PAGE課時(shí)素養(yǎng)檢測(cè)七誘導(dǎo)公式(二)(30分鐘60分)一、選擇題(每小題4分,共24分,多選題全部選對(duì)得4分,選對(duì)但不全對(duì)的得2分,有選錯(cuò)的得0分)1.已知cosQUOTE=-QUOTE,且α是其次象限角,則sinQUOTE的結(jié)果是 ()A.QUOTE B.-QUOTE C.±QUOTE D.QUOTE【解析】選B.因?yàn)閏osQUOTE=-QUOTE,所以-sinα=-QUOTE,所以sinα=QUOTE,又α是其次象限角,所以cosα=-QUOTE,所以sinQUOTE=cosα=-QUOTE.【補(bǔ)償訓(xùn)練】已知cosQUOTE=-QUOTE,且α是第四象限角,則cos(-3π+α)等于 ()A.QUOTEB.-QUOTEC.±QUOTED.QUOTE【解析】選B.因?yàn)閏osQUOTE=-QUOTE,所以sinα=-QUOTE,又α是第四象限角,所以cos(-3π+α)=-cosα=-QUOTE=-QUOTE.2.若α∈QUOTE,則QUOTE= ()A.sinα B.-sinαC.cosα D.-cosα【解析】選B.因?yàn)棣痢蔘UOTE,所以sinα<0,所以QUOTE=QUOTE=-sinα.【補(bǔ)償訓(xùn)練】已知sinQUOTE=QUOTE,α∈QUOTE,則sinα等于 ()A.-QUOTEB.QUOTEC.-QUOTED.QUOTE【解析】選C.sinQUOTE=sinQUOTE=cosα=QUOTE,又α∈QUOTE,所以sinα=-QUOTE=-QUOTE.3.若sin(3π+α)=-QUOTE,則cosQUOTE等于 ()A.-QUOTE B.QUOTE C.QUOTE D.-QUOTE【解析】選A.由已知得sinα=QUOTE,則cosQUOTE=-sinα=-QUOTE.【補(bǔ)償訓(xùn)練】已知sinα=QUOTE,則cosQUOTE等于 ()A.QUOTEB.QUOTEC.-QUOTED.-QUOTE【解析】選A.cosQUOTE=sinα=QUOTE.4.若sin(180°+α)+cos(90°+α)=-a,則cos(270°-α)+2sin(360°-α)的值是 ()A.-QUOTE B.-QUOTE C.QUOTE D.QUOTE【解析】選B.由sin(180°+α)+cos(90°+α)=-a,得:-sinα-sinα=-a,即sinα=QUOTE,cos(270°-α)+2sin(360°-α)=-sinα-2sinα=-3sinα=-QUOTEa.5.若sinQUOTE=QUOTE,則cosQUOTE的值為 ()A.QUOTE B.-QUOTEC.QUOTE D.-QUOTE【解析】選B.cosQUOTE=cosQUOTE=-sinQUOTE=-QUOTE.【補(bǔ)償訓(xùn)練】已知sinQUOTE=QUOTE,那么cosQUOTE的值是.

【解析】因?yàn)镼UOTE-QUOTE=QUOTE,所以α+QUOTE=QUOTE+QUOTE,所以cosQUOTE=cosQUOTE=-sinQUOTE=-QUOTE.答案:-QUOTE6.(多選題)下列與sinQUOTE的值相等的是 ()A.sinQUOTE B.cos(-θ)C.sinQUOTE D.sinQUOTE【解析】選AC.因?yàn)閟inQUOTE=-sinQUOTE=-cosθ,sinQUOTE=-cosθ,cos(-θ)=cosθ,sinQUOTE=-cosθ,sinQUOTE=cosθ.二、填空題(每小題4分,共8分)7.已知角α的終邊過點(diǎn)P(1,-2),則tanα=,QUOTE=.

【解析】因?yàn)榻铅恋慕K邊過點(diǎn)P(1,-2),所以tanα=QUOTE=-2.QUOTE=QUOTE=QUOTE=QUOTE=QUOTE.答案:-2QUOTE8.已知sinα是方程10x2+x-3=0的根,α是第三象限角,則sinα=,

QUOTE=.

【解析】由已知得sinα=-QUOTE或sinα=QUOTE,因?yàn)棣潦堑谌笙藿?所以sinα=-QUOTE,故cosα=-QUOTE=-QUOTE.所以原式=QUOTE=QUOTE=QUOTE.答案:-QUOTEQUOTE三、解答題(每小題14分,共28分)9.化簡:QUOTE.【解析】原式=QUOTE=QUOTE=QUOTE=tanα.10.化簡:QUOTE.【解析】原式==QUOTE=QUOTE=QUOTE.(35分鐘70分)一、選擇題(每小題4分,共16分,多選題全部選對(duì)得4分,選對(duì)但不全對(duì)的得2分,有選錯(cuò)的得0分)1.(2024·銀川高一檢測(cè))若cosQUOTE=QUOTE,則sin(π+α)= ()A.-QUOTE B.-QUOTE C.QUOTE D.QUOTE【解析】選B.cosQUOTE=sinα=QUOTE,sin(π+α)=-sinα=-QUOTE.2.已知sinQUOTE=QUOTE,那么cosQUOTE= ()A.QUOTE B.-QUOTE C.QUOTE D.-QUOTE【解析】選D.cosQUOTE=cosQUOTE=-sinQUOTE=-QUOTE.【補(bǔ)償訓(xùn)練】已知sinQUOTE=QUOTE,則cosQUOTE= ()A.-QUOTEB.QUOTEC.-QUOTED.QUOTE【解析】選A.因?yàn)閟inQUOTE=QUOTE,則cosQUOTE=cosQUOTE=-sinQUOTE=-QUOTE.3.若f(sinx)=3-cos2x,則f(cosx)等于 ()A.3-cos2x B.3-sin2xC.3+cos2x D.3+sin2x【解析】選C.f(cosx)=fQUOTE=3-cos2QUOTE=3-cos(π-2x)=3+cos2x.【補(bǔ)償訓(xùn)練】已知f(sinx)=cos3x,則f(cos10°)的值為 ()A.-QUOTEB.QUOTEC.-QUOTED.QUOTE【解析】選A.f(cos10°)=f(sin80°)=cos240°=cos(180°+60°)=-cos60°=-QUOTE.4.(多選題)若角A,B,C是△ABC的三個(gè)內(nèi)角,則下列等式中不成立的是 ()A.cos(A+B)=cosC B.sin(A+B)=-sinCC.cosQUOTE=sinB D.sinQUOTE=cosQUOTE【解析】選ABC.因?yàn)锳+B+C=π,所以A+B=π-C,所以cos(A+B)=-cosC,sin(A+B)=sinC.所以A,B都不正確;同理,A+C=π-B,B+C=π-A,所以cosQUOTE=cosQUOTE=sinQUOTE,sinQUOTE=sinQUOTE=cosQUOTE,因此C不正確,D是正確的.二、填空題(每小題4分,共16分)5.已知sinQUOTE=QUOTE,則sinQUOTE=.

【解析】因?yàn)閟inQUOTE=cosα=QUOTE,所以sinQUOTE=cosα=QUOTE.答案:QUOTE6.化簡QUOTE=.

【解析】原式==QUOTE=QUOTE=-1.答案:-1【補(bǔ)償訓(xùn)練】化簡:cos2QUOTE+sin2QUOTE=.

【解析】原式=sin2θ+cos2θ=1.答案:17.(2024·黑山高一檢測(cè))若cosQUOTE=QUOTE,則sinQUOTE=.

【解析】因?yàn)镼UOTE-θ+QUOTE+θ=QUOTE,由誘導(dǎo)公式得:sinQUOTE=sinQUOTE=cosQUOTE=QUOTE.答案:QUOTE8.已知sin(π-α)=-2sinQUOTE,則tanα=.

【解析】因?yàn)閟in(π-α)=sinα,sinQUOTE=cosα,所以sinα=-2cosα,即tanα=-2.答案:-2三、解答題(共38分)9.(12分)已知sin(π+α)=-QUOTE.計(jì)算:(1)cosQUOTE.(2)sinQUOTE.【解析】因?yàn)閟in(π+α)=-sinα=-QUOTE,所以sinα=QUOTE.(1)cosQUOTE=cosQUOTE=-sinα=-QUOTE.(2)sinQUOTE=cosα,cos2α=1-sin2α=1-QUOTE=QUOTE.因?yàn)閟inα=QUOTE,所以α為第一或其次象限角.①當(dāng)α為第一象限角時(shí),sinQUOTE=cosα=QUOTE.②當(dāng)α為其次象限角時(shí),sinQUOTE=cosα=-QUOTE.【補(bǔ)償訓(xùn)練】已知sinQUOTE=a,0<α<QUOTE,求sinQUOTE.【解析】因?yàn)?<α<QUOTE,所以-QUOTE<QUOTE-α<QUOTE,所以cosQUOTE>0,所以cosQUOTE=QUOTE=QUOTE,sinQUOTE=sinQUOTE=-sinQUOTE=-cosQUOTE=-cosQUOTE=-QUOTE.10.(12分)求證:QUOTE=-1.【證明】因?yàn)镼UOTE=QUOTE=QUOTE=QUOTE=-1=右邊.所以等式成立.11.(14分)已知tanα+QUOTE=QUOTE,求2sin2QUOTE-3cosQUOTEsinQUOTE+2的值.【解析】tanα+QUOTE=QUOTE