#VLJUBCOIFSIBLLTBLMESWF\":%*/:\":*/-\"3*OBBJUUJSTBZMZBTBOOIÐLÐNMFSJOF HËSFLJUBCOEÐ[FOJ NFUOJ TPSVWFõFLJMMFSJLTNFOEFPMTBIJ¿CJSõFLJMEFBMOQZBZNMBOB- NB[ GPUPLPQJZBEBCBõLBCJSUFLOJLMF¿PóBMUMBNB[ :BZO4PSVNMVTV $BO5&,÷/&- :BZO&EJUÌSÑ &TSB:·,4&-)BLBO\"ó$\" %J[HJ–(SBGJL5BTBSN \"ZEO:BZOMBS%J[HJ#JSJNJ *4#//P :BZOD4FSUJGJLB/P #BTN:FSJ &SUFN#BTN:BZO-UEõUJr ÷MFUJöJN \":%*/:\":*/-\"3* JOGP!BZEJOZBZJOMBSJDPNUS 5FMr 'BLT 0533 051 86 17 aydinyayinlari aydinyayinlari * www.aydinyayinlari.com.tr %¸O¾P.DSDáñ11.SINIF KARMA TEST - 1 Trigonometri Karma Testler 11. SINIF 1. MODÜL 1. x + N-O Z = n - 4. ²MÀÑTÑ - 59π SBEZBOPMBOBÀOOFTBTÌMÀÑTÑ Modülün sonunda 5 Alt bölümlerin EFOLMFNJCJSJNÀFNCFSCFMJSUUJôJOFHÌSF NO LBÀUS LBÀEFSFDFEJS \" # $ % & \" # $ % & TRİGONOMETRİ EDĜOñNODUñQñL©HULU 2. ôFLJMEFLJ\"OPLUBTCJSJN¿FNCFSÐ[FSJOEFEJS y tüm alt bölümleri A L©HUHQNDUPDWHVWOHU ³ Yönlü Açılar t 2 y \\HUDOñU 1 a ³ Esas Ölçü Birim Çember t 9 1 x D CO x ³ Trigonometrik Fonksiyonlar–I t 13 A B –1 O –1 ³ Trigonometrik Fonksiyonlar–II t 20 \"OPLUBTOOZFLTFOJOFV[BLMô 1 CJSJNPM- ôFLJMEFLJCJSJN¿FNCFSEF[AC] m [#%], ³ Trigonometrik Fonksiyonlar–III t 27 2 | | | |% ³ Kosinüs Teoremi t 47 ³ Sinüs Teo1r1e. SmINIiF t 501. MODÜL 53÷(0/0.&53÷ EVôVOB HÌSF Y FLTFOJOF PMBO V[BLMô LBÀ CJ- m ( BAC ) = aWF OC = $% EJS SJNEJS :VLBSEBLJWFSJMFSFHÌSF UBOaEFôFSJLBÀUS <D]ñOñ6RUXODUñ 3 B) 1 C) % & YAZILI SORULARIA) 1 A) 2 Trigonometri 7 6ñQñIð©LðĜOH\\LĜ B) 1 $ % 5 E) 7 2 5 www.aydinyayinlari.com.tr 1. 5BONMPMEVôVBSBMLUB 4. 1 + 1 = 8 sin3x - cos3x 1 - cos x 1 + cos x ³ Trigonometrik Fonksiyonların Grafikleri t:²5/8-·\"¦*-\"3 tan x. cot x + sin x. cos x PMEVôVOBHÌSF UBOYJOQP[JUJGEFôFSJLBÀUS ÷MJöLJMJ,B[BONMBS JGBEFTJOJOFOTBEFI»MJOJCVMVOV[ ³ Ters Trigonometrik Fonksiyonlar t 6511.1.1.1 : :ËOMÐB¿ZB¿LMBS 11.1.1.2 : \"¿ËM¿ÐCJSJNMFSJOJB¿LMBZBSBLCJSCJSJJMFJMJõLJMFOEJSJS 1 1 1 + cos x + 1 - cos x 2NXO\\D]ñOñVñQDYODUñQGD += =8 ©ñNDELOHFHNVRUXODUñL©HULU 1 - cos x 1 + cos x ^ 1 + cos x h^ 1 - cos x h %XE¸O¾PGHNL¸UQHN VRUXODUñQ©¸]¾POHULQH 3. a =hhh ^ sin x - cos x h.^ sin 2 + sin x. cos x + 2 x h ^ 1 + cos x h ^ 1 - cos x h DNñOOñWDKWDX\\JXODPDVñQGDQ x cos XODĜDELOLUVLQL] ³ Karma TTeAsNItMler t 72 ÖRNEK 2 b =hhh 1ta4n4x2. c4o4t x3 + sin x. cos x sin x 2 PMEVôVOBHÌSF a 1 - cos a =hhhPMN=BL1Ð-[cFoSsF x = 8 + x4a EFôFSJBöBôEBLJMFSEFOIBOH1JTJEJS #BõMBOH¿OPLUBMBSBZOPMBOJLJõOOCJSMFõJ- ^ sin x - cos x h.a 1 + sin x . cos x k bUPQMBNBöBôEBLJMFSEFO= a 1 + sin x . cos x k 3 \"öBôEBWFSJMFOBÀMBSEBOIBOHJMFSJQP[JUJGZÌOMÑBÀ- IBOHJTJEJS 2 2 2 1 sin x ³ Yazılı SorulNaJOrFı tBÀ 77EFOJS #V õOMBSB BÀOO LFOBSMBS, MBSES =8& = & sin x = ± 2 CBõMBOH¿OPLUBTOBBÀOOLÌöFTJEFOJS \" hhh # hhh A) 7hhh 114-4c2#os4 42x3h8hh ³ Yeni Nesil Sorular t 79 #JSB¿OOLFOBSMBSOEBOCJSJOJCBõMBOH¿LFOB- AA $ hhh 2. AöB%ô EBWFSJMFhOBhhÀMBSO FTBT ÌMÀÑMFS$JO JCVMV-hhh sin%2x 4hhh S EJóFSJOJCJUJNLFOBSPMBSBLBMEóN[EBFMEF O FEJMFOB¿ZBZÌOMÑBÀEFOJS & OV[hhh C 163π & hhh A 13 O a) - 7 tan x = = D - 72π 2k 33 ÖRNEK 1 B B E I. II. 1. D 2. \" 53. $ 72 4. D &x B k \"õBóEBLJõFLJMEF[0\"JMF[01#õOMBS B¿OOLFOBSMBS- B C 3k ES0OPLUBTB¿OOLËõFTJEJS A B B a) –3060° 360 b) 163 14 Ö –3240 –9 14 11 7 O 180° 23 III. 14 :BMO[** c) –72 10 9 360 \"õBóEB0WF0NFSLF[MJEõUBOUFóFU¿FNCFSMF- <HQL1HVLO6RUXODU –80 –8 d) 2870 SJOPSUBLUFóFUJWFSJMNJõUJS 0RG¾O¾QJHQHOLQGH\\RUXP OA 8 2520 7 A \\DSPDDQDOL]HWPHYE #VOBHÌSF öFLJMEFLJBÀZTFNCPMMFHÌTUFSFMJN Ö EHFHULOHUL¸O©HQNXUJXOX $OW%¸O¾P7HVWOHUL <(3510° m1(6m/6258/$53 4 3 B Trigonometri VRUXODUD\\HUYHULOPLĜWLU 5 $\\UñFDPRG¾OVRQXQGD Her alt bölümün VRQXQGDRE¸O¾POHLOJLOL 33 a 3 WHVWOHU\\HUDOñU : OA , : OB = % TEST - 2 :ÌOMÑ\"ÀMBS 1. \"õBóEBLJõFLJMEFHÐOÐOJMLTBBUMFSJOEFLJBQBSUNB-O1 53.3 \"õBOó2EBCJSPEBOOUBWBOOBõFLJMEFLJHJCJBTMBDBL AOB OOHËMHFTJJMFJMFSMFZFOTBBUMFSEFLJBQBSUNBOOHËM- MBNCBMBSHËTUFSJMNJõUJS 3. a =hhh HFTJHËTUFSJMNJõUJS 1. m (WA) = 24° 55hhhWFm (WB) = 11°hhh 4. 3r + r - 5r AB 7$1,0%m/*m 4 2 12 b =hhh | | | | 53° %37° 53° 37° 53° 37° 7$1,0%m/*m PMEVôVOB HÌSF m^WAh + m^WBh EFôFS#JJBSöBB¿ôOEOBLËJM-¿ÐMNFTJ B¿OUPOQMLBPNMMBOSOBSEBFTSFODEFBLDJJOTJOEFOFöJUJLBÀUS PMNBLÑ[FSF a + b JGBEFTJOJOTPOVDVOVCV- \"01 = 4 BO2 = 4 3 CS WF m ( AO1O2 ) = a 23 PMEVôVOBHÌSF TJOaEFôFSJLBÀUS (+) MFSEFOIBOHJTJOFFöJUUJS B¿LMóOCFMJSMFONFTJJMFZ\"B QMS\" ¿OOËM¿ÐTÐ#- $ Pozitif luOV[ # hOMFhÐhSJJGLBVEMMFBOFUMNSFLJ¿JOEFSFDFWFZBSBE%Z BOCJSJN- Yön \" hhh % hhh & ab 43 4 $ h(–) hh sin a=N = EJS 44° 112' 34' ' Negatif a = & 22° 56' 17' ' 53 5 Yön & hhh #JSUBN¿FNCFSZBZOOFõQBS¿BTOEBOCJ- 22 2 ,BCMPMBSBõBóEBLJHJCJLVMMBOMBDBLUS 18° 45' 174' ' SJOJHËSFONFSLF[B¿TOOËM¿ÐTÐOFEFSFDF b = & 6° 15' 58' ' | |r AB =DNEJS 33 3 EFOJS WF CJ¿JNJOEF HËTUFSJMJS #JS ¿FNCFSJO 2.gölgenin boyu ZBZËM¿ÐTÐPMVS%FSFDFOJO 1 JOFEB- 22° 56' 17'' r ,BCMPMBSJLJ¿FõJUUJS 60 \"¿OOCJSLËõFTJFUSBGOEBCBõMBOH¿LFOBSWF + 6° 15' 58'' 1.gölgenin boyu r :BUBZMBMJLB¿ZBQBOLBCMPMBSLFOEJJ¿JOEF CJUJõ LFOBS JLJ UÐSMÐ TF¿JMFCJMJS 4BBUJO EËONF LJLBEFOJS#VËM¿ÐhCJ¿JNJOEFHËTUFSJMJS%B- 29° 12' 15'' FõJUV[VOMVLUBWFNBWJSFOLUFEJS ZËOÐOÐOUFSTZËOÐOFEËOFOB¿ZBQP[JUJGZËO- (ÐOFõZÐLTFMEJL¿FBQBSUNBOOHËMHFTJOJOCPZVL- MÐ TBBUJOEËONFZËOÐOEFEËOFOB¿ZBOFHB- LJLBOO 1 JOFTBOJZFEFOJS#VËM¿ÐhhCJ¿J- TBMNõUS\"QBSUNBOZBUBZMBMJLB¿PMVõUVSBDBL r :BUBZB MJL B¿ ZBQBO LBCMPMBS LFOEJ J¿JOEF UJGZÌOMÑBÀEFOJS 60 FõJUV[VOMVLUBWFTBSSFOLUFEJS õFLJMEFEÐ[CJS[FNJOFZBQMNõUS NJOEFHËTUFSJMJS0IBMEF =h=hhEJS 4 1. TJOYmDPTY 9π 8π 3 r5 ,BCMPV[VOMVLMBSIFTBQMBOSLFO 2. B C 7HDÌ MHF5OJOE CPZV HÌ7MH7FOJO CPZVOVO ÑÀ LB-4. 3 TJO=DPT= 3. hhh sin a DPT=TJO= EFSFDFMJLBÀOOÌMÀÑTÑLBÀSBEZBOES U BQBSUNBOOCPZVNPMEVôVOBHÌSF BMOBDBLUS sin b % 2 2. :BMO[** 2π B) 4π C) 6π % 7π E) 8π LBÀUS 1. AOB A) 5 5 55 5 2. 13π SBEZBOMLBÀOOÌMÀÑTÑLBÀEFSFDFEJS A) 2 # $ 3 % 5 7 3 3 2 3 E) 4 \" # $ % & #VOBHÌSF LVMMBOMBOTBSSFOLMJLBCMPMBSNBWJ SFOLMJLBCMPMBSEBOUPQMBNEBLBÀDNGB[MBES 2. 4BBUUFLJI[MBSLNWFLNPMBOJLJCJTJLMFUMJBSB- \" # $ % & WDPDPñ\\HQLQHVLOVRUXODUGDQ MBSOEBLJB¿PMBDBLõFLJMEFBZOOPLUBEBOBZO ROXĜDQWHVWOHUEXOXQXU BOEBEPóSVTBMCJSZPMJ[MFZFSFLJMFSMJZPSMBS 4. \"õBóEBLJõFLJMEFHËTUFSJV¿VõVZBQBOEËSUV¿BóO PMVõUVSEVóVпHFOTFMCËMHFMFSHËSÐMNFLUFEJS )HQ/LVHOHULQH<¸QHOLN A 11. SINIF .0%·- 53÷(0/0.&53÷ :BS¿BQ CS PMBO EBJSFTFM CJS QJTUJO \" OPLUBwTwO-w.aydinyayinlari.com.tr )HQ/LVHVLP¾IUHGDWñQGD EBCVMVOBOJLJBSB¿BZOBOEBGBSLMZËOMFSEFIBSFLFU olup Anadolu Lisesi müfre- GDWñQGDROPD\\DQL©HULNYH 60° )(1/m6(/(5m1(<q1(/m. FEJZPSMBS#JSJODJBSÖB¿RNEKrC3SJ1LJODJBSB¿r CSHJU- BC UJLUFOTPOSBEVSVZPS D %m/*m 3. B Bir ABC üçgeninde A ABC üçgen % 60° m ( ABC ) = 68° A % = 52° TBBUTPOSBBSBMBSOEBLJV[BLMLYLNPMVZPS%B- && m ( ACB ) IBTPOSBI[MPMBO I[O 1 ünFEÐõÐSÐZPS EJóF- #VпHFOTFMCËMHFMFSEFOA^ ABC h = 3A^ ADC hEJS OA 4 SJBZOI[EBEFWBNFEFSFLTBBUEBIBJMFSMJZPS- A cC b | |68° 52° BC = 12 br MBSWFBSBMBSOEBLJV[BLMLZLNPMVZPS | | | |AB =LN AC =LN m(B%AD) = 30°, O B 12 C #VOBHÌSF Z-YLBÀLNEJS % CAD R \" # $ % & m ( ) = a PMEVôVOB HÌSF TJOa EFôFSJ LBÀ- ôFLJMEFLJ & EF m (WA) = 2r ,B m (WB) = raUÐS C :VLBSEBLJ WFSJMFSF HÌSF \"#$ ÑÀHFOJOJO ÀFWSFM US ABC #VBSBÀMBSOEVSÀEFVNLMCBFSSOJOPJOLUTBMOBSS#MBWEFô$BOMPBOLULBMBBÀ-ÖCSEJS 34 A) 1 B) 1 C) 1 3 E) 2 2 3 % SJTFm ( % ) LBÀSBEZBOPMBCJMJS 2 :VLBSEBLJWFSJMFSFHÌSF m (XC)LBÀUS BOC a = sπinb(WB)E=) c = 2R π 2π a 3π= 2R 5π sπin (WA) % sπin (XC) A) B) A π π C) sCin) % E) r A) B) 2 3 12 4 6 15 9 10 1. $ 2. $ 80 3. & 4. B 20 12 = 2R j3= 4 3 CS 3¥FWSFM¿FNCFSJOZBS¿BQ 3 1. \" 2. \" 3. B 8 4. $ 2 3 h2 =Ö VRUXODUSHPEH]HPLQOH Ö3&=Ö^4B ÖRNEK 29 | |\"#$ÑÀHFOJOEFm (XB)=WF \"$ =DNPMEV- ôVOBHÌSF \"#$ÑÀHFOJOJOÀFWSFMÀFNCFSJOJOZBS- ÖRNEK 32 A YHULOPLĜWLU ÀBQLBÀDNEJS 12 46 b C a = 2R HB sin (XB) 4 = 2R & R = 4 >sin 30° 1 2 ÖRNEK 30 ôFLJMEF\"#$пHFOJWF¿FWSFM¿FNCFSJWFSJMNJõUJS ,FOBSV[VOMVLMBSCS CSWFCSPMBOÑÀHFOJO | | | | | |[AH] m [BC], AB = 6 br, AH = 4 br ve AC = 12 br ÀFWSFMÀFNCFSJOJOZBSÀBQLBÀCSEJS EJS 4 A 4 = 2R :VLBSEBLJWFSJMFSFHÌSF ÀFNCFSJOZBSÀBQLBÀCS a 4 EJS B1 15 sin a 4 12 D 1C = 2R = 2R 15 4 sin a 12 8 15 R= = 2R 4 15 6 3=CS 8 15 54 4 15
www.aydinyayinlari.com.tr 11. SINIF 11. SINIF 1. MODÜL TRİGONOMETRİ ³ Yönlü Açılar t 2 ³ Esas Ölçü - Birim Çember t 9 ³ Trigonometrik Fonksiyonlar–I t 13 ³ Trigonometrik Fonksiyonlar–II t 20 ³ Trigonometrik Fonksiyonlar–III t 27 ³ Kosinüs Teoremi t 47 ³ Sinüs Teoremi t 50 ³ Trigonometrik Fonksiyonların Grafikleri t 58 ³ Ters Trigonometrik Fonksiyonlar t 65 ³ Karma Testler t 72 ³ Yazılı Soruları t 77 ³ Yeni Nesil Sorular t 79 1
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr :²/-·\"¦*-\"3 ÷MJöLJMJ,B[BONMBS 11.1.1.1 : :ËOMÐB¿ZB¿LMBS 11.1.1.2 : \"¿ËM¿ÐCJSJNMFSJOJB¿LMBZBSBLCJSCJSJJMFJMJõLJMFOEJSJS TANIM ÖRNEK 2 #BõMBOH¿OPLUBMBSBZOPMBOJLJõOOCJSMFõJ- \"öBôEBWFSJMFOBÀMBSEBOIBOHJMFSJQP[JUJGZÌOMÑBÀ- NJOF BÀ EFOJS #V õOMBSB BÀOO LFOBSMBS, MBSES CBõMBOH¿OPLUBTOBBÀOOLÌöFTJEFOJS A A #JSB¿OOLFOBSMBSOEBOCJSJOJCBõMBOH¿LFOB- O S EJóFSJOJCJUJNLFOBSPMBSBLBMEóN[EBFMEF O FEJMFOB¿ZBZÌOMÑBÀEFOJS B B I. ÖRNEK 1 II. \"õBóEBLJõFLJMEF[0\"JMF[0#õOMBS B¿OOLFOBSMBS- AB ES0OPLUBTB¿OOLËõFTJEJS B O III. :BMO[** OA #VOBHÌSF öFLJMEFLJBÀZTFNCPMMFHÌTUFSFMJN : OA , : OB = % 7$1,0%m/*m AOB #JS B¿OO ËM¿ÐMNFTJ B¿OO LPMMBS BSBTOEBLJ 7$1,0%m/*m (–) B¿LMóOCFMJSMFONFTJJMFZBQMS\"¿OOËM¿ÐTÐ- Negatif OÐJGBEFFUNFLJ¿JOEFSFDFWFZBSBEZBOCJSJN- (+) MFSJLVMMBOMS Pozitif Yön #JSUBN¿FNCFSZBZOOFõQBS¿BTOEBOCJ- SJOJHËSFONFSLF[B¿TOOËM¿ÐTÐOFEFSFDF Yön EFOJS WF CJ¿JNJOEF HËTUFSJMJS #JS ¿FNCFSJO \"¿OOCJSLËõFTJFUSBGOEBCBõMBOH¿LFOBSWF ZBZËM¿ÐTÐPMVS CJUJõ LFOBS JLJ UÐSMÐ TF¿JMFCJMJS 4BBUJO EËONF %FSFDFOJO 1 JOFEBLJLBEFOJS#VËM¿Ðh ZËOÐOÐOUFSTZËOÐOFEËOFOB¿ZBQP[JUJGZËO- MÐ TBBUJOEËONFZËOÐOEFEËOFOB¿ZBOFHB- 60 UJGZÌOMÑBÀEFOJS CJ¿JNJOEFHËTUFSJMJS %BLJLBOO 1 JOFTBOJZFEFOJS#VËM¿ÐhhCJ- 60 ¿JNJOEFHËTUFSJMJS =h=hhEJS % 2 2. :BMO[** 1. AOB
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ÖRNEK 3 ÖRNEK 7 m^ WA h =h m^ WB h =h ²MÀÑTÑ h PMBO BÀOO LBÀ EFSFDF WF LBÀ EBLJLB PMEVôVOVCVMVOV[ PMEVôVOBHÌSF m^ WA h - m^ WB hJöMFNJOJOTPOVDVOV 4730 60 78° 50' CVMVOV[ 420 78 41° h 530 h 480 – h 50 8° 33' ÖRNEK 4 ÖRNEK 8 hhh MJLBÀOOLBÀTBOJZFPMEVôVOVCVMVOV[ m^ WA h =hhh m^ WB h =hhh ==hh 22' == 1320'' PMNBLÑ[FSF BöBôEBLJJGBEFMFSJOFöJUJOJCVMVOV[ hh+ 1320'' + 33'' =hh a) m^ WA h + m^ WB h C m^ WA h - m^ WB h ÖRNEK 5 a) h + h TBOJZFMJL BÀOO LBÀ EFSFDF LBÀ EBLJLB WF h LBÀTBOJZFPMEVôVOVCVMVOV[ 70° 04' C 87' h – h h 43000 3600 3400 60 3600 11° 300 56' 7000 400 ÖRNEK 9 3600 360 3400 40\" m^ WA h =hhh m^ WB h =hhh h PMNBLÑ[FSF BöBôEBLJJGBEFMFSJOFöJUJOJCVMVOV[ ÖRNEK 6 m^ WA h m^ WB h a) C m^ WA h =h m^ WB h =h PMEVôVOBHÌSF m^ WA h + m^ WB hJöMFNJOJOTPOVDVOV 2 3 CVMVOV[ a) 19° 22' 30\" 18° 82' 30\" 32° 18' = 9° 41' 15\" + h 222 h C 27° 13' 21\" 27° 12' 81\" = 9° 4' 27\" 333 3. h 4. hh hhh 3 7. 8° 33' 8. B hhhC hhhB hhhC hhh
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr 7$1,0%m/*m ÖRNEK 11 #JS¿FNCFSEFZBS¿BQV[VOMVóVOBFõJUPMBOZB- \"öBôEBLJZBZÌMÀÑMFSJOJEFSFDFDJOTJOEFOZB[O[ ZHËSFONFSLF[B¿OOËM¿ÐTÐOFSBEZBOEF- OJS a) Õ C 3π D 2π 2 3 A r br yay 1 radyan E 11π r V[VOMVóV 2 F 7π G 5π 6 3 O 1 radyan r ÕSCS yay x radyan r V[VOMVóV B SY ÕS a) r = 3 · 180° C = 270° Y Õ 2 · 180° D = 120° 2 #VOBHËSF=ÕSBEZBOES 3 11· 180° #JS B¿OO ËM¿ÐTÐ EFSFDF UÐSÐOEFO % WF BZO E = 990° B¿OOËM¿ÐTÐSBEZBOUÐSÐOEFO3PMNBLÐ[FSF 7 · 180° F = 210° 2 DR = EJS 6 5 · 180° G = 300° 180° r 3BEZBODJOTJOEFOWFSJMFOCJSB¿EFSFDFZF¿FW- 3 SJMJSLFOr ZFSJOFZB[MS 3BEZBO π Õ 3π Õ ÖRNEK 12 22 %FSFDF \"öBôEBWFSJMFOUBCMPMBSEBLJCPöMVLMBSEPMEVSVOV[ \"¿OO 3BEZBO π π 2π 7π 11π Õ ÖlçüTÐ %FSFDF 6 334 6 ÖRNEK 10 \"öBôEBLJBÀÌMÀÑMFSJOJSBEZBODJOTJOEFOZB[O[ a) C D \"¿OO %FSFDF E F G ±M¿ÐTÐ 3BEZBO g) I 30° R r 45° R r a) r 180° r 180° a) = ( R = C = (R= = = 30°, = = 60° 66 33 180° r 6 180° r 4 64 2r 2 · 180° 7r 7 · 180° = = 120°, = = 315° 60° R r 90° R r 33 44 D) = ( R = E = (R= 180° r 3 180° r 2 11r 11· 180° = = 330°, 2r = 2 · 180° = 360° 32 66 1 2 180° R 120° R G 180° = r ( R = r F = ( R = 2r 180° r 3 C 3 30° R r 45° R r 31 = =R= = =R= 270° R 3r 300° R 5r 180° r 6 180° r 4 g) = ( R = I = (R= 3r 180° r 2 180° r 3 60° R r 135° R = =R= , = =R= 23 180° r 180° r 2 3 4 360° R 225° R 5r 340° R 17r = ( R = 2r = =R= , = =R= 180° r 180° r 4 180° r 9 π π π π 2π 3π 5π 11. B C D E F G 10. a) C D E F G ÖH I Ö 6432 3 23 4 π π π 3π 5π 17π 12. 643 4 4 9
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ÖRNEK 13 ÖRNEK 16 2 \"õBóEBLJõFLJMEF m ( % ) = h SBEZBOMLCJSBÀOOÌMÀÑTÑOÑOLBÀEFSFDFPMEV- ABD 5 % = hWFm ( % ) = hEJS ôVOVCVMVOV[ m ( ACD ) BDC D 2/5 A = D 180° r 2 · 180° D·r= 5 72° D= π BC :VLBSEBLJ WFSJMFSF HÌSF m ( % ) BÀTOO ÌMÀÑTÑ- BAC ÖRNEK 14 OÑCVMVOV[ \"öBôEBEFSFDFDJOTJOEFOWFSJMFOBÀÌMÀÑMFSJOJSBE- m^ A%BD h + ma A%CD k + ma B%AC k = ma B%DC kPMEVôVJÀJO ZBOB SBEZBODJOTJOEFOWFSJMFOBÀÌMÀÑMFSJOJEFSFDF- ZFEÌOÑöUÑSÑOÑ[ ma % k = ma % k – a m^ % h + ma % k k PMVS BAC BDC ABD ACD a) - C - 2π D - 5π E m h 143° h 72'' 3 3 h + h - 330° R - 2.180° h – h a) 180° = π C = - 120° h h 11π 3 R=- 135° R 6 E - 180° = π - 5.180° D = - 300° 3π R =- 3 4 ÖRNEK 15 ÖRNEK 17 #JS\"#$ÑÀHFOJOEFm^ WA h = r SBEZBO a =hBÀTOOUÑNMFSWFCÑUÑOMFSBÀMBSOO 3 UPQMBNO EFSFDF EBLJLB WF TBOJZF DJOTJOEFO CVMV- OV[ m^ WB h = h PMEVôVOB HÌSF m^ WC h LBÀ EFSFDF- aBÀTOOUÑNMFSJ- a EJS aBÀTOOCÑUÑOMFSJ- a ES r 180° h - a + 180° - a = 270° - 2a ES = = 60° 180° 00 2a =p h 33 + 132° 43' =h 47° 17' 270° 72° 43' + – h h 132° 43' 72° 14. a) - 11π C mD mE - 3π 47° 17' hhh17. h 13. π 64
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr ÖRNEK 18 ÖRNEK 20 #JS\"#$%EÌSUHFOJOEF \"õBóEBLJõFLJMEFCJSEPóVNHÐOÐQBTUBTOO 1 TOO m (WA) = 2r SBEZBO m (WB) =hWFm (XC) = 6 3 LFTJMJQ BMONBTOEBO TPOSB LBMBO QBS¿BT HËTUFSJMNFL- PMEVôVOBHÌSF m (XD) EFôFSJOJEFSFDF EBLJLBWFTB- UFEJS OJZFDJOTJOEFOCVMVOV[ m^ XA h + m^ XB h + m^ XC h + m^ XD h = 360° m^ XA h = m^ XB h =h – – 120 18000 – – 24 m^ XC h =h 1BTUBOO ÐTU LTN EBJSF õFLMJOEFEJS ,BMBO QBS¿BZ NFSLF[EFOFõJUEBJSFEJMJNMFSJPMBDBLõFLJMEFQBS¿BMBSB 120° h hh BZSNBLJTUFZFO\"MQ:Bó[ IFSCJSEJMJNJONFSLF[B¿T- h OhhPMBDBLõFLJMEFEJMJNMJZPS h + h + hhh h 1BTUBEB HÌSÑOFO NVN EBJSFOJO NFSLF[JOEF PMEV- ôVOBHÌSF \"MQ:Bô[QBTUBZLBÀQBSÀBZBBZSS h m^ XD h =h ÖRNEK 19 A 108.000' ' = 108.000' ' = 30° 300° 3600 60° D 300° 300° = UBOF 30° 30° 30° 30° BC & EFm ( A%BD ) = 2m ( % ), [ BD ]B¿PSUBZWF ABC BCA m^ % h =hhhEJS BDA :VLBSEBLJWFSJMFSFHÌSF m ( % ) OCVMVOV[ DBC A 3 a =h 12' 84' ' 3a D 93° 13 ' 24 ' ' a= 3 a = 31h 2a 2a =p h 2a a =h B C 18. hhh hhh 20. 10
:ÌOMÑ\"ÀMBS TEST - 1 1. TBOJZFMJLCJSBÀOOEFSFDFEBLJLBWFTB- 4. h hh MJL CJS BÀOO 3 J BöBôEBLJMFSEFO 5 OJZFDJOTJOEFOEFôFSJBöBôEBLJMFSEFOIBOHJTJ- IBOHJTJEJS EJS \" hhh # hh \" åhhh # hhh $ hh % hhh $ hhh % hhh & hhh & hhh 2. m (WA) =hhWFm (WB) =hhh #JS\"#$ÑÀHFOJOJOJLJJÀBÀTOOÌMÀÑMFSJ PMEVôVOBHÌSF m (XA) - m (XB)EFôFSJBöBôEBLJ- WF MFSEFOIBOHJTJEJS PMEVôVOB HÌSF ÑÀÑODÑ BÀTOO ÌMÀÑTÑ BöBô- \" hhh # hh EBLJMFSEFOIBOHJTJEJS $ hhh % hh \" h # h & hhh $ h % h & h 3. #JS\"#$ÑÀHFOJOEF TBOJZFMJLBÀOOUÑNMFSBÀTLBÀTBOJ- m (WA) = 90° ve m (WB) = 48°hhh ZFEJS PMEVôVOBHÌSF m (XC)BöBôEBLJMFSEFOIBOHJTJ- EJS \" hh # hh \" # $ $ hhh % hhh % & & hhh 1. \" 2. $ 3. D 7 4. B D D
TEST - 2 :ÌOMÑ\"ÀMBS 1. m (WA) = 24° 55hhhWFm (WB) = 11°hhh 4. 3r + r - 5r PMEVôVOB HÌSF m^WAh + m^WBh EFôFSJ BöBôEBLJ- 4 2 12 MFSEFOIBOHJTJOFFöJUUJS UPQMBNOOEFSFDFDJOTJOEFOFöJUJLBÀUS \" # $ \" hhh # hhh % & $ hhh % hhh & hhh EFSFDFMJLBÀOOÌMÀÑTÑLBÀSBEZBOES A) 2π B) 4π 6π D) 7π 8π 5 5 C) E) 2. 13π SBEZBOMLBÀOOÌMÀÑTÑLBÀEFSFDFEJS 5 55 3 \" # $ % & 3. B :BS¿BQ CS PMBO EBJSFTFM CJS QJTUJO \" OPLUBTO- EBCVMVOBOJLJBSB¿BZOBOEBGBSLMZËOMFSEFIBSFLFU FEJZPS#JSJODJBSB¿rCSJLJODJBSB¿r CSHJUUJL- UFOTPOSBEVSVZPS OA AC ôFLJMEFLJ & EFm (WA) = 2r , m (WB) = r UÐS ABC #VBSBÀMBSOEVSEVLMBSOPLUBMBS#WF$OPLUBMB- 34 :VLBSEBLJWFSJMFSFHÌSF m (XC)LBÀUS SJTF m ( % ) LBÀSBEZBOPMBCJMJS BOC A) π B) π C) π D) π E) π A) π B) 2π C) 3π D) 5π E) r 20 12 15 9 10 2 3 4 6 1. \" 2. \" 3. B 8 4. $ & B
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF &4\"4²-¦·#÷3÷.¦&.#&3 ÷MJöLJMJ,B[BONMBS 11.1.1.2 : \"¿ËM¿ÐCJSJNMFSJOJB¿LMBZBSBLCJSCJSJJMFJMJõLJMFOEJSJS 7$1,0%m/*m ÖRNEK 3 L ` ; WF # i < PMNBL Ð[FSF ËM¿ÐTÐ ²MÀÑTÑ 124π SBEZBO PMBO BÀOO FTBT ÌMÀÑTÑ LBÀ i +LpPMBOB¿OOFTBTÌMÀÑTÑ i EFSF- 3 DFEJS SBEZBOES L ` ; WF # i < r PMNBL Ð[FSF ËM¿ÐTÐ i +LprPMBOB¿OOFTBTÌMÀÑTÑ i SBEZBOES Ö Ö \"¿MBSOFTBTËM¿ÐMFSJOFHBUJGPMBNB[ – Ö 20 Ö 4r 3 ÖRNEK 1 \"öBôEBLJBÀMBSOFTBTÌMÀÑMFSJOJCVMVOV[ ÖRNEK 4 a) C ²MÀÑTÑ - 49 r SBEZBOPMBOBÀOOFTBTÌMÀÑTÑLBÀ 6 D - E - SBEZBOES B C 1340 -Ö 12Ö D 1-1420°2+43 – 1080 3 – +Ö - 240° Ö C m 11r 6 – 1800 m 70° ÖRNEK 5 \"öBôEBWFSJMFOUBCMPEBLJCPöMVLMBSEPMEVSVOV[ ÖRNEK 2 \"¿OO±M¿ÐTÐ 45π - 47π - 43 ²MÀÑTÑ 21π SBEZBO PMBO BÀOO FTBT ÌMÀÑTÑ LBÀ \"¿OO&TBT 2 ±M¿ÐTÐ SBEZBOES 45π 47π - \"ÀOO²MÀÑTÑ 4 3 m 300° Ö 4Ö r \"ÀOO&TBT 5π π 2 ²MÀÑTÑ 4 3 300° – Ö Ö 1. B C D E r 4r 11r 5π π 2. 3. 4. 36 2 43
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr ÖRNEK 6 ÖRNEK 9 &TBTÌMÀÑMFSJPMBOGBSLMJLJOFHBUJGBÀOOFOCÑ- -hMJLBÀOOFTBTÌMÀÑTÑOÑCVMVOV[ ZÑLEFôFSMFSJOJCVMVOV[ = h 22° +pL + mh L= -JÀJO-= -338° L= -JÀJO- 720° = - h ÖRNEK 7 ÖRNEK 10 &TBTÌMÀÑTÑ 2r PMBO 2π UFOGBSLMFOLÑÀÑLQP[J- ÷LJJÀBÀTOOÌMÀÑMFSJ 25r WFBÀMBSOOFTBT 33 3 UJGBÀJMFFOCÑZÑLOFHBUJGBÀOOÌMÀÑMFSJOJCVMVOV[ ÌMÀÑMFSJBMOBSBLPMVöUVSVMBOCJSÑÀHFOJO ÑÀÑODÑJÀ BÀTLBÀSBEZBOES 2r 8r 2r - 4r Ö Ö r Ö+ = - 2r = j = 60° – 2880 8 33 33 – Ö 4 3 Ö ++a = 180° r a == 4 ÖRNEK 8 \"öBôEBLJMFSEFO IBOHJTJOJO FTBT ÌMÀÑTÑ EJôFSMFSJO- EFOGBSLMES a) 37r C - 29r ÖRNEK 11 3 3 D :BOEBLJTBBUUBNZJHËTUFSJS- E r F m LFO¿BMõUSMZPS :FMLPWBO JMFSMFEJôJOEF a) Ö Ö r = 60° C -Ö Ö r TBBULBÀHÌTUFSJS 3 = 60° – Ö – -Ö - 3 Ö Ö D r 180° E = = 60° 3000 – 7 33 – 2880 8 30° 120° 120° 120° =EBLJLB F – m m 8.20 1440 mm 7. 8π 4π 8. 10 π /- hhh 10. 11. 8.20 33 4
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF #JSJN¦FNCFS ÖRNEK 14 7$1,0%m/*m L`3+WF Af 1 , k pOPLUBTCJSJN¿FNCFSÐ[FSJOEFEJS 2 \"OBMJUJL EÐ[MFNEF NFSLF[J CBõMBOH¿ OPLUBT WF ZBS¿BQ CJSJN PMBO ¿FNCFSF CJSJN ÀFN- #VOBHÌSF# WF0 PMNBLÑ[FSF m ( % ) CFSEFOJS AOB y BÀTLBÀEFSFDFEJS 1 y A( 1 , :3 ) x2 + y2 = 1 L(x, y) 1 2 2 1 2 1 :3 f p +L2 = 1 2 2 x 60 Bx L= 3 ma % k = O 1/2 AOB –1 O M1 2 –1 ÖRNEK 15 - Y Z CJSJN¿FNCFSÐ[FSJOEFCJSOPLUBPMNBL 22 Ð[FSF 0.-EJLпHFOJOEF #JSJN¿FNCFSÐ[FSJOEFLJAf , a pWFBf b, pOPL- 2 2 I I I I I I0. + .- = OL j x +Z =PMVS UBMBSWFSJMJZPS a ` R+ C` R-PMNBLÑ[FSFLÌöFMFSJ\" #WFPSJKJO PMBOÑÀHFOJOBMBOLBÀCS2EJS ÖRNEK 12 2 2 2 2 2 p =1 & b=- 2 A f a , 1 pOPLUBTCJSJNÀFNCFSÑ[FSJOEFPMEVôVOB f p + 2 = 1 & a= , 2 +f 3 a b HÌSF BOOBMBCJMFDFôJEFôFSMFSJCVMVOV[ 22 22 B ( – :2 , :2 ) y A( :2 , :2 ) 2 2 2 2 x A^ AOB h = 1·1 O 2 a2 + d 1 2 2 82 2 & = ise a = \" 1 n =1 a = 3 93 2 ÖRNEK 13 ÖRNEK 16 ( a - Y + C+ Z = #JSJNÀFNCFSÑ[FSJOEFLJ Af - 3 , - 1 pWF 22 EFOLMFNJOJOCJSJNÀFNCFSCFMJSUNFTJJÀJOBpCLBÀPM- NBMES B f 3 , - 1 pOPLUBMBSBSBTV[BLMLLBÀCSEJS 22 y CSEJS x2 + y2 =PMNBMES 30 30 x 1 120 1 a - 2 = C+ 1 = 1 A(– :3 , –1 ) B(:32 , –1 ) a = C= 0 2 2 2 BpC= 3 · 0 = 0 22 13. 0 11 14. 1 3 12. ± 2 3
TEST - 3 &TBT²MÀÑm#JSJN¦FNCFS 1. 38r 4. -MJLBÀOOFTBTÌMÀÑTÑLBÀSBEZBOES 3 A) r B) r C) 2r 5 3 SBEZBOMLBÀOOFTBTÌMÀÑTÑLBÀSBEZBOES E) 11r A) r B) 2r $ Õ D) 5r 6 3 6 D) 4r E) 5r 3 3 2. -EFSFDFMJLBÀOOFTBTÌMÀÑTÑLBÀSBEZBO- - 21r ES 2 A) 5π B) 13π C) 4π D) 7π E) 8π SBEZBOMLBÀOOFTBTÌMÀÑTÑLBÀSBEZBOES 3 93 69 A) r B) r C) 3r D) 3r & Õ 4 2 3. #JSJNÀFNCFSÑ[FSJOEF 5π SBEZBOMLZBZOCJ- \"öBôEBLBSöMBSOEBFTBTÌMÀÑMFSJWFSJMFOBÀ- 6 MBSEBOLBÀUBOFTJEPôSVWFSJMNJöUJS UJö OPLUBTOO LPPSEJOBUMBS BöBôEBLJMFSEFO IBOHJTJEJS * 197π $ 5π 88 A) f 3 , 1 p B) f 1 , 3 22 p ** –4470° $ 7π 22 6 C) f - 3 , - 1 p D) f - 3 , 1 p *** 25π $ 30° 22 22 6 E) f - 1 , 3 *7 - 26π $ π p 33 22 \" # $ % & 1. B 2. B 3. D 12 4. D D D
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF 53÷(0/0.&53÷,'0/,4÷:0/-\"3* ÷MJöLJMJ,B[BONMBS 11.1.2.1 : 5SJHPOPNFUSJLGPOLTJZPOMBSCJSJN¿FNCFSZBSENZMBB¿LMBS Kosinüs ve Sinüs Fonksiyonları 7$1,0%m/*m #FO[FS õFLJMEF \" # $ m WF % m OPLUBMBSJ¿JOVZHVOFõMFNFMFSZBQMS- y TBBõBóEBLJEFóFSMFSCVMVOVS B(0, 1) P(x, y) x DPTY y = sina 1 TJOY m a C(–1, 0) A(1, 0) x m O x = cosa D(0,–1) Trigonometrik Özdeşlikler #JSJN ¿FNCFS Ð[FSJOEF [01 OO CJSJN ¿FNCFSJ %m/*m y LFTUJóJOPLUB1 Y Z WF[01OOYFLTFOJJMFZBQ- C B x = cosa UóQP[JUJGZËOMÐB¿aPMTVO P(x, y) y = sina 1 Y Z OPLUBTOOBQTJTJOF aHFSÀFLTBZTOO M 1 NA x LPTJOÑTÑ EFOJS WF DPTa JMF HËTUFSJMJS a HFS¿FL a TBZTO DPTa ZB EËOÐõUÐSFO GPOLTJZPOB LPTJ- OÑTGPOLTJZPOVEFOJS O 1 Y Z OPLUBTOOPSEJOBUOB aHFSÀFLTBZT- OOTJOÑTÑEFOJSWFTJOaJMFHËTUFSJMJSaHFS¿FL D TBZTO TJOa ZB EËOÐõUÐSFO GPOLTJZPOB TJOÑT GPOLTJZPOVEFOJS I I I I I I0. = PN =TJOaWF ON =DPTa :VLBSEBLJ õFLJMEF 1 OPLUBT ¿FNCFS Ð[FSJOEF WF¿FNCFSJOZBS¿BQCJSJNPMEVóVJ¿JO1OPL- PMEVóVOBHËSF 0/1EJLпHFOJOEF1JTBHPSUF- UBTOOIFNBQTJTJIFNEFPSEJOBU aOOTJOÐTÐ PSFNJOEFO WF DPTJOÐTÐ - EFO LпÐL ZB EB EFO CÐZÐL PMBNB[ I I ION + I NP = 0IBMEFTJOÐTWFLPTJOÐTGPOLTJZPOMBSOO DPTa ) + TJOa ) = 5BONLÐNFTJ3 DPTa +TJOa =EJS (ËSÐOUÐLÐNFTJ<m >EJS :BOJr! `3J¿JO m#DPTa #WFm#TJOa # WFZB DPT3Z<m > TJO3Z<m >EJS 13
4*/*' 1. MODÜL 53÷(0/0.&53÷ www.aydinyayincilik.com.tr ÖRNEK 1 ÖRNEK 4 a =-TJOa \"öBôEBLJJöMFNMFSJTPOVÀMBOESO[ PMEVôVOBHÌSF BHFSÀFLTBZTOOBMBCJMFDFôJEFôFS- a) DPT TJO MFSJOLÑNFTJOJCVMVOV[ C cos r + sin 3r – cosr -ãTJOaã 22 -ãTJOaã D cos 7r - 3 sin 5r + cos 3r - cos 0° ã- 3TJOaã& [ ] 22 E TJO-DPT +TJODPT ÖRNEK 2 F cosf - 5r p - sinf - 3r p + sin^ –3r h Y-DPTa + 1 = 22 2 a) -1 + ( -1 ) = -2 FöJUMJôJOEFYHFSÀFLTBZTOOBMBCJMFDFôJEFôFSMFSJO C + ( -1 ) - ( -1 ) = 0 LÑNFTJOJCVMVOV[ D - 3 ( 1 ) + ( -1 ) -1 = - E - 3( -1)2 + ( -1 ) · ( -1 ) = 2 -3 + 1 = 0 1 F - 1 - 0 = -1 2cosa – 12 3x =DPTa - Y= 23 -ãDPTaã& -ãDPTaã ÖRNEK 5 5 13 0 # a #ÖPMNBLÑ[FSF – ãDPTa – ã DPTa = 4 5 2 22 PMEVôVOBHÌSF TJOa EFôFSMFSJOJCVMVOV[ 2 cos a – 1 5 21 –# # 632 51 TJO2a +DPT2a = 1 – #x# TJO2a + d 4 2 62 n =- 5 1 G 5 =1 , 62 TJO2a = 9 EJS 25 ÖRNEK 3 33 TJOa = WFZBTJOa = – PMVS a =TJOY-DPTZ+ PMEVôVOBHÌSF BHFSÀFLTBZTOOBMBCJMFDFôJEFôFS- 55 MFSJOLÑNFTJOJCVMVOV[ ÖRNEK 6 -ãTJOYã -ãDPTZã \" - OPLUBTiBÀTOOCJUJöLFOBSOOCJSJN ÀFNCFSJ LFTUJôJ OPLUB PMEVôVOB HÌSF TJOi EFôFSJ- -ãTJOYã -ãDPTZã OJCVMVOV[ -ãTJOYã y TJOi A(–0,6, 0,8) 1 1 + -ãDPTZã 0,8 -ãTJOY+DPTZã –1 0,6 -ãTJOY+DPTZ+ã –1 a [ - ] 1. [ ] 2. = - 5 , 1 G 3. [m ] 14 33 62 4. B m C D E F m WFZB – 55
www.aydinyayincilik.com.tr 53÷(0/0.&53÷ 1. MODÜL 4*/*' ÖRNEK 7 ôFLJMEFLJ CJSJN ¿FN- ÖRNEK 9 CFS Ð[FSJOEF NPEFM- y MFONJõ CJS SBEBS TJT- 5BONMPMEVLMBSBSBMLMBSEBBöBôEBLJJGBEFMFSJOFO UFNJ HËSÐMNFLUFEJS TBEFCJÀJNMFSJOJCVMVOV[ x 3BEBSEB HËSÐOFO a) 1 - sin2 i V¿BóO SPUBTOO CJ- SJN ¿FNCFS Ð[FSJO- 1 + cos i EFLJ OPLUBTOO PSEJ- 2 6 44co7s 4i48 sin2 i cos i + 1– sin2 i 1 a) – = OBU 4 PMEVóVOBHË- 1 1+ cos i 1+ cos i 5 ( 1+ cos i) SF VÀBôOSPUBTOOYFLTFOJJMFZBQUôBÀOOLPTJ- cos i ^ 1+ cos i h OÑTÑOÑCVMVOV[ = = cos i ^ 1+ cos i h 2 + d 4 2 3 j x= x n =1 55 ÖRNEK 8 C sin3 i - cos3 i (1 + sin i cos i) (sin i - cos i) 5BONM PMEVôV BSBMLMBSEB BöBôEBLJ JGBEFMFSJO FO TBEFCJÀJNMFSJOJCVMVOV[ 6 4 4 4 4 4 4 4471 4 4 4 4 4 4 448 2 2 sin2 i a sin i - cos i k^ i + sin i. cos i + i h a) sin cos 1+ cos i C ^ 1 + sin i. cos i ha sin i - cos i k 1 + sin i. cos i =1 1 + sin i. cos i B TJO2i +DPT2 i = 1 TJO2 i = 1 -DPT2 i 1 – 2 i ^ 1– cos i h^ 1+ cos i h D 2 - 3 + cos2 x 2 + sin x cos = j 1 -DPTi 1+ cos i ^ 1+ cos i h C 1 + 1 3 + ^ 1– sinx h 1+ sin i 1- sin i 2– =2– 4 – sin x 2 + sin x 2 + sin x D 2 – ^ 2 – sin x h^ 2 + sin x h = 2 – 2 + sin x C 1 1 1– sin i + 1+ sin i 2 ^ 2 + sin x h += = = sin x 1 + sin i 1 - sin i 2 2 (1- sin i) ^ 1+ sin i h 1– sin i cos i D cos i - cos i E 1+ sin x – cos2 x 1 - sin i 1 + sin i 1+ sin x cos i cos i D - 1– sin i 1 + sin i (1+ sin i) ^ 1 - sin i h 2 cos i + cos i sin i – cos i + cos i sin i 6 44si7n 4x48 = 1– cos2x + sin x sin x^ sin x + 1 h E = = sin x 1– sin2 i 1+ sin x ^ 1+ sin x h 2 cos i sin i 2 sin i == cos2 i cos i 3 8. a) mDPTi 2 2sini B DPTi C D TJOY E TJOY 7. C D 5 cos2 i cosi
11. SINIF .0%·- 53÷(0/0.&53÷ www.aydinyayinlari.com.tr ÖRNEK 10 ÖRNEK 11 y C 5BONMPMEVLMBSBSBMLMBSEBBöBôEBLJJGBEFMFSJOFO A TBEFCJÀJNMFSJOJCVMVOV[ B a) sin4 x + cos2 x . sin2 x - sin2 x 20° x a) = sin4x - sin2x +DPT2x · sin2x O = sin2x ( sin2x - 1) +DPT2x · sin2x -DPT2x | |:VLBSEBLJCJSJNÀFNCFSEF BC OVCVMVOV[ = -sin2YDPT2x+DPT2x · sin2x =0 y b) 3 ( cos4 x + sin4 x ) - 2 ( cos6 x + sin6 x ) = 1 sin70° 20° 1 1 | BC | = 1 - sin70° –1 70° b) =DPT4x + 3sin4x - DPT2x)3 + ( sin2x)3) ÖRNEK 12 = DPT4x + 3sin4x - DPT2x + sin2Y DPT4x-sin2x + sin4x)) #JSJNÀFNCFSÑ[FSJOEFLJ\" DPT TJO WF # DPT TJO OPLUBMBSBSBTV[BLMôCVMVOV[ 1 =DPT4x + 3sin4x -DPT4x +DPT2x + 2 sin2x - 2sin4x y =DPT4x + sin4x +DPT2x · sin2x 1 = DPT2x + sin2x )2 A 1 1 50° ]\"#]= 2 CSEJS –1 40° 1 1 B –1 c) 1– cos4 x + sin4 x cos2 x - 1 ÖRNEK 13 + 4 4 #JSJNÀFNCFSÑ[FSJOEFLJ\" -DPT -TJO WF c) 1 sin x – cos x # DPT -TJO OPLUBMBSOWF0 OPLUBTO 2 % cos x – 1 AOB LÌöFLBCVMFEFO\"#$ÑÀHFOJOEFm ( ) LBÀEFSF- 2 2 6 4 4 4471 4 4 448 DFEJS 2 2 1+ ^ sin x – cos h^ sin + x h x cos = 2 y cos x – 1 2 6 44si7n 4x48 1– cos2x + sin2x 2 O 30° 50° x 100° 2 sin x = = = –2 100° 2 2 AB cos – 1 – sin x 10. a) 0 b) 1 c) –2 16 11. 1–sin70° 12. 2 13. 100°
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF %m/*m ÖRNEK 16 #JSJN ¿FNCFSJO IFSIBOHJ CJS CËMHFTJOEFLJ B¿OO 3π < a < 2πPMEVôVOBHÌSF LPTJOÐTWFTJOÐTÐOÐOJõBSFUJ CVCËMHFEFLJCJSOPL- 2 UBOOBQTJTWFPSEJOBUOOJõBSFUJJMFBZOES y 2 sin a + 3 cos a cos – 1 cos + sin a cos a sin + sin + JGBEFTJOJOFöJUJOJCVMVOV[ –1 II I 1x - 2 sin a 3 cos a IV + cos – O cos + sin a sin – III sin – cos a -2 + 3 = 1 –1 ÖRNEK 14 \"öBôEBLJUSJHPOPNFUSJLEFôFSMFSJOJöBSFUMFSJOJCVMV- OV[ a) DPT C TJO D DPT E DPTf - 7r p ÖRNEK 17 6 F DPTc - r m #JS\"#$ÑÀHFOJOJOJÀBÀMBSa b iPMNBLÑ[FSF BöB- 5 G TJOc 3r m ôEBLJJGBEFMFSJOLBÀUBOFTJEPôSVPMBCJMJS 7 * TJOaDPTb g) DPTf - 135r p ** TJOaTJObTJOi 4 I TJO – *** TJOaDPTi = *7 DPTa +TJOi a) - C - D + E - F + G + g) + I - * b >JTF TJOaDPTb < 0 DPTb >PMVS *EPôSVPMBCJMJS ÖRNEK 15 ** a bWFiJMFBSBTOEBQP[JUJGPMEVôVJÀJO LFTJOEPôSVEVS B DPT C TJOWFD DPT PMEVôVOBHÌSF B CWFDOJOJöBSFUMFSJOJCVMVOV[ *** i =JTFEPôSVPMBCJMJS *7 DPTa <PMVQDPTa +TJOi <PMBCJMJS D 3 540 a =DPT& 180 = r & r = B= - C=TJO& D 5 900 = C= - = r& 180 r D 7 1260 D=DPT& 180 = r & r = D= + 14. B mC mD E mF G H I m B C D 17 1 17. 4
TEST - 4 5SJHPOPNFUSJL'POLTJZPOMBS* sin (18r) 4. cosf - 703r p 1. 3 EFôFSJLBÀUS sin 18° JGBEFTJBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS \" # $ - A) - # - 1 $ D) 1 & 2 2 D) 1 sin 18° & DPU 2. DPTY-TJOZ sin A - sin B + cos A - cos B cos A + cos B sin A + sin B JGBEFTJOJOFOCÑZÑLEFôFSJLBÀUS JöMFNJOJOTPOVDVLBÀUS \" # $ % & # - 1 $ 2 \" m % & 3. sin 205r + cos 2550° + tan^ - 26π h sin a = 2x - 1 3 7 UPQMBNOOTPOVDVLBÀUS PMEVôVOB HÌSF Y JO FO CÑZÑL UBN TBZ EFôFSJ LBÀUS \" # C) 3 D) E) 2 3 A) - # - $ % & 2 1. \" 2. \" 3. D 18 4. D $ &
5SJHPOPNFUSJL'POLTJZPOMBS* TEST - 5 1. 4 - cosf x + 5r p 4. 5BONMPMEVôVBSBMLUB 6 - cos i - 1 - sin i - 1 JGBEFTJOJ FO CÑZÑL ZBQBO Y EFôFSMFSJOJO FTBT cos i - 1 - sin i - 1 JGBEFTJBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS ÌMÀÑTÑLBÀSBEZBOES r r r r r A) - # $ A) B) C) D) E) % DPTi & TJOi 5 2. 9 + 7 sin2 i - cos2 i 3π < x < 2πPMEVóVOBHËSF 5 sin2 i + 3 cos2 i - 1 2 JGBEFTJOJOFöJUJLBÀUS sin2x - cos2 x - sin x - 1 cos x JGBEFTJBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS A) - # $ % & A) - # - $ - % & 3. 5BONMPMEVôVBSBMLUB 5BONMPMEVôVBSBMLUB sin3x + cos3x + sin3x - cos3x cos2 i + cos i – sin2 i 1+ sin i 1- cos i sin x + cos x sin x - cos x JGBEFTJBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS JGBEFTJOJOEFôFSJLBÀUS A) -TJOi B) -DPTi $ A) 1 # $ - % & % TJOi & DPTi 2 1. \" 2. D 3. B 4. \" B \"
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr 53÷(0/0.&53÷,'0/,4÷:0/-\"3** ÷MJöLJMJ,B[BONMBS 11.1.2.1 : 5SJHPOPNFUSJLGPOLTJZPOMBSCJSJN¿FNCFSZBSENZMBB¿LMBS Tanjant Fonksiyonu Kotanjant Fonksiyonu 7$1,0%m/*m 7$1,0%m/*m y y 1 cota y=1 1 T(1, tan a) K(cota,1) P P tana –1 a 1 x –1 a 1 x O O –1 –1 x=1 [01OOZ=EPóSVTVOVLFTUJóJ,OPLUBTOO #JSJN ¿FNCFS Ð[FSJOEF [01 OO CJSJN ¿FNCF- SJ LFTUJóJ OPLUB 1 Y Z WF [01 OO Y FLTFOJ JMF BQTJTJOF a BÀTOO LPUBOKBOU EFOJS WF DPUa CJ¿JNJOEFHËTUFSJMJS ZBQUóQP[JUJGZËOMÐB¿aPMTVO %FOLMFNJZ=PMBOEPóSVZBLPUBOKBOUFLTF- [01OOY=EPóSVTVOVLFTUJóJ5OPLUBTOO OJEFOJS PSEJOBUOB aBÀTOOUBOKBOUEFOJSWFUBOaCJ- Y FLTFOJ JMF LPUBOKBOU FLTFOJ QBSBMFM PMEVóVO- ¿JNJOEFHËTUFSJMJS EBO a =WFZBa =ÕPMEVóVOEB [01JMF %FOLMFNJ Y = PMBO EPóSVZB UBOKBOU FLTFOJ EFOJS Z=EPóSVTVLFTJõNF[#VSBEBO ZFLTFOJJMFUBOKBOUFLTFOJQBSBMFMPMEVóVOEBO DPU=DPU=UBONT[ a = π WFZBa = 3π PMEVóVOEB[01JMF DPU=DPUÖ=UBONT[PMVS 22 5BONLÐNFTJ 3- {LÕL` Z }PMBOWFUBON x =EPóSVTVLFTJõNF[#VSBEBO LÐNFTJOEFLJIFSCJSBSFFMTBZTODPUaZBFõ- tan 90° = tan π =UBONT[ MFZFOGPOLTJZPOBLPUBOKBOUGPOLTJZPOVEFOJS :BOJ 2 tan 270° = tan 3π =UBONT[PMVS DPU3- {LÖL` Z } Z R UBONMES 2 5BOKBOU WF LPUBOKBOUO UBONOEBO BõBóEBLJ 5BONLÐNFTJR - ' π + kπ, k ! Z 1PMBOWFUB- UBCMPZVPMVõUVSBCJMJSJ[ 2 x ONLÐNFTJOEFLJIFSCJSaHFS¿FLTBZTOUBOa ZBFõMFZFOGPOLTJZPOBUBOKBOUGPOLTJZPOVEF- UBOY UBONT[ UBONT[ OJS:BOJ DPUY UBONT[ UBONT[ UBONT[ tan: R – ' π + kπ, k ! Z 1 \" R 2 UBONMES 20
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF %m/*m ÖRNEK 1 tan a = sin a ve cot a = cos a \"öBôEBLJJöMFNMFSJOTPOVDVOVCVMVOV[ cos a sin a a) DPU+UBO C DPUc 3r m +UBO -Õ -DPUf - 7r p y 22 B cota 123 M y=1 D DPTf - 5r p -TJOf - 3r p +UBO -Õ P L sina K tana x 22 1 E DPUc r m -UBOÕ-UBO+DPTÕ a C O cosa N A a) 0 + 0 = 0 D C + 0 + 0 = 0 x=1 D + (- 1) + 0 = -1 E - 0 - 0 - 1 = -1 O&NK ` O&AL PMEVóVOEBO ON = NK & cos a = sin a ÖRNEK 2 1 tan a OA AL 3 cos a - sin a = 1 2 sin a + 5 cos a 2 & tan a = sin a olur. PMEVôVOBHÌSF UBOaEFôFSJOJCVMVOV[ cos a 3cosa - sina 1 & + & PMEVóVOEBO = OPK OBM 2sina + 5cosa 2 OP = PK & sin a = cos a DPTa -TJOa =TJOa +DPTa 1 cot a DPTa =TJOa OB BM 1 & cot a = cos a PMVS UBOa = sin a 4 #VJLJFõJUMJLUFOZBSBSMBOBSBLDPTaáWF TJOaáPMNBLÐ[FSF tan a. cot a = 1 ÖRNEK 3 FõJUMJóJFMEFFEJMJS tan i + cot i = 3 2 #VOBHËSF PMEVôVOBHÌSF UBn2 i +DPU2 iEFôFSJOJCVMVOV[ tan a = 1 , cot a = 1 cot a tan a f tani + 1 2 32 p =f p tani 2 tan2 i + 2 · tani · 1 + 1 9 = 14 4 4 2 4ta4n4i3 tan2 i 4 2 UBO2 i +DPU2 i = 9 - 2 4 UBO2 i +DPU2 i = 1 4 21 1. B C D mE m 1 1 2. 3. 44
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr ÖRNEK 4 %m/*m coseca y sina UBOi +DPUi = C N PMEVôVOBHÌSF UBOi +DPUiUPQMBNOOQP[JUJGEFôF- B x SJOJCVMVOV[ LP UBOi +DPUi =LPMTVO& UBOi +DPUi)2 =L2 1i UBO2i +UBOiDPUi +DPU2i =L2 a iM 2 O cosa K A UBO2i +DPU2i )2 = L2 - 2 )2 seca UBO4i +DPU4i +UBO2i DPU2i = L2 - 2 )2 D 22 4 = L2 - 2 )2 & L2 - 2 =WFZBL2 - 2 = -2 & ` & PMEVóVOEBO OKP OPM & L2 =WFZBL2 = 0 & L= \" WFZBL= 0 & L= 2 OK = OP & cos a = 1 OP OM 1 sec a Sekant ve Kosekant Fonksiyonları & sec a = 1 olur. cos a 7$1,0%m/*m & + & PMEVóVOEBO y OLP OPN K B OL = OP & sin a = 1 1 cosec a L OP ON aM C OA x & coseca = 1 PMVS sina D #JSJN ¿FNCFS Ð[FSJOEF % = a PMNBL m ( MOL ) Ð[FSF -OPLUBTOEBLJUFóFUJOYFLTFOJOJLFTUJóJ ÖRNEK 5 OPLUBOOBQTJTJOFaB¿TOOTFLBOUEFOJSWF sin i + cosec i = 5 2 TFDaJMFHËTUFSJMJS PMEVôVOBHÌSF TJO2 i +DPTFD2 iEFôFSJOJCVMVOV[ -OPLUBTOEBLJUFóFUJOZFLTFOJOJLFTUJóJOPLUB- OO PSEJOBUOB a B¿TOO LPTFLBOU EFOJS WF DPTFDaJMFHËTUFSJMJS #WF%OPLUBMBSOEBTFLBOUEFóFSMFSJUBONT[ sini + 15 = PMBDBóOEBOTFLBOUGPOLTJZPOVOVO sini 2 5BONLÑNFTJ R – ' π + kπ : k ! Z 1 2 1 2 52 f sini + p =f p (ÌSÑOUÑLÑNFTJ3- ( - PMVS sini 2 2 1 1 25 sin i + 2 · sini · + = \" WF $ OPLUBMBSOEB LPTFLBOU EFóFSMFSJ UB- 14 4 4 2 4s4in4i3 sin2 i 4 ONT[PMBDBóOEBOLPTFLBOUGPOLTJZPOVOVO 2 5BONLÑNFTJ3-\\LÕL! Z } sin2 i + cosec2 i = 25 - 2 = 17 44 (ÌSÑOUÑLÑNFTJ3- ( - PMVS 4. 2 22 17 4
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ÖRNEK 6 ÖRNEK 8 y 5BONM PMEVôV BSBMLMBSEB BöBôEBLJ JGBEFMFSJO FO A TBEFCJÀJNMFSJOJCVMVOV[ a) UBOi C cot i. sec i 2 1 + cot i B sin2 i 6 4 4471 4 448 x 2 2 a) + 1= sin i + cos i = 1 =TFD2i 2 i cos2 i cos2 i cos cosi 1 1 1 sin2 i = sini · sini &· C sini cosi = :VLBSEBLJ CJSJN ÀFNCFSEF \" DPTFD WF \"# 2 64 4 4471 4 4 448 sini 1 ÀFNCFSF UFôFU PMEVôVOB HÌSF # OPLUBTOO LPPSEJ- sin2 i + cos2 i OBUMBSOCVMVOV[ cos i 1+ sin2 i sin2 i y 1442443 D TJOiTFDiDPUi E 1 + tan i A sin i + cos i cosec 22° 1 x # TFD 1242°243B 1 cosi sec22° D sini · · = 1 cosi sini sini cosi + sini 1 1+ cosi = = seci cosi sini + cosi cosi E = sini + cosi ÖRNEK 7 F 1 + 1 G sec x - cos x cosec x - sin x < a < r WFDPTFDa = 3 22 x-2 1 + tan i 1 + cot i PMEVôVOBHÌSF YJOUBNTBZEFôFSMFSJOJCVMVOV[ 22 1 1 cos i sin i + =+ 2 13 , x-2 2 22 22 = sin a = F 1 + sin i cos i 1+ 1s4in44i2+ c4os44i3 1s4in44i2+ c4os44i3 sin a x - 2 3 2 cos i 2 1 1 sin i x-2 22 = sin i + cos i = 1 0 <TJOa < 1 & 0 < 3 <1 1- cos2x 2 6 44si7n 4x48 1 cos x - cos x 2 &0<x-2<3 cos x sin x 1- cos x = =· & 2 < x <& x =WFCVMVOVS 1 - sin x 2 cos x 2 sin x G 1- sin x 114-4 2sin44x3 sin x 2 cos x 3 sin x 3 = = tan x 3 cos x # TFD 7. WF 23 8. a) TFD2 i C TJOi D 1 E TFDi F G UBO3 x
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr ÖRNEK 9 %m/*m 5BONMPMEVLMBSBSBMLMBSEBBöBôEBLJJGBEFMFSJOFO #JSJN¿FNCFSJOIFSIBOHJCJSCËMHFTJOEFLJB¿OO TBEFCJÀJNMFSJOJCVMVOV[ UBOKBOUWFLPUBOKBOUOOJõBSFUJPCËMHFEFLJTJOÐT a) sin i. cos i WFLPTJOÐTGPOLTJZPOMBSOOJõBSFUMFSJOJOPSBOOB FõJUUJS (1 - sin i) tan i a) sini · cosi 2 tan – y tan + cot – 1 cot + ^ 1- sini h · sini cos i cosi = 1- sini 2 ^ 1- sini h^ 1+ sini h –1 II I x IV 1 1- sin i = = 1+ sini tan + O cot + III tan – 1- sini ^ 1- sini h cot – –1 C cosec2 i. sec2 i #JSJN ¿FNCFSJO IFSIBOHJ CJS CËMHFTJOEFLJ B¿- cot2 i + tan2 i + 2 OOTFLBOULPTJOÐTJMFLPTFLBOUTJOÐTJMFBZO JõBSFUMJEJS 11 1 · sin2 i cos2 i 2 i · cos 2 i C = sin cos2 i 2 i sin i cos i 2 f+p sin + +2 cos i sin i sin2 i cos2 i ÖRNEK 10 11 \"öBôEBLJUSJHPOPNFUSJLEFôFSMFSJOJöBSFUMFSJOJCVMV- OV[ sin2 i cos2 i 2 i cos 2 i sin = =1 J 644471 4 448 N 12 K sin2 i + cos2 i O dn K O cos i sin i KO L cos i · sin i P a) DPTFD C UBO D TFD E DPUf 5r p D mDPUi DPTFDi F UBOf - 135r p 6 4 G DPUc - r m D 1 - cos2 i + 1 a) + C + D + E - F + G - 2 i sin2 i sin 2 6 4 44si7n 4i 448 sin2 i - cos2 i + 1 sin2 i + sin2 i = =2 sin2 i sin2 i B TJOi C D 24 10. B C D E mF G m
5SJHPOPNFUSJL'POLTJZPOMBS** TEST - 6 1. ãYã r PMEVôVOBHÌSF 4. sin x + 2 cos x = 3 PMEVôVOBHÌSF 4 cos x - 2 sin x 2 1- tan x 1+1 5 cot x tan x UPQMBNLBÀUS JGBEFTJOJOEFôFSJIBOHJBSBMLUBCVMVOVS A) >0, 1 H B) >- 1 , 0 h A) - 65 B) - 1 C) 1 D) 63 & 3 5 88 8 8 C) >0, 1 H D) >- 1 , 0H 5 3 E) >- 1 , 1 H 55 5BONMPMEVôVBSBMLUB TJOY DPTFDY-TJOY 2. 5BONMPMEVôVBSBMLUB JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS cosec i - sin i . tan2 i \" DPTY # TJOY $ DPTx sec i - cos i % -TJOY & +DPTY JGBEFTJOJOEFôFSJBöBôEBLJMFSEFOIBOHJTJEJS \" TFDi # TJOi $ DPTi % UBOi & DPUi 3. sin x + cos x = 5 5BONMPMEVôVBSBMLUB sin x - cos x 4 cos2 i + 1 - sin2 i 1 + sin i sec i 1 - cos i PMEVôVOBHÌSF UBOYLBÀUS JGBEFTJBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS A) 9 B) 9 C) 5 % & A) -TJOi B) -DPTi $ UBOi 4 2 % TJOi & DPTi 1. $ 2. & 3. D 4. \" $ \"
TEST - 7 5SJHPOPNFUSJL'POLTJZPOMBS** 1. UBOY+DPUY=PMEVôVOBHÌSF 4. DPT sin 7r UBOÕ sec 29r DPU UBOx +DPUx 9 18 JGBEFTJOJOEFôFSJLBÀUS JGBEFMFSJOEFOLBÀUBOFTJQP[JUJGUJS \" # $ % & \" # $ % & DPT TJO UBO DPU 2. 5BONMPMEVôVBSBMLUB EFôFSMFSJOJOJöBSFUMFSJTSBTZMBBöBôEBLJMFSEFO IBOHJTJEJS cosec2 i - sec2 i tan2 i - cot2 i A) ( –, –, –, – ) B) ( –, –, –, + ) JGBEFTJOJOEFôFSJLBÀUS C) ( –, –, +, – ) D) ( –, +, –, – ) A) - # - $ % & E) ( +, –, –, – ) 3. TFD x +=UBOY TJO UBO TFD DPTFD PMEVôVOB HÌSF UBOY EFôFSJ BöBôEBLJMFSEFO EFôFSMFSJOJOJöBSFUMFSJTSBTZMBBöBôEBLJMFSEFO IBOHJTJEJS IBOHJTJEJS A) ( +, –, –, – ) B) ( +, –, +, – ) \" # $ % & C) ( +, +, –, – ) D) ( +, –, –, + ) E) ( –, –, +, + ) 1. $ 2. B 3. B 4. B \" \"
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF 53÷(0/0.&53÷,'0/,4÷:0/-\"3*** ÷MJöLJMJ,B[BONMBS 11.1.2.1 : 5SJHPOPNFUSJLGPOLTJZPOMBSCJSJN¿FNCFSZBSENZMBB¿LMBS %JL·ÀHFOEF%BS\"ÀMBSO5SJHPOPNFUSJL ÖRNEK 1 0SBOMBS 0 < a < π PMNBLÑ[FSF TANIM 2 y sin a = 5 L 13 K PMEVôVOB HÌSF DPTa UBOa WF DPUa EFôFSMFSJOJ CV- MVOV[ P x A 12 1 cos a = a 13 O cosa P1 A K1 L1 5 tan a = 13k 5k 12 a JTFaEBSB¿ES a 12 C aB¿TOOCJUJõLPMV¿FNCFSJ1OPLUBTOEBLFTTJO 12k cot a = 5 B I I r OP =DPTa I I r PP =TJOa I I r OP =EJS & + & B¿-B¿CFO[FSMJóJ PMEVóVOEBO OPP OKK 11 OP OP1 1 = cos a =& OK OK1 OK OK1 ÖRNEK 2 OK1 0 < a < π PMNBLÑ[FSF & cosa = 2 OK UBOa = Komflu dik kenar›n uzunlu€u PMEVôVOBHÌSF TJOa DPTaÀBSQNOOEFôFSJOJCVMV- & cos a = OV[ Hipotenüsün uzunlu€u A 2 sin a = PMVS#FO[FSõFLJMEFTJOa UBOa DPUaPSBOMBS EBUBONMBOS r cos a = Komflu dik kenar›n uzunlu€u 5 Hipotenüsün uzunlu€u 5k 1 cos a = r sin a = Karfl› dik kenar›n uzunlu€u 2k 5 Hipotenüsün uzunlu€u a 2 Karfl› dik kenar›n uzunlu€u C k sin a. cos a = Komflu dik kenar›n uzunlu€u B5 r tan a = r cot a = Komflu dik kenar›n uzunlu€u Karfl› dik kenar›n uzunlu€u 27 12 5 12 2 1. cos a = tan a = cot a = 2. 13 12 5 5
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr ÖRNEK 3 ÖRNEK 5 DPT=BPMEVôVOBHÌSF D 8C sin 24° 68 tan 66° + cot 66° 6 JGBEFTJOJOBDJOTJOEFOFöJUJOJCVMVOV[ A 2 A 8 E 18 i 66° 10 B a 1 – a2 sin 24° = 1 - 1 B tan 66° = a \"#$%ZBNVL [DC] // [AB], maA%BCk = i 1 - a2 I I I I I I I IBC = CD =DN AB =DNWF AD =DNEJS 1 - a 24° a cot 66° = a :VLBSEBLJWFSJMFSFHÌSF TJOiEFôFSJOJCVMVOV[ C 1 - a2 1 - a2 3 = 63 sin i = = a 1 - a2 a 1 - 2 10 5 + a + 5 a 1 - a2 2 a 1-a ^ a h a 1 - a2 k = 1 - a2 = 1 - a2 .a 1 - 2 = a^ 1 - 2 h = a - 3 a a a + 1 - a a a 1 - a2 ÖRNEK 6 A b ÖRNEK 4 c A Ba C & 2 3 cm2 A#$EJLпHFOJOEF A^ ABC h = 17 17 sin (WA) . sin (XC) = 3 UÐS 15 4 a | |:VLBSEBLJWFSJMFSFHÌSF \"$ LBÀDNEJS B3D 5 E 13 8 C I I I I\"#$пHFO maA%DCk = a AB = AC =DN sin XA. sin XC = 3 ac 3 I I I IBD =DNWF DC =DNEJS j ·= 4 bb 4 :VLBSEBLJWFSJMFSFHÌSF UBOaEFôFSJOJCVMVOV[ a.c 3 = EJS b 4 \" \"#$ = a.c = 2 3 jBD= 4 3 PMVS 2 15 4@3 3 43 3 tan a = = 3 ac &= = 5 2 4 2 4 j C2 jC b b 3. BmB3 4. 3 28 3 4 5
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ÖRNEK 7 ÖRNEK 10 A \"#$пHFO A \"#$ÑÀHFOJOEF x BDPT( XC ) DDPT( WA ) 5 I IAB DN H 7h I IAC DN JGBEFTJOJOFöJUJOJ I IBC DN cb CVMVOV[ b–x tan XC = 5 tan WB xC 6 Ba C mæY B :VLBSEBLJWFSJMFSFHÌSF cos XB EFôFSJOJCVMVOV[ a.>cos XC + c.>cos XA .. tan XC = 5 tan XB & h = 5· h b-x x x 6-x a · + c · =C- x + x =C ac - x =Yj x = 1 j cos XB = 5 7 ÖRNEK 8 ÖRNEK 11 D [ AB ] m [ BC ] [ DC ] m [ BC ] A \"#$EJLпHFO 6=x 3 3 a [ AB ] m [ AC ] B 4 8 I IDC =CS [ AH ] m [ BC ] . 55 10 I IC b I IAB =CS AB =CS B2 I IBH =CS I IAD =CS C H :VLBSEBLJWFSJMFSFHÌSF TJOaEFôFSJOJCVMVOV[ A :VLBSEBLJ WFSJMFSF HÌSF % EFôFSJOJ CVMV- tan ( BAD ) 1 OV[ 21 a + b = sin a = = 3 42 tan ( B%AD ) = 6 = 3 2 84 4 ÖRNEK 9 ÖRNEK 12 I I I I\"#$пHFOJOEF AB = AC , tan(WA) = 3 WF #JS\"#$пHFOJOEFm (WB) = 45°, m (XC) = 60°WF 4 I I I IAC =CSPMEVôVOBHÌSF \"# LBÀCJSJNEJS I IAB = 4 10 br PMEVôVOBHÌSF #$ LBÀCJSJNEJS A tan XA = 3 A x2 = ^ 3 3 h2 + ^ 3 3 h2 4 x2 = 54 4k ]\"#]=L= 4 10 45° 30° x= 3 6 x 5k 33 6 H k= 4 10 3k 5 k 4 10 BC = 10 k = 10 · 45° 5 B 33 60° C 10 k B D 3C j]#$]=CS 5 1 8 10. C 3 12. 3 6 7. 8. 11. 4 7 2
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr ÖRNEK 13 ÖRNEK 16 H G ôFLJMEFLJ EJL- A EËSUHFOMFS QSJ[- cosa F NBTOEB E 10 | |10 AB =CS 5 D | |C BC =CS B H1 C A x=13 | |5 GC =CS 12 % B m ( DHB ) = a \"#$EJLпHFOJOEF [ AB > m [ AC >, [ AH > m [ BC > PMEVôVOBHÌSF UBOaEFôFSJLBÀUS I IBC =CSEJS 2 + 122 = x2 j x = 13 I I:VLBSEBLJWFSJMFSFHÌSF \") nun aDJOTJOEFOFöJ- 13 UJOJCVMVOV[ tan a = 10 & AC ACB ÖRNEK 14 EF cos a = 1 A \"#$JLJ[LFOBSпHFO ]\"$]=DPTa a I AB I = I AC I =CS & EF sin a = AH & AH = sin a. cos a AHC cos a 5 K I IBC =CS [$,] m [AB] % = a m ( BCK ) b H 4C ÖRNEK 17 B4 8 A :VLBSEBLJWFSJMFSFHÌSF TJOaEFôFSJOJCVMVOV[ i/2 cb 4 a + b = TJOa = 5 i i/2 B aC b D ÖRNEK 15 | |\"#$WF\"#%EJLпHFO m (WB) = 90°, AC =DCS | | | | | |BC =BCS AC = CD =CCS m(A%CB) = i sin x - cos x = 1 3 :VLBSEBLJWFSJMFSFHÌSF cot i EFôFSJOJOB C DUÑ- 2 PMEVôVOBHÌSF UBOY+DPUYEFôFSJOJCVMVOV[ SÑOEFOFöJUJOJCVMVOV[ TJOY-DPTY 2 = 1 3 22 1 1s4in444x 2+ c4o4s44x3 - 2 sin x. cos x = 9 1 1 i a+b PMVS 1 - = 2 sin x. cos x cot = 2 c 9 4 64 4 4471 4 4 448 = sin x. cos x 9 22 sin x cos x sin x + cos x 9 += = cos x sin x 1s4in4x2. co4s4x3 4 ^ sin x h ^ cos x h 4 9 13 4 9 30 TJOaDPTa a+b 13. 14. 17. c 10 5 4
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ÖRNEK 18 ²[FM\"ÀMBSO5SJHPOPNFUSJL0SBOMBS \"õBóEBEJLEËSUHFOõFLMJOEFLJJLJGBSLMNBEEFEFOZBQM- %m/*m NõQMBUGPSNB0OPLUBTOEBCVMVOBOCJSBUDUBSBGOEBO BUõ ZBQMNBLUBES #JSJODJ BUõUB : OPLUBT JLJODJ BUõUB OJO5SJHPOPNFUSJL0SBOMBS YOPLUBTWVSVMNVõUVS A %JLLFOBSV[VOMVL- GF 45° 2 MBS CS PMBO JLJ[- LFOBS EJL пHFOJ 1 JODFMFZFMJN O 4 Y1 a + b = 90° 45° D Cy B 1C 3 ax B E I I I I I IAB = BC =CSj AC = 2 br EJS A Y2 \"#$EJLпHFOJOEF | |O YYCJSEJLпHFO m ( E%Y2Y1 ) = a, AD =NFUSF sin 45° = AB = 1 = 2 | | | |BE =NFUSFPMEVôVOBHÌSF Y1 Y2 LBÀNFUSFEJS AC 2 2 cos 45° = BC = 1 = 2 4 AC 2 2 cos a = y & y = 4 sec a tan 45° = AB = 1 = 1 BC 1 3 sin a = x & x = 3 cosec a cot 45° = BC = 1 = 1 AB 1 x + y =TFDa +DPTFDa WFOJO5SJHPOPNFUSJL0SBOMBS A ÖRNEK 19 2x H xC 30° 30° 22 D 3 ya 60° 60° B1 H 1C a #JSLFOBSOOV[VOMVóVCSPMBOFõLFOBSпHF- ab OJJODFMFZFMJN A 2x E xB I I I I I IAB = AC = BC =CSPMNBLÐ[FSF I I I I I IAH = 3 br WF BH = HC =CSPMVS \"#$%EJLEËSUHFO m ( % ) = % = a WF BCE m ( AED ) | BE | = | AE | EJS \")$EJLпHFOJOEF TJO= 3 = cos 30° :VLBSEBLJWFSJMFSFHÌSF UBOaEFôFSJLBÀUS 2 a + b = x1 DPT= 1 =TJO y2 = 2x.x = 2 y = 2 x & tan a = 2x 2 UBO= =DPU 3 DPU= = tan 30° 3 18. TFDa DPTFDa 1 31 2
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr %m/*m 0° 30° ÖRNEK 22 TJO 1 3 sin2 5π + sin2 π + cot 2π cot 3π 22 12 12 7 14 3 1 JöMFNJOJOTPOVDVOVCVMVOV[ 22 π 5π 2π 3π sin = cos WF cot = tan DPT 12 12 7 14 2 5π 2 5π 3π 3π sin + cos + tan . cot = 2 144414242 4 4441423 14441442 4441443 UBO UBONT[ 11 DPU UBONT[ 5PQMBNMBSPMBOJLJB¿EBOCJSJOJOTJOÐTÐEJ- ÖRNEK 23 óFSJOJOLPTJOÐTÐOF CJSJOJOUBOKBOUEJóFSJOJOLP- UBOKBOUOBFõJUUJS A 30° 15° 62 6 43 ÖRNEK 20 60a° 45° C TJO+TJO+TJO++TJO B 23 D JöMFNJOJOTPOVDVOVCVMVOV[ TJO=DPT \"#$CJSEJLпHFO [AB] m [BC], % = a, m ( ADB ) TJO=DPT | |%= =6 2 CSWFsin a = 3 EJS m ( DAC ) 15°, AC h 2 TJO=DPT | |:VLBSEBLJWFSJMFSFHÌSF \"% LBÀCSEJS TJO2 1 +TJO2 3° + .... +DPT2 3° +DPT2 1° 2 + 2 = 1 _ 3 b sin a = JTFa = sin 1° cos 1° b b 2 22 bb j 22 + 1 = 45 ]\"%]= 4 3 CS ` 2 2 sin 3° + cos 3° = 1 h 2 + 2 = 1 b b sin 43° cos 43° 21 b sin 45° = bb a 2 ÖRNEK 21 ÖRNEK 24 tan 45° UBOYUBOY= cos 45° + sin 60° LPöVMVOV TBôMBZBO FO LÑÀÑL QP[JUJG Y BÀTO CVMV- JGBEFTJOJOEFôFSJOJCVMVOV[ OV[ H1 1 tan 45° 1 tan 5x = = >cos 45° + >sin 60° 3+ 2 tan x 2 UBOY=DPUY 23 x +Y= Y= 22 x = 2 2.^ 3 - 2 h = =2 3-2 2 3-2 3+ 2 45 32 22. 2 23. 4 3 24. 20. 21. 2 3 - 2 2 2
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ÖRNEK 25 ±M¿ÐMFSJÕ÷aPMBOB¿MBSOUSJHPOPNFUSJLPSBO- MBSOOaB¿TOOUSJHPOPNFUSJLPSBOMBSDJOTJO- 18x =ÖPMEVôVOBHÌSF EFOFõJUJ tan 4x. tan 5x - 3 TJO Õ- a ) = -TJOa TJO( 2Õ+ a ) =TJOa 2 + cos26x + cos23x DPT Õ- a ) =DPTa DPT( 2Õ+ a ) =DPTa JGBEFTJOJOEFôFSJLBÀUS UBO Õ- a ) = -UBOa UBO( 2Õ+ a ) =UBOa DPU Õ- a ) = -DPUa DPU( 2Õ+ a ) =DPUa UBOY=DPUY DPTY=TJOY 64 4471 4 448 cot 5x. tan 5x - 3 -2 = 22 3 2 + 1s4in4434x 2+ c4o4s434x3 1 ±M¿ÐMFSJ Õ- a WF -a PMBOB¿MBSOFTBTËM- ÖRNEK 26 ¿ÐMFSJBZOPMEVóVJ¿JO 0 < a < π PMNBLÑ[FSF TJO -a ) = -TJOa 2 DPT -a ) =DPTa UBO -a ) = -UBOa 2 sind π + a n - cosd π - a n = 3 DPU -a ) = -DPUaPMVS 6 32 ±M¿ÐMFSJ c π ± a m WF f 3π ± a p PMBO B¿MBSO PMEVôVOBHÌSF aBÀTLBÀSBEZBOES 22 sind π + a n = cosd π - a n USJHPOPNFUSJL PSBOMBSOO a B¿TOO USJHPOP- 63 NFUSJLPSBOMBSDJOTJOEFOFõJUJ 2 sind π + a n - sind π + a n = 3 sinc π - a m DPTa sinc π + a m DPTa 6 62 2 2 sind π + a n = 3π cosc π - a m TJOa cosc π + a m mTJOa = sin 2 2 6 23 tanc π - a m DPUa tanc π + a m mDPUa π 2 2 a= cotc π - a m UBOa cotc π + a m mUBOa 6 2 2 #JS \"ÀOO 5SJHPOPNFUSJL %FôFSMFSJOJO %BS \"À $JOTJOEFO:B[MNBT %m/*m 3π 3π sinf - a p = -DPTa sinf + a p = -DPTa ±M¿ÐMFSJÕ÷aPMBOB¿MBSOUSJHPOPNFUSJLPSBO- MBSOOaB¿TOOUSJHPOPNFUSJLPSBOMBSDJOTJO- 2 2 EFOFõJUJ cosf 3π - a p = -TJOa cosf 3π + a p =TJOa TJO Õ- a ) =TJOa TJO Õ+ a ) = -TJOa 2 2 DPT Õ- a ) = -DPTa DPT Õ+ a ) = -DPTa UBO Õ- a ) = -UBOa UBO Õ+ a ) =UBOa tanf 3π - a p =DPUa tanf 3π + a p = -DPUa DPU Õ- a ) = -DPUa DPU Õ+ a ) =DPUa 2 2 cotf 3π - a p =UBOa cotf 3π + a p = -UBOa 2 2 2 π 33 - 3 6
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr ÖRNEK 27 ÖRNEK 29 \"öBôEBLJCPöMVLMBSVZHVOöFLJMEFEPMEVSVOV[ D 18 C \"#$%ZBNVL a. 5PQMBNMBSPMBOB¿MBSOTJOÐTMFSJ UBOKBOU- a 6 14 <\"#><$%> MBSFõJUUJS 12 maA%DCk = a 6 C a + b =jTJOa = 180°–a | |AD =CS | DC | =CS DPTa = A 18 32 14 B | |BC =CS | |AB =CS UBOa = DPUa =PMVS :VLBSEBLJWFSJMFSFHÌSF DPTaEFôFSJLBÀUS B ,PTJOÑTMFSJOFmLPUBOKBOUMBSOB cos^ 180° - a h = 6 C TJO a) =TJO - b) =TJOb 14 DPT a) =DPT - b) = -DPTb 3 -3 UBO a) =UBO - b) = -UBOb - cos a = & cos a = DPU a) =DPU - b) = -DPUb 77 ÖRNEK 30 \"öBôEBLJJGBEFMFSJOFOTBEFIBMMFSJOJCVMVOV[ ÖRNEK 28 a) sinc r - x m C cosc r + x m 2 2 \"öBôEBLJJGBEFMFSJOFOTBEFCJÀJNMFSJOJCVMVOV[ a. sin 50°. tan 70° D sinc r + 2x m E cosc r - 3x m 2 2 sin 310°. cot 160° C. sin 28°. cot 36° F tanc r - 4x m G cotc5x + rm 2 2 sin 152°. cot 144° cos^ 2π - a h. sin^ 7π - a h. cotf 11π - a p g)TJO Õ+ a ) I DPT ÕmB 2 K UBO Õ+B L DPU Õmi ) D l) sinc 2r - x m N cosf 3r + i p tan^ 3π + a h. cos^ 7π + a h. sin^ 11π + a h 2 2 P DPT Õ+ x ) n) UBO ÕmY S cotf 3r + 2i p sin 50° . tan 70° cos 40° . tan 70° Q tanf 3r - 2i p 2 a) = 2 sin 310° . cot 160° ^ - sin 50° h^ – cot 20° h sin 50° . cot 20° DPTY -TJOY =1 sin 50° . cot 20° sin 28° . cot 36° sin 28° . cot 36° C = = - 1 sin 152° . cot 144° sin 28° .^ - cot 36° h DPTY TJOY cos^ 2π - a h. sin^ 7π - a h. cotd 11π - a n DPUY -UBOY 2 -TJOB -DPTB D tan^ 3π + a h. cos^ 7π + a h. sin^ 11π + a h UBOB -DPUi 6 4 44co7s a4 44 8 6 44s7in a44 8 6 4 44ta7n a4 44 8 x i cos^ 2π - a h. sin^ π - a h. cotd 3π - a n - sin - cos 2 =1 2 2 1ta4n4^2π +4 3h. 1c4os4–4^c2oπs+4a 3h. 1s4in4^2π +4a4 3h -UBOY DPTY tan a a4 4a4 – sin a DPUi -UBOi 27. B ,PTJOÑTMFSJOFmLPUBOKBOUMBSOBC TJObmDPTbmUBObmDPUb 34 - 3 28. B C mD 7
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ÖRNEK 31 ÖRNEK 33 \"öBôEBLJÌ[EFöMJLMFSEFOEPôSVPMBOMBSJÀJOCPöLV- ôFLJMEF UFOJT PZOBZBO CJS TQPSDVOVO WVSEVóV UPQ ZFS UVMBSB% ZBOMöPMBOMBSJÀJO:ZB[O[ EÐ[MFNJJMFMJLB¿ZBQZPS a. TJO Õ- a ) =TJOB B C DPT Õ- a ) = -DPTa 32° D tanf 19π + a p =DPUa YA 2 5PQ ZFSF ¿BSQULUBO TPOSB TBQBSBL [\"# ZPMVOV J[- E cotf a – 7π p = -UBOa MJZPS 2 TJO=YPMEVôVOBHÌSF ( Y%AB )BÀTOOLPTJOÑT EFôFSJOJYDJOTJOEFOCVMVOV[ Y TJO Ö- a) = -TJOB D DPT Ö-a) =DPT Ö- a) = -DPTa DPT=DPT + 32°) = -TJO= - x Y tand 19π + a n = tand 3π + a n = - cot a 22 D cotd a - 7π n = cotd π + a n = - tan a 22 ÖRNEK 32 ÖRNEK 34 \"öBôEBLJUBCMPZVVZHVOöFLJMEFEPMEVSBMN DPT= x PMEVôVOBHÌSF TJOEFôFSJOJOYDJOTJOEFOFöJUJ- OJCVMVOV[ TJO=TJO + =TJO DPT=DPT + = -TJO= -x ¦Ì[ÑN ÖRNEK 35 EFSFDF 120° 210° 240° 300° #JS\"#$пHFOJOJOJ¿B¿MBS WA , WB ve XC EJS 2π 3π 5π 7π 5π 4π 5π #VOBHÌSF SBEZBO 3 4 6 3 4 3 3 tan WA · tan WB + XC 22 DPT 1 2 332 1 1 - - --- - 2 JGBEFTJOJOFöJUJOJCVMVOV[ 2 2 222 2 \"+ B +$= 180° 3 2 1 1 2 33 B +$= 180°-\" 2 2 - --- B + C 180° - A TJO 2 2 22 = 2 22 UBO - 3 1 1 3 -3 tand A n. tand 90° - A n - - 3 1 22 DPU - 3 AA tan · cot = 1 3 - - 3 3 11 22 - 33 31. :%:% 33. mY 34. mY1
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr ÖRNEK 36 ÖRNEK 39 x + y = π PMEVôVOBHÌSF C :BOEBLJ ZBSN ¿FN- 2 CFSEF[AB]¿BQUS UBO Y+Z JGBEFTJOJOFöJUJOJCVMVOV[ 10 45° D \"#$пHFO 45° 135° | |AB =CS tan (3^>x +π y h + y) = tand 3· π +yn | |A B CD =CS 2 | |DB =CS 2 :VLBSEBLJ WFSJMFSF HÌSF cota B%DA k EFôFSJOJ CVMV- tand 3π + y n = - cot y OV[ 2 ÖRNEK 37 ]\"$]2 + 242 =2 ]\"$]= 10 0 < a < π WF tanf 3π - a p = 3 PMEVôVOBHÌSF DPU= -1 2 24 cosecd π + a n. cot_ π - a i 2 JGBEFTJOJOFöJUJOJCVMVOV[ tand 3π - a n = cot a = 3 ÖRNEK 40 2 4 DPTFD d π \"õBóEBCJSIBWBTBIBTOOMB[FSõOMBSZMBË[EFõLBSF- A 2 + a n. cot^ π - a h MFSFCËMÐONÐõCJSLFTJUJWFSJMNJõUJS#VMB[FSõOMBSOEBO CJSJEEJS\"WF#OPLUBMBSOEBCVMVOBOV¿BLMBSEPóSVTBM 1 ·^ - cot a h CJSSPUBJ[MFZFSFL$OPLUBTOBHFMJZPSMBS 5k π a sind + a n B 4k 1 4 44co22s a4 443 C 3k B 1 ·d - 3 n = - 5 34 4 5 A 180°–a ÖRNEK 38 b 4k 3k d1 cosf 3π + x p 180°–a b < x <ÕWF 2 = - 3 cotd x - π n 2 2k C 3k 2 \"EBOIBSFLFUFEFOVÀBôOE1öOJMFZBQUôQP[JUJG ZÌOMÑBÀaWF#EFOIBSFLFUFEFOVÀBôOE1öOJMF PMEVôVOBHÌSF YJOBMBCJMFDFôJEFôFSMFSJCVMVOV[ ZBQUôQP[JUJGZÌOMÑBÀbPMEVôVOBHÌSF UBOaUBOb 6 4 44s7in x4 448 EFôFSJLBÀUS cosd 3π + x n -3 2 -3 j - cos x = = cotd x - π n 2 2 4 14442 42443 UBOb = – tan x 3 3 3 cos x = UBOa = - 2 2 x = Y= 330° d 4 n.d - 3 n = - 2 EJS 32 mDPUZ 5 38. m40. m 37. – 4
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ÖRNEK 41 5SJHPOPNFUSJL 'POLTJZPOMBSO \"À %FôFSMFSJOF (ÌSF4SBMBONBT #JS\"#$пHFOJOEF m (WA) > 90°, A ( ABC ) =CSWF %m/*m | |AB =CSPMEVôVOBHÌSF cot XA + cot XB EFôFSJOJCV- 7FSJMFO B¿MBSO USJHPOPNFUSJL EFóFSJ EBS B¿ MVOV[ DJOTJOEFOCVMVOVS h=12C h.8 5SJHPOPNFUSJL EFóFSMFS BSBTOEBLJ CÐZÐLMÐL DxA = 48 LпÐLMÐLJMJõLJTJOJOCFMJSMFOFCJMNFTJJ¿JOUSJHPOP- NFUSJLEFóFSMFSCJSJN¿FNCFSEFFLTFOMFSFUBõ- 2 OS I= 12 cot XA + cot%B .. 2 x x+8 8 2 -+ == 8 B 12 12 12 3 3 ÖRNEK 42 ÖRNEK 43 \"OBMJUJL EÐ[MFNEF \" WF # OPLUBMBSOEB CV- \"öBôEBLJ TBZMBS LÑÀÑLUFO CÑZÑôF EPôSV TSBMBZ- MVOBOJLJLBSODBEPóSVTBMCJSZPMJ[MFZFSFL$ L OPL- O[ UBTOEBCVMVõVZPS a) x =TJO Z=TJO [=TJO 0 PMNBLÐ[FSF CVLBSODBMBSOZFLTFOJJMFPMVõ- C x =TJO Z=TJO [=TJO UVSEVLMBSB¿MBSBSBTOEB m % = 2m ( % )CBóO- ( OCB ) OCA D x =TJO Z=DPT [=UBO UTWBSES E x =TJO Z=UBO [=DPU #VOBHÌSF tan ( % ) EFôFSJOJCVMVOV[ F x =TJO Z=TJO [= -TJO BAC B [=TJO=TJO ÷ÀBÀPSUBZUFPSFNJOEFO x < y <[ OC BC C Z=TJO= -TJO [=TJO= -TJO = & 5 OC = 3 OA olur. < < y <[< x OA AB D Z=DPT=TJO 3k 5k x <[< y y L 2 + 82 = L E Y=TJO=TJO C L= 2 y =UBO= -UBO aa [=DPU=UBO> 1 3k tan ( % ) = - 6 =-2 BAC y < x <[ 3 F Y=TJO= -TJO 5k y =TJO= -TJO [= -TJO= -TJO x < y <[ 35x A(3, 0) B(8, 0) 2 42. m 37 43. a) Y Z [ C Z [ Y D Y [ Z E Z Y [ F Y Z [ 41. 3
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr ÖRNEK 44 ÖRNEK 46 \"öBôEBLJFöJUMJLMFSJOLBÀUBOFTJEPôSVEVS aWFbEBSBÀPMNBLÑ[FSF * DPT<TJO TJOa >TJOb ** UBO<DPU PMEVôVOB HÌSF BöBôEBLJMFSEFO IBOHJMFSJ EBJNB EPôSVEVS *** DPT<TJO * DPUb >DPUa **DPTa >DPTb *7 DPT>TJO *** UBOa >DPUb *7DPTa >DPUb 7 UBO>DPT 7 UBOa >UBOb * TJO<TJO ** UBO<UBO y *** -TJO< -TJO *7 DPT> -TJO cotb 7 UBO> -DPT cota a tanb b tana cosa cosb x ÖRNEK 45 *%PôSV **:BOMö ***#JMFNFZJ[ *7#JMFNFZJ[ 7%PôSV Õ < a < b < 3π PMNBLÐ[FSF ÖRNEK 47 2 a =TFD BöBôEBLJMFSEFOIBOHJMFSJEPôSVEVS * TJOa >TJOb C=DPTFD ** DPTb >DPTa *** UBOa >UBOb D=DPU *7 DPUb >DPUa 7 TJOa >UBOb y E=UBO cota PMEVôVOB HÌSF B C D E BSBTOEBLJ TSBMBNBZ CV- cotb MVOV[ cosa a = sec 1° = 11 cosb tanb = sina cos 1° sin 89° sinb tana 1 b = cosec 1° = x sin 1° D=UBO E=UBO 1 >TJO>TJO 11 1< < sin 89° sin 1° * %PôSV **%PôSV ***:BOMö *7:BOMö UBO44° <UBO= 1 7 <sin– a > <ta+n b ZBOMö C> a >E>D 44. * ** 38 * 7 47. C B E D
5SJHPOPNFUSJL'POLTJZPOMBS*** TEST - 8 1. \"öBôEBLJMFSEFOIBOHJTJ ZBOMöUS 4. TFD=BPMEVóVOBHËSF A) TJO=DPT TJO EFôFSJ BöBôEBLJMFSEFO IBOHJTJOF FöJU- B) UBO=DPU UJS C) TFD=DPTFD D) DPT+DPT= A) -a B) 1 C) a D) a & B E) UBODPU= a 2. tan 10° + cos 20° TJOY-DPTY = cot 80° sin 70° PMEVôVOBHÌSF UBOYEFôFSJLBÀUS JöMFNJOJOTPOVDVLBÀUS A) 1 B) 2 C) 4 D) - 3 E) - 3 \" # $ 1 D) E) 3 3 3 42 2 3. r < x < 3r WFUBOY-= r < x < 3r ve tan x = 1 PMEVôVOBHÌSF 2 22 PMEVôVOBHÌSF DPTYEFôFSJLBÀUS sin x + cos x sin x - cos x A) - 3 B) - 4 C) - 3 5 5 4 JöMFNJOJOTPOVDVBöBôEBLJMFSEFOIBOHJTJEJS E) - 3 D) - 3 2 4 A) - B) - $ % & 1. & 2. B 3. \" 4. B & \"
TEST - 9 5SJHPOPNFUSJL'POLTJZPOMBS*** 1. G Y =DPTY+DPTY+DPTY 4. \"#$ÑÀHFOJOEF PMEVôVOBHÌSF fc π mEFôFSJLBÀUS | |cos(WB) = 3 , cos(XC) = - 3 WF AC =DN 6 25 3 3 3 | |PMEVôVOBHÌSF \"# LBÀDNEJS A) B) + 1 C) - 1 2 2 2 3-1 \" # $ D) E) 3 + 1 D) 8 3 E) 10 3 2 2 2. \"#$ÑÀHFOJOEF UBO B | |tan(WA) = 12 , tan(WB) = 1WF BC =CS PMEVôVOB HÌSF UBO EFôFSJ BöBôEBLJMFS- 5 EFOIBOHJTJOFFöJUUJS | |PMEVôVOBHÌSF \"$ LBÀCSEJS \" # $ 12 2 % & 13 2 \" # B $ 1 a E) - 1 D) -a a 3. A x 24x =ÖPMEVôVOBHÌSF BH C sin 7x. cos 9x cos 5x. sin 3x JöMFNJOJOTPOVDVLBÀUS | | \"#$пHFOJOEF[AH] m [BC], AH =CS \" # C) 3 | |BC =CSWF tan(WB) = 3 tan(XC)ES 2 | |:VLBSEBLJWFSJMFSFHÌSF \"$ = x LBÀCJSJNEJS 5 D) E) \" # $ % & 2 1. & 2. & 3. B 40 4. B & B
5SJHPOPNFUSJL'POLTJZPOMBS*** TEST - 10 1. D C \"#$%LBSF 4. 0NFSLF[MJ¿FNCFSEF |AB| |= AE | % = 5 EJS cot ( AOB ) O 12 AB AE B :VLBSEBLJ WFSJMFSF HÌSF cot ( O%AB ) EFôFSJ LBÀUS :VLBSEBLJWFSJMFSFHÌSF tan ( % ) LBÀUS CDE \" # $ 1 D) 1 E) 2 \" # $ % 2 E) 1 2 33 3 3 2. A \"#$EJLпHFO 3 [ AB ] m [ BC ] H [ BH ] m [ AC ] | |AB =CS a + i =WFsin (2a + 3i) = 5 e | |BC =CS 13 B4 C % PMEVôVOBHÌSF UBOiEFôFSJLBÀPMBCJMJS m ( HBC ) = i :VLBSEBLJWFSJMFSFHÌSF UBOiLBÀUS A) 5 B) 5 C) 12 D) 12 E) 13 13 12 13 5 5 22 32 22 A) B) C) 5 8 3 3 E) 4 D) 3 5 3. ôFLJMEFLJ0\"#EJLпHFOJOEF\" EJS T y 12 A(1, 2) P 8A O B [15 0NFSLF[MJZBSN¿FNCFSF5OPLUBTOEBUF- x OB óFUUJS m ( A%BO ) = a PMEVôVOBHÌSF UBOaLBÀUS | | | |PT =DN PA =DNWFm(T%PB) = aES :VLBSEBLJWFSJMFSFHÌSF TJOaEFôFSJLBÀUS 1 3 2 5 12 A) 3 B) 2 C) 2 D) 1 E) 1 A) B) C) D) E) 4 5 3 3 2 2 2 3 13 13 41 1. \" 2. & 3. D 4. D D &
TEST - 11 5SJHPOPNFUSJL'POLTJZPOMBS*** 1. A :BOEBLJõFLJMEF 4. DPT - <\"#>m<#$> EFôFSJBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS D <#%>m<\"$> | |AB =CS 3 1 % A) B) C) m ( BCA ) = i 2 2 E) - 3 B C D) - 1 2 2 | |:VLBSEBLJWFSJMFSFHÌSF %$ BöBôEBLJMFSEFO IBOHJTJEJS A) sin i B) cot i C) tan i cot i sin i cos i D) cos i & TJOi tan i A TJO - 2. EFôFSJBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS cb \" DPT # -DPT $ TJO D) -TJO & -TJO 30° a 40° B C õFLJMEFWFSJMFOMFSFHÌSF cos 20° PSBOÑÀHFOJO cos 50° LFOBSMBSDJOTJOEFOBöBôEBLJMFSEFOIBOHJTJOF FöJUUJS c a a2 4b2 4c2 A) B) C) D) E) a c bc ac ab \"öBôEBLJMFSEFOIBOHJTJDPTc x - r mEFôFSJOF 2 FöJUEFôJMEJS \" TJOY # -TJO Õ+ x ) 3. \"öBôEBLJMFSEFO IBOHJTJOJO EFôFSJ EJôFSMFSJO- C) - cosf 3r - x p D) cosc r + x m 2 2 EFOGBSLMES \" DPT # TJO - $ -DPT E) cosc r - x m 2 % DPT - & TJO - 1. D 2. B 3. & 42 4. & \" D
5SJHPOPNFUSJL'POLTJZPOMBS*** TEST - 12 1. TJO+TJO - +DPT 4. \"öBôEBLJMFSEFO LBÀ UBOFTJ DPT EFôFSJOF UPQMBNBöBôEBLJMFSEFOIBOHJTJOFFöJUUJS FöJUEFôJMEJS r -TJO A) - 1 B) - 3 C) 1 r DPT 2 2 2 r TJO r TJO D) 3 r DPT - E) 2 \" # $ % & 2. sin 5° . cos 5° . tan 5° \"öBôEBLJMFSEFOLBÀUBOFTJZBOMöUS sin 85° . cos 95° . tan 175° r sinc π - a m = - cos^ π + a h 2 JGBEFTJOJOEFôFSJBöBôEBLJMFSEFOIBOHJTJEJS r DPT Õ+ a ) =DPT Õ- a ) \" DPU # TFD $ r tanf 3π + a p = tan^ π – a h % 2 & - r cot^ π – a h = tanf 3π + a p 2 r sinf 3π - a p = sin^ π + a h 2 \" # $ % & 3. \"öBôEBLJMFSEFO IBOHJTJ DPT - EFôFSJOF cos^ 489π h + sinf – 9π p - tan^ –15π h FöJUEFôJMEJS 2 \" DPT # -DPT $ -DPT cotf 21π p 4 % TJO & -DPT JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" # C) - % - & - 1. $ 2. $ 3. & 43 4. B B $
TEST - 13 5SJHPOPNFUSJL'POLTJZPOMBS*** 1. \"õBóEBË[EFõLBSFMFSEFOPMVõBOõFLJMWFSJMNJõUJS 4. A \"#$пHFO H [AB] m [BC] [BH] m [AC] a AH = 2 b HC 3 BC :VLBSEBLJWFSJMFSFHÌSF cos ( % ) EFôFSJLBÀ- ACB #VOBHÌSF UBOa +UBObEFôFSJLBÀUS US A) 1 # $ 1 % & 5 A) 15 B) 15 C) 10 D) 10 E) 6 2 4 2 7 5 5 75 2. DPT+DPT+DPT++DPT B [AC>¿BQ JöMFNJOJOTPOVDVLBÀUS C 0NFSLF[ O I IA BC =CS \" # 31 $ D) 33 & I IAB =CS 2 2 D :VLBSEBLJWFSJMFSFHÌSF tan aB%DCk EFôFSJLBÀ- US 5 5 C) 12 13 E) 12 A) B) 5 D) 12 13 5 13 3. π < x < πPMNBLÑ[FSF 2 sin2x – cos2x = 2 cos x 3 PMEVôVOBHÌSF DPTYEFôFSJLBÀUS DPU + a ) =C 3 10 2 10 10 PMEVôVOBHÌSF DPU -B EFôFSJOJOCUÑSÑO- A) - B) - C) - EFOFöJUJOFEJS 10 10 10 A) - 1 $ 1 % C & b b b 2 35 2 10 B) -C D) - E) - 5 5 1. D 2. B 3. \" 44 4. B \" $
5SJHPOPNFUSJL'POLTJZPOMBS*** TEST - 14 1. UBOY-DPUY=PMEVôVOBHÌSF 4. UBO Y+ UBO Y- = tan3 x EFOLMFNJOJ TBôMBZBO FO LÑÀÑL Y EBS BÀT LBÀ 1- tan6 x EFSFDFEJS JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" # $ % & A) - 1 B) - 1 C) - 1 18 21 24 D) - 1 E) - 1 30 36 < x < π WF 1 – 1 = –3 2 cos x – 1 cos x + 1 2. a +C=PMEVôVOBHÌSF PMEVôVOBHÌSF UBOYEFôFSJLBÀUS UBO B+C UBO B+C C) A) 2 3 B) D) 2 JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS E) A) - # $ UBOC 3 % DPUB & -UBOBDPUC 3. cot x = 1 PMEVôVOBHÌSF BJMFCEBSBÀMBS a + b = π WF 2 5 tan2 x - sin2 x TJO B+C = - 1 cot2 x - cos2 x 3 JöMFNJOJOTPOVDVLBÀUS PMEVôVOBHÌSF tanc π + b mEFôFSJLBÀUS 2 \" # $ D) 1 E) 1 A) B) 3 2 $ 16 64 D) –2 2 E) - 3 2 1. & 2. & 3. \" 4. \" D D
TEST - 15 5SJHPOPNFUSJL'POLTJZPOMBS*** 1. x =DPT 4. r < x < y < 3r PMNBLÐ[FSF Z=TJO 2 BöBôEBLJMFSEFOLBÀUBOFTJEPôSVEVS [=DPT PMEVôVOB HÌSF BöBôEBLJ TSBMBNBMBSEBO IBO- r DPTY<DPTZ HJTJEPôSVEVS r TJOY<TJOZ r UBOY<UBOZ A) x <Z<[ # Y<[<Z $ Z< x <[ r DPUY<DPUZ r TFDY<TFDZ % Z<[<Y & [<Z< x r DPTFDY<DPTFDZ \" # $ % & 2. a =UBO C=UBO D=UBO \"öBôEBLJMFSEFOIBOHJTJFOCÑZÑLUÑS PMEVôVOBHÌSF B C DBSBTOEBLJTSBMBNBBöB- ôEBLJMFSEFOIBOHJTJEJS \" C< a <D # C<D< a C) a <C<D \" DPT # TJO $ DPU D) a <D<C & D<C< a % UBO & TJO 3. a =TJO C=DPT 0 < a <PMNBLÑ[FSF D=UBO E=DPU a = cosd π – a n 2 PMEVôVOB HÌSF BöBôEBLJ TSBMBNBMBSEBO IBO- HJTJEPôSVEVS b = sind π + a n 2 D=DPU a +Õ A) a <C<D<E # D<E< a <C PMEVôVOBHÌSF B C DBSBTOEBLJTSBMBNBBöB- ôEBLJMFSEFOIBOHJTJEJS $ D<E<C<B % E<D< a <C A) a <C<D # B<D<C $ D< a <C & E<D<C< a % D<C<B & C< a <D 1. \" 2. \" 3. D 4. B $ \"
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ,04÷/·47&4÷/·45&03&.-&3÷ ÷MJöLJMJ,B[BONMBS 11.1.2.2 : ,PTJOÐTUFPSFNJZMFJMHJMJQSPCMFNMFSJ¿Ë[FS 11.1.2.3 : 4JOÐTUFPSFNJZMFJMHJMJQSPCMFNMFSJ¿Ë[FS ,PTJOÑT5FPSFNJ ÖRNEK 1 m63$7 A \"#$пHFO A x 23 2 13 | |AB = 2 3 CS B 10 | |AC = 2 13 CS H b | |C BC =CS c–x hc PMEVôVOBHÌSF m ( WB )LBÀEFSFDFEJS B aC ^ 2 13 h2 = ^ 2 3 h2 + ^ 10 h2 - 2.2 3 .10. cos XB \"#$пHFOJOEFB =C +D -CDPT WA FõJUMJ- = 12 + 100 - 40 2 cos XB óJOF,PTJOÑT5FPSFNJBEWFSJMJS 40. 3 cos XB = 60 :VLBSEBLJ\"#$пHFOJOJO $LFOBSOBBJUZÐL- TFLMJóJIDPMTVO 3 \")$ WF #)$ EJL пHFOMFSJOEF 1JTBHPS UFPSF- 60 3 3 NJOEFOGBZEBMBOBSBL cos b = = = #= 30° 40 3 2 3 2 ID) =C- xn ID) = a- D- x )o 2 CVMVOVS ÖRNEK 2 n WFoEFOLMFNMFSJCJSMJLUF¿Ë[ÐMEÐóÐOEF A C - x = a - D- x )FMEFFEJMJS x3 #VSBEBO 60° 2 C a -D +DY- x =C - x B a =C+D -DYpCVMVOVS | |\"#$пHFOJOEFm (WB) = 60°, AC = CSWF | |BC =CSEJS \")$EJLпHFOJOEF | |:VLBSEBLJWFSJMFSFHÌSF \"# =YLBÀCJSJNEJS cos WA = x & x = b cos WA b #VEFóFSp EFOLMFNJOEFZFSJOFZB[MSJTF a2=C2 +D2 -CDDPT X\" FMEFFEJMJS ,PTJOÑTUFPSFNJOEFO ^ 3 h2 = 2 + 2 - 2.2.x. cos 60° 2 x %m/*m 0 = x2 - 2x + 1 C = a +D -BDcos WB 0 = (x - 1)2 Z x =EJS D = a +C -BCcos XC 47 1. 30° 2. 1
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr ÖRNEK 3 ÖRNEK 6 A x \"#$пHFO A 8 ABC üçHFO 2 5 <%&>m<#$> 2 D D | |BD =CS | |AD =CS | |DA =CS 4 | |BD =CS 3x | |BE =CS | |C BE =CS | |C EC =CS a a B3 E B 4 E3 | |EC =CS | |AC =CS | |:VLBSEBLJWFSJMFSFHÌSF \"$ LBÀCJSJNEJS | |:VLBSEBLJWFSJMFSFHÌSF %& =YLBÀCJSJNEJS ,PTJOÑTUFPSFNJOEFOY2 =2 + 82 -DPTa \"#$ÑÀHFOJOEFLPTJOÑTUFPSFNJOEFO =2+ 72 -DPTa 2 3 x2 = 28 j x = 2 7 x2 = 100 - 8 · DPTa = 10 1 4 cos a = ÖRNEK 4 7 D 12 C \"#$%QBSBMFMLFOBS #%&ÑÀHFOJOEFLPTJOÑTUFPSFNJZB[BSTBL 120° x x2 = 32 + 42-DPTa 6 4 m ( A%DC ) = 120° x2= - 2.3.4 1 E F | |AE =CS 7 4 120° 2 A x = 151 & x = 151 G 77 | |DE =CS ÖRNEK 7 B | |CF =CS | | | |DC =CSPMEVôVOBHÌSF &' =YLBÀCJSJNEJS A ôFLJMEF &'(ÑÀHFOJOEFLPTJOÑTUFPSFNJOJZB[BSTBL 10 | |AC =CS x2 = 122+ 22 -DPT | |BC =CS x C 5 D | |CD =CS a | |CE =CS a x2= 144 + 4 - 2.2.12. d - 1 n 12 7 2 B 6 x2 = 172 x = 2 43 E ÖRNEK 5 | |DE =CSWF[ AE ] a[ BD ] = { C }EJS \"#$ÑÀHFOJOJOLFOBSV[VOMVLMBSBSBTOEB | |#VOBHÌSF \"# = x LBÀCJSJNEJS D = a +C -BC \"#$WF$%&ÑÀHFOMFSJOEFLPTJOÑTUFPSFNJZB[MSTB CBôOUTPMEVôVOBHÌSF m ( WC )LBÀEFSFDFEJS 72 =2 +2 -DPTa ,PTJOÑTUFPSFNJOEFOD2 = a2 +C2 -BCDPT X$ 1 a2 +C2 -BCDPT X$ = a2+C2 -BC 2 ab . cos XC = ab 12 =DPTa jDPTa = cos XC = 1 j XC = 60°EJS 5 2 x2 = 102+ 122 -DPTa x2= 100 + 144 - 2.10 . 12 . 1 5 x2 = 244 - 48 j x2=j x = 14 3. 2 7 4. 2 43 48 151 7 14 7
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