區(qū)間eq\b\lc\[\rc\](\a\vs4\al\co1(2kπ＋\f(π,2)，2kπ＋\f(3π,2)))(k∈Z)上都單調(diào)遞減在每一個(gè)閉區(qū)間[2kπ－π，2kπ](k∈Z)上都單調(diào)遞增，在每一個(gè)閉區(qū)間[2kπ，2kπ＋π](k∈Z)上都單調(diào)遞減最值x＝eq\f(π,2)＋2kπ(k∈Z)時(shí)，ymax＝1；x＝－eq\f(π,2)＋2kπ(k∈Z)時(shí)，ymin＝－1x＝(k∈Z)時(shí)，ymax＝1；x＝(k∈Z)時(shí)，ymin＝－1三年真題一、單選題1．已知SKIPIF1<0，關(guān)于該函數(shù)有下列四個(gè)說(shuō)法：①SKIPIF1<0的最小正周期為SKIPIF1<0；②SKIPIF1<0在SKIPIF1<0上單調(diào)遞增；③當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0的取值范圍為SKIPIF1<0；④SKIPIF1<0的圖象可由SKIPIF1<0的圖象向左平移SKIPIF1<0個(gè)單位長(zhǎng)度得到．以上四個(gè)說(shuō)法中，正確的個(gè)數(shù)為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0的圖象大致為（

）A． B．C． D．3．已知函數(shù)SKIPIF1<0，則（

）A．SKIPIF1<0在SKIPIF1<0上單調(diào)遞減 B．SKIPIF1<0在SKIPIF1<0上單調(diào)遞增C．SKIPIF1<0在SKIPIF1<0上單調(diào)遞減 D．SKIPIF1<0在SKIPIF1<0上單調(diào)遞增4．在SKIPIF1<0中，SKIPIF1<0．P為SKIPIF1<0所在平面內(nèi)的動(dòng)點(diǎn)，且SKIPIF1<0，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．設(shè)函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0恰有三個(gè)極值點(diǎn)、兩個(gè)零點(diǎn)，則SKIPIF1<0的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．如圖是下列四個(gè)函數(shù)中的某個(gè)函數(shù)在區(qū)間SKIPIF1<0的大致圖像，則該函數(shù)是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．將函數(shù)SKIPIF1<0的圖像向左平移SKIPIF1<0個(gè)單位長(zhǎng)度后得到曲線(xiàn)C，若C關(guān)于y軸對(duì)稱(chēng)，則SKIPIF1<0的最小值是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．記函數(shù)SKIPIF1<0的最小正周期為T(mén)．若SKIPIF1<0，且SKIPIF1<0的圖象關(guān)于點(diǎn)SKIPIF1<0中心對(duì)稱(chēng)，則SKIPIF1<0（

）A．1 B．SKIPIF1<0 C．SKIPIF1<0 D．39．函數(shù)SKIPIF1<0是A．奇函數(shù)，且最大值為2 B．偶函數(shù)，且最大值為2C．奇函數(shù)，且最大值為SKIPIF1<0 D．偶函數(shù)，且最大值為SKIPIF1<010．函數(shù)SKIPIF1<0的最小正周期和最大值分別是（

）A．SKIPIF1<0和SKIPIF1<0 B．SKIPIF1<0和2 C．SKIPIF1<0和SKIPIF1<0 D．SKIPIF1<0和211．下列函數(shù)中最小值為4的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<012．已知命題SKIPIF1<0﹔命題SKIPIF1<0﹐SKIPIF1<0，則下列命題中為真命題的是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<013．下列區(qū)間中，函數(shù)SKIPIF1<0單調(diào)遞增的區(qū)間是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題14．已知函數(shù)SKIPIF1<0的圖像關(guān)于點(diǎn)SKIPIF1<0中心對(duì)稱(chēng)，則（

）A．SKIPIF1<0在區(qū)間SKIPIF1<0單調(diào)遞減B．SKIPIF1<0在區(qū)間SKIPIF1<0有兩個(gè)極值點(diǎn)C．直線(xiàn)SKIPIF1<0是曲線(xiàn)SKIPIF1<0的對(duì)稱(chēng)軸D．直線(xiàn)SKIPIF1<0是曲線(xiàn)SKIPIF1<0的切線(xiàn)三、填空題15．記函數(shù)SKIPIF1<0的最小正周期為T(mén)，若SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的零點(diǎn)，則SKIPIF1<0的最小值為_(kāi)___________．【答案】SKIPIF1<0【詳解】解：因?yàn)镾KIPIF1<0，（SKIPIF1<0，SKIPIF1<0）所以最小正周期SKIPIF1<0，因?yàn)镾KIPIF1<0，又SKIPIF1<0，所以SKIPIF1<0，即SKIPIF1<0，又SKIPIF1<0為SKIPIF1<0的零點(diǎn)，所以SKIPIF1<0，解得SKIPIF1<0，因?yàn)镾KIPIF1<0，所以當(dāng)SKIPIF1<0時(shí)SKIPIF1<0；故答案為：SKIPIF1<016．已知函數(shù)SKIPIF1<0的部分圖像如圖所示，則SKIPIF1<0_______________.17．已知函數(shù)SKIPIF1<0的部分圖像如圖所示，則滿(mǎn)足條件SKIPIF1<0的最小正整數(shù)x為_(kāi)_______．四、解答題18．設(shè)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在SKIPIF1<0上的最大值.19．小明同學(xué)用“五點(diǎn)法”作某個(gè)正弦型函數(shù)SKIPIF1<0在一個(gè)周期內(nèi)的圖象時(shí)，列表如下：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<00SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0030-30根據(jù)表中數(shù)據(jù)，求：（1）實(shí)數(shù)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0的值；（2）該函數(shù)在區(qū)間SKIPIF1<0上的最大值和最小值.20．在銳角△ABC中，角A，B，C的對(duì)邊分別為a，b，c，且SKIPIF1<0．（I）求角B的大小；（II）求cosA+cosB+cosC的取值范圍．三年模擬1．函數(shù)SKIPIF1<0的圖象如圖所示，將函數(shù)SKIPIF1<0的圖象向右平移SKIPIF1<0個(gè)單位長(zhǎng)度，得到函數(shù)SKIPIF1<0的圖象，則（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．下列四個(gè)函數(shù)中，在區(qū)間SKIPIF1<0上為增函數(shù)的是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．函數(shù)SKIPIF1<0的部分圖象大致為（

）A． B． C． D．4．已知函數(shù)SKIPIF1<0，函數(shù)SKIPIF1<0的圖象由SKIPIF1<0圖象向右平移SKIPIF1<0個(gè)單位得到，則下列關(guān)于函數(shù)SKIPIF1<0的圖象說(shuō)法正確的是（

）A．關(guān)于y軸對(duì)稱(chēng) B．關(guān)于原點(diǎn)對(duì)稱(chēng)C．關(guān)于直線(xiàn)SKIPIF1<0對(duì)稱(chēng) D．關(guān)于點(diǎn)SKIPIF1<0對(duì)稱(chēng)5．對(duì)于函數(shù)SKIPIF1<0，給出下列四個(gè)命題：（1）該函數(shù)的值域是SKIPIF1<0；（2）當(dāng)且僅當(dāng)SKIPIF1<0時(shí)，該函數(shù)取最大值SKIPIF1<0；（3）該函數(shù)的最小正周期為SKIPIF1<0；（4）當(dāng)且僅當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0；其中所有正確命題個(gè)數(shù)是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．對(duì)于函數(shù)SKIPIF1<0，給出下列五個(gè)命題：（1）該函數(shù)的值域是SKIPIF1<0；（2）當(dāng)且僅當(dāng)SKIPIF1<0或SKIPIF1<0時(shí)，該函數(shù)取最大值1；（3）該函數(shù)的最小正周期為2π；（4）當(dāng)且僅當(dāng)SKIPIF1<0