第02講5.2.1三角函數(shù)的概念課程標(biāo)準(zhǔn)學(xué)習(xí)目標(biāo)①理解結(jié)合單位圓定義三角函數(shù)的意義。②結(jié)合任意角終邊與單位圓的交點(diǎn)會求任意角的正弦、余弦、正切值。③根據(jù)任意角終邊所在象限的位置，會判斷任意角三角函數(shù)值的符號。1.掌握三角函數(shù)的定義；2會求任意角的三個三角函數(shù)值；3.能準(zhǔn)確判斷任意角的三角函數(shù)值的符號；知識點(diǎn)01：任意角的三角函數(shù)定義1、單位圓定義法：如圖,設(shè)SKIPIF1<0是一個任意角,SKIPIF1<0,它的終邊SKIPIF1<0與單位圓相交于點(diǎn)SKIPIF1<0①正弦函數(shù)：把點(diǎn)SKIPIF1<0的縱坐標(biāo)SKIPIF1<0叫做SKIPIF1<0的正弦函數(shù),記作SKIPIF1<0,即SKIPIF1<0②余弦函數(shù)：把點(diǎn)SKIPIF1<0的橫坐標(biāo)SKIPIF1<0叫做SKIPIF1<0的余弦函數(shù),記作SKIPIF1<0,即SKIPIF1<0

③正切函數(shù)：把點(diǎn)SKIPIF1<0的縱坐標(biāo)與橫坐標(biāo)的比值SKIPIF1<0叫做SKIPIF1<0的正切,記作SKIPIF1<0,即SKIPIF1<0(SKIPIF1<0)

我們將正弦函數(shù)、余弦函數(shù)和正切函數(shù)統(tǒng)稱為三角函數(shù)【即學(xué)即練1】（2023春·北京·高一北京四中?？计谥校┮阎铅鹊慕K邊經(jīng)過點(diǎn)SKIPIF1<0，則SKIPIF1<0等于（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【答案】B【詳解】SKIPIF1<0，故SKIPIF1<0在單位圓上，根據(jù)三角函數(shù)值的定義，SKIPIF1<0的橫坐標(biāo)的值即為SKIPIF1<0，故SKIPIF1<0.故選：B2、終邊上任意一點(diǎn)定義法：在角SKIPIF1<0終邊上任取一點(diǎn)SKIPIF1<0，設(shè)原點(diǎn)到SKIPIF1<0點(diǎn)的距離為SKIPIF1<0①正弦函數(shù)：SKIPIF1<0②余弦函數(shù)：SKIPIF1<0

③正切函數(shù)：SKIPIF1<0(SKIPIF1<0)

【即學(xué)即練2】（2023春·廣西欽州·高一校考期中）若點(diǎn)SKIPIF1<0在角SKIPIF1<0的終邊上，則SKIPIF1<0.【答案】SKIPIF1<0/0.8【詳解】點(diǎn)SKIPIF1<0在角SKIPIF1<0的終邊上，所以SKIPIF1<0.故答案為：SKIPIF1<0.知識點(diǎn)02：三角函數(shù)值在各象限的符號SKIPIF1<0,SKIPIF1<0,SKIPIF1<0在各象限的符號如下:（口訣“一全正,二正弦,三正切,四余弦”）知識點(diǎn)03：特殊的三角函數(shù)值角度SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0弧度SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0正弦值SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0余弦值SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0正切值SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0知識點(diǎn)04：誘導(dǎo)公式一(1)語言表示:終邊相同的角的同一三角函數(shù)的值相等.

(2)式子表示:①SKIPIF1<0②SKIPIF1<0③SKIPIF1<0其中SKIPIF1<0.

知識點(diǎn)05：三角函數(shù)線設(shè)SKIPIF1<0角的終邊與單位圓相交點(diǎn)SKIPIF1<0；④由點(diǎn)SKIPIF1<0向SKIPIF1<0軸做垂線，垂足為點(diǎn)SKIPIF1<0；⑤由點(diǎn)SKIPIF1<0作單位圓的切線與終邊相交于點(diǎn)SKIPIF1<0。如下圖所示：在SKIPIF1<0中：SKIPIF1<0SKIPIF1<0為正弦線，長度為正弦值。SKIPIF1<0SKIPIF1<0為余弦線，長度為余弦值。在SKIPIF1<0中：SKIPIF1<0。SKIPIF1<0為正切線，長度為正切值。題型01利用三角函數(shù)的定義求三角函數(shù)值【典例1】（2023春·陜西西安·高一校考階段練習(xí)）已知角SKIPIF1<0的頂點(diǎn)在原點(diǎn)，始邊與SKIPIF1<0軸的非負(fù)半軸重合，終邊在第三象限且與單位圓交于點(diǎn)SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023·全國·高一專題練習(xí)）已知角SKIPIF1<0的終邊與單位圓交于點(diǎn)SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例3】（2023·浙江嘉興·高一統(tǒng)考期末）在平面直角坐標(biāo)系SKIPIF1<0中，角SKIPIF1<0的頂點(diǎn)與原點(diǎn)SKIPIF1<0重合，它的始邊與SKIPIF1<0軸的非負(fù)半軸重合，終邊SKIPIF1<0交單位圓SKIPIF1<0于點(diǎn)SKIPIF1<0，則SKIPIF1<0的值為A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式1】（2023春·湖南·高二統(tǒng)考學(xué)業(yè)考試）設(shè)角SKIPIF1<0的終邊與單位圓的交點(diǎn)坐標(biāo)為SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．1【變式2】（2023·福建泉州·高一統(tǒng)考期末）已知角SKIPIF1<0的頂點(diǎn)與原點(diǎn)重合，始邊與x軸的非負(fù)半軸重合，若SKIPIF1<0的終邊與圓心在原點(diǎn)的單位圓交于SKIPIF1<0，且SKIPIF1<0為第四象限角，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型02由終邊或終邊上點(diǎn)求三角函數(shù)值【典例1】（2023秋·云南大理·高二大理白族自治州民族中學(xué)?？奸_學(xué)考試）已知角SKIPIF1<0的終邊落在直線SKIPIF1<0上，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例2】（多選）（2023秋·江西贛州·高二江西省全南中學(xué)?？奸_學(xué)考試）已知角SKIPIF1<0的終邊經(jīng)過點(diǎn)SKIPIF1<0，則SKIPIF1<0的值可能為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【典例3】（2023春·新疆塔城·高一塔城地區(qū)第一高級中學(xué)?？茧A段練習(xí)）已知角SKIPIF1<0的終邊過點(diǎn)SKIPIF1<0，且SKIPIF1<0，求SKIPIF1<0及SKIPIF1<0的值．【變式1】（2023春·四川達(dá)州·高一四川省萬源中學(xué)?？茧A段練習(xí)）若角SKIPIF1<0的終邊經(jīng)過點(diǎn)SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【變式2】（多選）（2023春·江西萍鄉(xiāng)·高一統(tǒng)考期中）已知角SKIPIF1<0的終邊上有一點(diǎn)SKIPIF1<0，若SKIPIF1<0，則（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【變式3】（2023秋·北京·高三北京市第六十六中學(xué)校考開學(xué)考試）若SKIPIF1<0的終邊所在射線經(jīng)過點(diǎn)SKIPIF1<0，則SKIPIF1<0，SKIPIF1<0.題型03由三角函數(shù)值求終邊上的點(diǎn)或參數(shù)【典例1】（2023·全國·高一假期作業(yè)）角SKIPIF1<0的終邊經(jīng)過點(diǎn)SKIPIF1<0且SKIPIF1<0，則b的值為（

）A．3 B．SKIPIF1<0 C．SKIPIF1<0 D．5【典例2】（2023·全國·高一專題練習(xí)）已知角SKIPIF1<0的終邊上有一點(diǎn)SKIPIF1<0，且SKIPIF1<0，則實數(shù)m取值為．【典例3】（2023秋·高一課時練習(xí)）已知角SKIPIF1<0的終邊上一點(diǎn)SKIPIF1<0，且SKIPIF1<0，求SKIPIF1<0值.【變式1】（2023·上海·高一專題練習(xí)）已知角SKIPIF1<0的終邊經(jīng)過點(diǎn)SKIPIF1<0，且SKIPIF1<0，則實數(shù)m的值是（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<0【變式2】（2023春·黑龍江大慶·高一大慶中學(xué)?？茧A段練習(xí)）已知角SKIPIF1<0的終邊經(jīng)過點(diǎn)SKIPIF1<0，且SKIPIF1<0，則實數(shù)SKIPIF1<0.【變式3】（2023春·廣西欽州·高一?？计谥校┮阎c(diǎn)SKIPIF1<0角SKIPIF1<0的終邊上，且SKIPIF1<0，求m，SKIPIF1<0，SKIPIF1<0．題型04三角函數(shù)值符號的運(yùn)用【典例1】（2023·全國·高一假期作業(yè)）求SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【典例2】（2023春·貴州畢節(jié)·高一校考期中）若SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0是（

）A．第一象限角 B．第二象限角 C．第三象限角 D．第四象限角【典例3】（2023·全國·高一專題練習(xí)）SKIPIF1<0所有可能取值的集合為．【變式1】（2023春·遼寧沈陽·高一沈陽市第十一中學(xué)?？茧A段練習(xí)）已知SKIPIF1<0，則點(diǎn)P所在象限為（

）A．第一象限 B．第二象限 C．第三象限 D．第四象限【變式2】（2023春·廣東湛江·高一雷州市第一中學(xué)?？茧A段練習(xí)）已知點(diǎn)SKIPIF1<0是第三象限的點(diǎn)，則SKIPIF1<0的終邊位于（

）A．第一象限 B．第二象限 C．第三象限 D．第四象限【變式3】（2023秋·高一課時練習(xí)）點(diǎn)SKIPIF1<0位于第象限．題型05畫三角函數(shù)線【典例1】（2023春·山東威?！じ咭恍？茧A段練習(xí)）不等式SKIPIF1<0在區(qū)間SKIPIF1<0上的解集為．【典例2】（2023春·高一課時練習(xí)）利用三角函數(shù)線，寫出滿足下列條件的角x的集合：(1)SKIPIF1<0且SKIPIF1<0；(2)SKIPIF1<0．【變式1】（2023·全國·高一專題練習(xí)）使SKIPIF1<0成立的x的一個變化區(qū)間是（）A．[SKIPIF1<0，SKIPIF1<0] B．[SKIPIF1<0，SKIPIF1<0]C．[SKIPIF1<0，SKIPIF1<0] D．[0，SKIPIF1<0]【變式2】（2023·高一課時練習(xí)）已知SKIPIF1<0，則SKIPIF1<0的取值范圍是．【變式3】（2023·高一課時練習(xí)）利用單位圓分別寫出符合下列條件的角α的集合：（1）SKIPIF1<0；（2）SKIPIF1<0；（3）SKIPIF1<0.題型06三角函數(shù)線的應(yīng)用【典例1】（2023秋·遼寧撫順·高一撫順一中?？计谀┮阎猄KIPIF1<0是SKIPIF1<0的一個內(nèi)角，且SKIPIF1<0，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·全國·高三專題練習(xí)）（1）設(shè)SKIPIF1<0，試證明：SKIPIF1<0；（2）若SKIPIF1<0，試比較SKIPIF1<0與SKIPIF1<0的大小.

【變式1】（2023·高一課時練習(xí)）如圖，已知點(diǎn)SKIPIF1<0是單位圓與SKIPIF1<0軸的交點(diǎn)，角SKIPIF1<0的終邊與單位圓的交點(diǎn)為SKIPIF1<0，SKIPIF1<0軸于SKIPIF1<0，過點(diǎn)SKIPIF1<0作單位圓的切線交角SKIPIF1<0的終邊于SKIPIF1<0，則角SKIPIF1<0的正弦線、余弦線、正切線分別是（

）A．SKIPIF1<0，SKIPIF1<0，SKIPIF1<0B．SKIPIF1<0，SKIPIF1<0，SKIPIF1<0C．SKIPIF1<0，SKIPIF1<0，SKIPIF1<0D．SKIPIF1<0，SKIPIF1<0，SKIPIF1<0A夯實基礎(chǔ)B能力提升A夯實基礎(chǔ)一、單選題1．（2023春·廣東佛山·高一?？计谥校┤艚荢KIPIF1<0的終邊經(jīng)過點(diǎn)SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2023秋·重慶渝中·高三重慶巴蜀中學(xué)?？茧A段練習(xí)）角SKIPIF1<0的終邊上一點(diǎn)SKIPIF1<0的坐標(biāo)為SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2023秋·天津武清·高三?？茧A段練習(xí)）已知角SKIPIF1<0的頂點(diǎn)與原點(diǎn)SKIPIF1<0重合，始邊與SKIPIF1<0軸的非負(fù)半軸重合，它的終邊過點(diǎn)SKIPIF1<0，則SKIPIF1<0的值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（2023春·新疆·高一八一中學(xué)?？计谥校┤鬝KIPIF1<0，SKIPIF1<0，則SKIPIF1<0的終邊在（

）A．第一、三象限B．第二、四象限C．第一、三象限或在x軸的非負(fù)半軸上D．第二、四象限或在x軸上5．（2023春·河南南陽·高一南陽中學(xué)?？茧A段練習(xí)）SKIPIF1<0的符號為（

）A．正 B．0 C．負(fù) D．無法確定6．（2023·北京·高三專題練習(xí)）在平面直角坐標(biāo)系中，SKIPIF1<0是圓SKIPIF1<0上的四段弧（如圖），點(diǎn)P在其中一段上，角SKIPIF1<0以SKIPIF1<0為始邊，OP為終邊，若SKIPIF1<0，則P所在的圓弧是A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<07．（2023·全國·高一假期作業(yè)）質(zhì)點(diǎn)P和Q在以坐標(biāo)原點(diǎn)O為圓心，半徑為1的圓周上順時針作勻速圓周運(yùn)動，同時出發(fā).P的角速度為3rad/s，起點(diǎn)為射線SKIPIF1<0與圓的交點(diǎn)；Q的角速度為5rad/s，起點(diǎn)為圓與x軸正半軸交點(diǎn)，則當(dāng)質(zhì)點(diǎn)Q與P第二次相遇時，Q的坐標(biāo)為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．（2023秋·內(nèi)蒙古巴彥淖爾·高一巴彥淖爾中學(xué)?？计谀┮阎瘮?shù)SKIPIF1<0（SKIPIF1<0，且SKIPIF1<0）的圖像恒過點(diǎn)P，若點(diǎn)SKIPIF1<0是角SKIPIF1<0終邊上的一點(diǎn)，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題9．（2023春·四川眉山·高一校考期中）已知SKIPIF1<0，則SKIPIF1<0可能是（

）A．第一象限角 B．第二象限角 C．第三象限角 D．第四象限角10．（2023·全國·高二專題練習(xí)）若SKIPIF1<0是第四象限角，則點(diǎn)SKIPIF1<0在第（

）象限.A．第一象限 B．第二象限 C．第三象限 D．第四象限三、填空題11．（2023秋·福建漳州·高三校考階段練習(xí)）已知角SKIPIF1<0的終邊上一點(diǎn)SKIPIF1<0的坐標(biāo)為SKIPIF1<0，則角SKIPIF1<0的最小正值為12．（2023·高一課時練習(xí)）如果SKIPIF1<0是第二象限角，則SKIPIF1<0的符號為（選填“正”或“負(fù)”）．四、解答題13．（2023·高一課時練習(xí)）已知SKIPIF1<0，試判斷SKIPIF1<