第四章　三角函數(shù)、解三角形

第四章三角函數(shù)、解三角形[知識網(wǎng)絡(luò)][命題方向]1．從題型和題量上看，一般是兩小(選擇題或填空題)一大(解答題)，總的分值是20分左右，也有可能和其它內(nèi)容綜合命題；(1)高考試題中主要考查三角函數(shù)的圖象及其變換、性質(zhì)及其應(yīng)用，以及正弦、余弦定理在解三角形中的應(yīng)用，有時(shí)也以化簡求值為背景考查三角恒等變換等問題；(2)在處理三角函數(shù)與解三角形有關(guān)問題時(shí)，熟記公式是解決此類問題的前提，同時(shí)注意換元法在解決與三角函數(shù)性質(zhì)有關(guān)問題中的應(yīng)用．2．本章考查的主要內(nèi)容有：(1)三角函數(shù)的定義、圖象和性質(zhì)；(2)利用三角函數(shù)公式進(jìn)行三角恒等變換及化簡、求值等；(3)函數(shù)y＝Asin(ωx＋φ)的圖象變換、求解析式與性質(zhì)應(yīng)用；(4)以解三角形為載體考查正弦、余弦定理以及三角形面積公式的應(yīng)用(如2023年新高考Ⅱ卷第17題，2023年全國乙卷理科第18題，2023年全國甲卷文科第17題)；(5)以函數(shù)、不等式、向量為載體與三角函數(shù)有關(guān)的綜合性問題仍要關(guān)注．同時(shí)需要注意數(shù)形結(jié)合思想和函數(shù)方程思想在解題中的應(yīng)用．探究1(人教A版必修第一冊P172)如圖，在射線OA上任取一點(diǎn)Q(不同于點(diǎn)O)，OQ＝r1.在旋轉(zhuǎn)過程中，點(diǎn)Q所形成的圓弧eq\o(QQ1,\s\up8(︵))的長為l1，l1與r1的比值是多少？你能得出什么結(jié)論？________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究2(人教A版必修第一冊P203)函數(shù)y＝Asin(ωx＋φ)，x∈R及函數(shù)y＝Acos(ωx＋φ)，x∈R(其中A，ω，φ為常數(shù)，且A≠0，ω＞0)的周期僅與自變量的系數(shù)有關(guān)，那么，如何用自變量的系數(shù)表示上述函數(shù)的周期呢？事實(shí)上，令z＝ωx＋φ，那么由x∈R得z∈R，且函數(shù)y＝Asinz，z∈R及函數(shù)y＝Acosz，z∈R的周期都是2π.因?yàn)閦＋2π＝(ωx＋φ)＋2π＝ωeq\b\lc\(\rc\)(\a\vs4\al\co1(x＋\f(2π,ω)))＋φ，所以自變量x增加eq\f(2π,ω)，函數(shù)值就重復(fù)出現(xiàn)；并且增加量小于eq\f(2π,ω)時(shí)，函數(shù)值不會重復(fù)出現(xiàn)，即T＝eq\f(2π,ω)是使等式Asin[ω(x＋T)＋φ]＝Asin(ωx＋φ)，Acos[ω(x＋T)＋φ]＝Acos(ωx＋φ)成立的最小正數(shù)．從而，函數(shù)y＝Asin(ωx＋φ)，x∈R及函數(shù)y＝Acos(ωx＋φ)，x∈R的周期T＝eq\f(2π,ω).根據(jù)這個(gè)結(jié)論，我們可以由這類函數(shù)的解析式直接寫出函數(shù)的周期．想一想：上述求函數(shù)y＝Asin(ωx＋φ)，x∈R及函數(shù)y＝Acos(ωx＋φ)，x∈R周期的方法是否能推廣到求一般周期函數(shù)的周期？即命題“如果函數(shù)y＝f(x)的周期是T，那么函數(shù)y＝f(ωx)(ω＞0)的周期是eq\f(T,ω)是否成立？__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究3(人教A版必修第一冊P207)你能求出函數(shù)y＝sineq\b\lc\(\rc\)(\a\vs4\al\co1(－\f(1,2)x＋\f(π,3)))，x∈[－2π，2π]的單調(diào)遞增區(qū)間嗎？__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究4(人教A版必修第一冊P219)sineq\b\lc\(\rc\)(\a\vs4\al\co1(\f(π,4)－α))＝coseq\b\lc\(\rc\)(\a\vs4\al\co1(\f(π,4)＋α)).那么對于任意角α，此等式成立嗎？若成立，你會用幾種方法予以證明？_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究5(人教A版必修第一冊P236)你能總結(jié)一下從正弦函數(shù)圖象出發(fā)，通過圖象變換得到y(tǒng)＝Asin(ωx＋φ)(A＞0，ω＞0)的圖象的過程與方法嗎？_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題1(人教A版必修第一冊P176習(xí)題5.1T12)已知相互嚙合的兩個(gè)齒輪，大輪有48齒，小輪有20齒．(1)當(dāng)大輪轉(zhuǎn)動(dòng)一周時(shí)，求小輪轉(zhuǎn)動(dòng)的角度；(2)如果大輪的轉(zhuǎn)速為180r/min(轉(zhuǎn)/分)，小輪的半徑為10.5cm，那么小輪周上一點(diǎn)每1s轉(zhuǎn)過的弧長是多少？_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題2(人教A版必修第一冊P179例2)如圖，設(shè)α是一個(gè)任意角，它的終邊上任意一點(diǎn)P(不與原點(diǎn)O重合)的坐標(biāo)為(x，y)，點(diǎn)P與原點(diǎn)的距離為r.求證：sinα＝eq\f(y,r)，cosα＝eq\f(x,r)，tanα＝eq\f(y,x)._______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題3(人教A版必修第一冊P183例7)求證eq\f(cosx,1－sinx)＝eq\f(1＋sinx,cosx).__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題4(人教A版必修第一冊P186習(xí)題5.2T16)化簡eq\r(\f(1＋sinα,1－sinα))－eq\r(\f(1－sinα,1＋sinα))，其中α為第二象限角．__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題5(人教A版必修第一冊P186習(xí)題5.2T17)從本節(jié)的例7可以看出，eq\f(cosx,1－sinx)＝eq\f(1＋sinx,cosx)就是sin2x＋cos2x＝1的一個(gè)變形．你能利用同角三角函數(shù)的基本關(guān)系推導(dǎo)出更多的關(guān)系式嗎？__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題6(人教A版必修第一冊P186習(xí)題5.2T18)(1)分別計(jì)算sin4eq\f(π,3)－cos4eq\f(π,3)和sin2eq\f(π,3)－cos2eq\f(π,3)的值，你有什么發(fā)現(xiàn)？(2)任取一個(gè)α的值，分別計(jì)算sin4α－cos4α，sin2α－cos2α，你又有什么發(fā)現(xiàn)？(3)證明：?x∈R，sin2x－cos2x＝sin4x－cos4x.__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題7(人教A版必修第一冊P193例4)化簡eq\f(sin（2π－α）cos（π＋α）cos\b\lc\(\rc\)(\a\vs4\al\co1(\f(π,2)＋α))cos\b\lc\(\rc\)(\a\vs4\al\co1(\f(11π,2)－α)),cos（π－α）sin（3π－α）sin（－π－α）sin\b\lc\(\rc\)(\a\vs4\al\co1(\f(9π,2)＋α))).__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題8(人教A版必修第一冊P193例5)已知sin(53°－α)＝eq\f(1,5)，且－270°＜α＜－90°，求sin(37°＋α)的值．__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題9(人教A版必修第一冊P195習(xí)題5.3T9)化簡下列各式，其中n∈Z：(1)sineq\b\lc\(\rc\)(\a\vs4\al\co1(\f(nπ,2)＋α))；(2)coseq\b\lc\(\rc\)(\a\vs4\al\co1(\f(nπ,2)－α)).__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題10(人教A版必修第一冊P206例4)不通過求值，比較下列各組數(shù)的大?。?1)sineq\b\lc\(\rc\)(\a\vs4\al\co1(－\f(π,18)))與sineq\b\lc\(\rc\)(\a\vs4\al\co1(－\f(π,10)))；(2)coseq\b\lc\(\rc\)(\a\vs4\al\co1(－\f(23π,5)))與coseq\b\lc\(\rc\)(\a\vs4\al\co1(－\f(17π,4))).__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題11(人教A版必修第一冊P207例5)求函數(shù)y＝sineq\b\lc\(\rc\)(\a\vs4\al\co1(\f(1,2)x＋\f(π,3)))，x∈[－2π，2π]的單調(diào)遞增區(qū)間．_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題12(人教A版必修第一冊P207練習(xí)T5)求函數(shù)y＝3sineq\b\lc\(\rc\)(\a\vs4\al\co1(2x＋\f(π,4)))，x∈[0，π]的單調(diào)遞減區(qū)間．_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2021·新高考Ⅰ卷)下列區(qū)間中，函數(shù)f(x)＝7sineq\b\lc\(\rc\)(\a\vs4\al\co1(x－\f(π,6)))單調(diào)遞增的區(qū)間是()A.eq\b\lc\(\rc\)(\a\vs4\al\co1(0，\f(π,2))) B.eq\b\lc\(\rc\)(\a\vs4\al\co1(\f(π,2)，π))C.eq\b\lc\(\rc\)(\a\vs4\al\co1(π，\f(3π,2))) D.eq\b\lc\(\rc\)(\a\vs4\al\co1(\f(3π,2)，2π))點(diǎn)評本題和教材習(xí)題都是求三角函數(shù)的單調(diào)區(qū)間，解決此類問題，首先化簡成y＝Asin(ωx＋φ)形式，再求y＝Asin(ωx＋φ)的單調(diào)區(qū)間，只需把ωx＋φ看作一個(gè)整體代入y＝sinx的相應(yīng)單調(diào)區(qū)間內(nèi)即可，注意要先把ω化為正數(shù)．典題13(人教A版必修第一冊P212例6)求函數(shù)y＝taneq\b\lc\(\rc\)(\a\vs4\al\co1(\f(π,2)x＋\f(π,3)))的定義域、周期及單調(diào)區(qū)間．_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題14(人教A版必修第一冊P214習(xí)題5.4T11)根據(jù)正弦函數(shù)、余弦函數(shù)的圖象，寫出使下列不等式成立的x的取值集合：(1)sinx≥eq\f(\r(3),2)(x∈R)；(2)eq\r(2)＋2cosx≥0(x∈R)．__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題15(人教A版必修第一冊P214習(xí)題5.4T16)已知函數(shù)f(x)＝eq\f(1,2)sineq\b\lc\(\rc\)(\a\vs4\al\co1(2x－\f(π,3)))，x∈R.(1)求f(x)的最小正周期；(2)求f(x)在區(qū)間eq\b\lc\[\rc\](\a\vs4\al\co1(－\f(π,4)，\f(π,4)))上的最大值和最小值．__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題16(人教A版必修第一冊P214習(xí)題5.4T19)容易知道，正弦函數(shù)y＝sinx是奇函數(shù)，正弦曲線關(guān)于原點(diǎn)對稱，即原點(diǎn)是正弦曲線的對稱中心．除原點(diǎn)外，正弦曲線還有其他對稱中心嗎？如果有，那么對稱中心的坐標(biāo)是什么？另外，正弦曲線是軸對稱圖形嗎？如果是，那么對稱軸的方程是什么？你能用已經(jīng)學(xué)過的正弦函數(shù)性質(zhì)解釋上述現(xiàn)象嗎？對余弦函數(shù)和正切函數(shù)，討論上述同樣的問題．_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題17(人教A版必修第一冊P222例6)在△ABC中，cosA＝eq\f(4,5)，tanB＝2，求tan(2A＋2B)的值．_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題18(人教A版必修第一冊P225例8)求證：(1)sinαcosβ＝eq\f(1,2)[sin(α＋β)＋sin(α－β)]；(2)sinθ＋sinφ＝2sineq\f(θ＋φ,2)coseq\f(θ－φ,2)._______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題19(人教A必修第一冊P227例9)求下列函數(shù)的周期，最大值和最小值：(1)y＝sinx＋eq\r(3)cosx；(2)y＝3sinx＋4cosx.__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題20(人教A版必修第一冊P227例10)如圖，在扇形OPQ中，半徑OP＝1，圓心角∠POQ＝eq\f(π,3)，C是扇形弧上的動(dòng)點(diǎn)，矩形ABCD內(nèi)接于扇形，記∠POC＝α，求當(dāng)角α取何值時(shí)，矩形ABCD的面積最大？并求出這個(gè)最大面積．__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題21(人教A版必修第一冊P230習(xí)題5.5T16)是否存在銳角α，β，使α＋2β＝eq\f(2π,3)，taneq\f(α,2)tanβ＝2－eq\r(3)同時(shí)成立？若存在，求出α，β的度數(shù)；若不存在，請說明理由．__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題22(人教A版必修第一冊P230習(xí)題5.5T17)(1)求函數(shù)f(x)＝sineq\b\lc\(\rc\)(\a\vs4\al\co1(\f(π,3)＋4x))＋sineq\b\lc\(\rc\)(\a\vs4\al\co1(4x－\f(π,6)))的周期和單調(diào)遞增區(qū)間；(2)求函數(shù)f(x)＝asinx＋bcosx(a2＋b2≠0)的最大值和最小值．_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題23(人教A版必修第一冊P230習(xí)題5.5T18)觀察以下各等式：sin230°＋cos260°＋sin30°cos60°＝eq\f(3,4)，sin220°＋cos250°＋sin20°cos50°＝eq\f(3,4)，sin215°＋cos245°＋sin15°cos45°＝eq\f(3,4).分析上述各式的共同特點(diǎn)，寫出能反映一般規(guī)律的等式，并對等式的正確性作出證明．猜想：sin2α＋cos2(α＋30°)＋sinαcos(α＋30°)＝eq\f(3,4).__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題24(人教A版必修第一冊P238例2)摩天輪是一種大型轉(zhuǎn)輪狀的機(jī)械建筑設(shè)施，游客坐在摩天輪的座艙里慢慢地往上轉(zhuǎn)，可以從高處俯瞰四周景色．如圖，某摩天輪最高點(diǎn)距離地面高度為120m，轉(zhuǎn)盤直徑為110m，設(shè)置有48個(gè)座艙，開啟后按逆時(shí)針方向勻速旋轉(zhuǎn)，游客在座艙轉(zhuǎn)到距離地面最近的位置進(jìn)艙，轉(zhuǎn)一周大約需要30min.(1)游客甲坐上摩天輪的座艙，開始轉(zhuǎn)動(dòng)tmin后距離地面的高度為Hm，求在轉(zhuǎn)動(dòng)一周的過程中，H關(guān)于t的函數(shù)解析式；(2)求游客甲在開始轉(zhuǎn)動(dòng)5min后距離地面的高度；(3)若甲、乙兩人分別坐在兩個(gè)相鄰的座艙里，在運(yùn)行一周的過程中，求兩人距離地面的高度差h(單位：m)關(guān)于t的函數(shù)解析式，并求高度差的最大值(精確到0.1)．_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題25(人教A版必修第一冊P239練習(xí)T3)函數(shù)y＝eq\f(2,3)sineq\b\lc\(\rc\)(\a\vs4\al\co1(\f(1,2)x－\f(π,4)))的圖象與正弦曲線有什么關(guān)系？__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2021·全國乙卷)把函數(shù)y＝f(x)圖象上所有點(diǎn)的橫坐標(biāo)縮短到原來的eq\f(1,2)倍，縱坐標(biāo)不變，再把所得曲線向右平移eq\f(π,3)個(gè)單位長度，得到函數(shù)y＝sineq\b\lc\(\rc\)(\a\vs4\al\co1(x－\f(π,4)))的圖象，則f(x)＝()A．sineq\b\lc\(\rc\)(\a\vs4\al\co1(\f(x,2)－\f(7π,12))) B．sineq\b\lc\(\rc\)(\a\vs4\al\co1(\f(x,2)＋\f(π,12)))C．sineq\b\lc\(\rc\)(\a\vs4\al\co1(2x－\f(7π,12))) D．sineq\b\lc\(\rc\)(\a\vs4\al\co1(2x＋\f(π,12)))點(diǎn)評本題和教材習(xí)題考查角度相同，都屬于三角函數(shù)圖象的變換，解決此類問題的關(guān)鍵是熟練掌握其變換規(guī)則．典題26(人教A版必修第一冊P241習(xí)題5.6T4)函數(shù)y＝Asin(ωx＋φ)(A＞0，0＜φ＜π)在一個(gè)周期內(nèi)的圖象如圖所示，此函數(shù)的解析式為________．____________________________________________________________________________________________________________________________________________________________________________________________________________(2023·新高考Ⅱ卷)已知函數(shù)f(x)＝sin(ωx＋φ)，如圖，A，B是直線y＝eq\f(1,2)與曲線y＝f(x)的兩個(gè)交點(diǎn)，若|AB|＝eq\f(π,6)，則f(π)＝________．點(diǎn)評本題和教材習(xí)題高度相似，已知f(x)＝Asin(ωx＋φ)(A＞0，ω＞0)的部分圖象求其解析式時(shí)，A比較容易看圖得出，困難的是求待定系數(shù)ω和φ，常用如下兩種方法：(1)由ω＝eq\f(2π,T)即可求出ω；確定φ時(shí)，若能求出離原點(diǎn)最近的右側(cè)圖象上升(或下降)的“零點(diǎn)”橫坐標(biāo)x0，則令ωx0＋φ＝0(或ωx0＋φ＝π)，即可求出φ.(2)代入點(diǎn)的坐標(biāo)，利用一些已知點(diǎn)(最高點(diǎn)、最低點(diǎn)或“零點(diǎn)”)坐標(biāo)代入解析式，再結(jié)合圖形解出ω和φ，若對A，ω的符號或?qū)Ζ盏姆秶幸?，則可用誘導(dǎo)公式變換使其符合要求．典題27(人教A版必修第一冊P245例1)如圖，某地一天從6～14時(shí)的溫度變化曲線近似滿足函數(shù)y＝Asin(ωx＋φ)＋b.(1)求這一天6～14時(shí)的最大溫差；(2)寫出這段曲線的函數(shù)解析式．_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題28(人教A版必修第二冊P47例8)在△ABC中，已知B＝30°，b＝eq\r(2)，c＝2，解這個(gè)三角形．___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題29(人教A版必修第二冊P50例10)如圖，AB是底部B不可到達(dá)的一座建筑物，A為建筑物的最高點(diǎn)．設(shè)計(jì)一種測量建筑物高度AB的方法，并求出建筑物的高度．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________