www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ÖRNEK 8 ÖRNEK 10 H 1K 1 G ôFLJMEFLJLÐQUF D C \"#$%LJSJõMFSEËSUHFOJ a 5 |),| = |,(| WF 180°–a 3 2 2 | |AB =CS 5 2x a | |BC =CS EF % | |AD =CS 2 m ( BKD ) = a ES A6 B | |CD =CS 2 2D C 22 A2B :VLBSEBLJWFSJMFSFHÌSF DPTaEFôFSJLBÀUS | |:VLBSEBLJWFSJMFSFHÌSF \"$ =YLBÀCJSJNEJS \"#$%LJSJöMFSEÌSUHFOJPMEVôVJÀJO ,ÑQÑOCJSLFOBSOCSLBCVMFEFMJN m ( % ) + m ( % ) = 180° EJS ])%]2 + |),]2 =],%]2 ],']2 +]'#]2 =],#]2 ABC ADC \"#$WF\"%$ÑÀHFOMFSJOEFLPTJOÑTUFPSFNJZB[MSTB 22 + 12 =],%]2 5 2 + 22=],#]2 x2 = 22+2 -DPTa ],%]= 3 =],#] x2= 22+ 32 - 2.2.3. 1c4os4^4412804° -4 4a43h ,%#ÑÀHFOJOEFLPTJOÑTUFPSFNJZB[BSTBL – cos a 2 h2 = ^ h2 2 40 - DPTa = 13 + DPTa ^2 5 + 3 - 2.2. 5 cos a 27 =DPTa 8 = 5 + 9 - 6 5 cos a 5 cos a = 1 & cos a = 3 1 DPTa= 5 4 x2= 22+2 - 2. =cos a x2 = 4 +- 18 3 4 x2= 22 j x = 22 ÖRNEK 9 ÖRNEK 11 A8 D 7 76 #JS\"#$пHFOJOEF B CWFDпHFOJOLFOBSV[VOMVLMB- SES 8 5a B E13 C C+D+B C+D- a ) =CD PMEVôVOBHÌSF m (XA)LBÀEFSFDFEJS | | | |\"#$%CJSZBNVL [AD] // [ BC ], AD =CS BC =CS | | | |CD =CSWF AB =CSEJS C+D+F C+D- a) =CD C2 +D2 +CD - a2 =CD :VLBSEBLJWFSJMFSFHÌSF TJOaEFôFSJOJCVMVOV[ a2 =C2 +D2 +CD \"#$ÑÀHFOJOEFDPTJOÑTUFPSFNJZB[BSTBL %&$ÑÀHFOJOEFDPTJOÑTUFPSFNJZB[MSTB a2 =C2 +D2 -CDDPT X\" =+-DPTa C2+D2 +CD=C2+D2 -CDDPT X\" DPTa = 12 CD= -CDDPT X\" - 1 = cos XA 1 cos a = 2 XA = 120°EJS 5 5 26 sin a = a 1 5 26 8. 22 26 1 10. 11. 120 5 5
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr 4JOÑT5FPSFNJ %m/*m TANIM & 1 BCTJO XC A^ ABC h = 2 A = 1 CDTJn WA 2 cb = 1 BDTJO WB ha 2 BH C ÖRNEK 12 a A :VLBSEBLJ\"#$пHFOJOJO \"LFOBSOBBJUZÐL- 6 32 \"#$пHFO TFLMJóJIaPMTVO A_ ABC i = a.ha ES B 45° | |AB =CS 2 C | |AC = 3 2 CS \"#)EJLпHFOJOEF m (XC) = 45° 0 < m (WB ) < 90° sin WB = ha & ha = c. sin WB PMVS :VLBSEBLJWFSJMFSFHÌSF #BÀTLBÀEFSFDFEJS c A_ ABC i = 1 a.ha = 1 ·a.c. sin WB \"#$ÑÀHFOJOEFTJOÑTUFPSFNJZB[MSTB 2 2 6 32 #FO[FSõFLJMEF = A_ ABC i = 1 .b.c. sin WA 2 sin 45° sin XB A_ ABC i = 1 .a.b. sin XC CVMVOVS 6 32 2 = 0IBMEF 2 sin XB 1 a.b. sin XC = 1 b.c. sin WA = 1 a.c. sin WB 2 2 22 12 sin XB = 6 sin XB = 1 ise XB = 30°EJS FõJUMJLMFSJOJ 2 JMF¿BSQBSTBL 2 abc ÖRNEK 13 sin XC = sin WA = sin WB cab A \"#$пHFO PMVS#VSBEBO | |AC =CS abc 4 == | |BC =CS sin XA sin XB sin XC CVMVOVS B 6 60° m (XC ) = 60° C :VLBSEBLJWFSJMFSFHÌSF ÑÀHFOJOBMBOLBÀCS2 EJS A^ ABC h = 1 j A^ ABC h = 6 2 2 ·4.6.>sin 60° 3 br 3 2 12. 30° 13. 6 3
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ .0%·- 11. SINIF ÖRNEK 14 ÖRNEK 17 #JS\"#$ÑÀHFOJOEF ,FOBSV[VOMVLMBSCS CSWFCSPMBOÑÀHFOJO BMBOLBÀCS2EJS & | |A^ ABC h = 14 2 cm2, m (WB) = 45° ve BC = 8 cm A =+-DPTa | |PMEVôVOBHÌSF \"# LBÀDNEJS DPTa= 1 7 cos a = 6 5 1 ·a.c. sin XB = 14 2 a 2 B 5C 12 3 8 .c = 14 2 jD=CS 22 26 5k sin a = a 5 k ÖRNEK 15 2 6k & 1 ·6.5.<sin a A^ ABC h = 2 D C ABCD paralelkenar 26 6 |BC | = 5 br 5 | |5 CD = 6 br & m (WA ) = 60° A^ ABC h = 6 6 5 6 B ÖRNEK 18 60° A :VLBSEBLJ WFSJMFSF HÌSF HÌSF QBSBMFMLFOBSO BMB- x A ABC üçgen OLBÀCS2EJS 105° 30° |AC | = 4 br A^ ABD h = 1 B 4 ·5.6. sin 60° m (WA ) = 105° 2 45° m (WB ) = 30° C 13 | |:VLBSEBLJWFSJMFSFHÌSF \"# =YLBÀCSEJS ·5.6. 22 & 15 3 CSEJS A^ ABD h= & &2 x4 A^ ABD h = A^ BDC hj\" \"#$% =15 3 CS2 = ÖRNEK 16 sin 45° sin 30° | |A AE = 4 br x4 = j x = 4 2 21 22 4a 6 | EB | = 3 br E | AD | = 6 br ÖRNEK 19 3 D && | | | |#JS\"#$ÑÀHFOJOEF \"# =CS \"$ =CSWF B x A^ ABD h = A^ AEC h | |#$ =CSPMEVôVOBHÌSF C sin WA + sin WB sin XC | |:VLBSEBLJWFSJMFSFHÌSF %$=YLBÀCSEJS JGBEFTJOJOEFôFSJLBÀUS & 1 ·7.6. sin a A^ ABD h = 2 & 1 ·4.^ 6 + x h. sin a A^ AEC h = 2 sin A + sin B sin C = a+b C && 1 1 A^ ABD h = A^ AEC h & ·7.6. sin a = ·4.^ 6 + x h. sin a 22 F15 sin XA + sin XB a+b =+YjY= 9 = =3 2 sin XC 5c 5 9 51 6 6 4 2 3 715 3 2
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr ÖRNEK 20 ÖRNEK 22 A A \"#$пHFO \"#$%LJSJõMFSEËSUHFOJ | |AB =CS x 45° | |CD = 3 2CS ab | |AC =CS Ba 6 |BD| = |DC| % 9 m ( ACB ) 180°–a = 30° D % m ( B%AD ) = a m ( CAD ) B DC 32 = 45° 30° % m ( DAC ) = b C :VLBSEBLJWFSJMFSFHÌSF sin a EFôFSJLBÀUS sin b | |:VLBSEBLJWFSJMFSFHÌSF \"# =YLBÀCJSJNEJS && \"#$%LJSJöMFSEÌSUHFOJPMEVôVJÀJO A^ ABD h = A^ ADC h \"#% WF \"%$ ÑÀHFOMFSJOEF TJ- OÑTMÑPMBOGPSNÑMÑOÑZB[BSTBL m ( % ) + m ( % ) = 180° EJS 11 ABC ADC ·6. AD . sin a = ·9. AD . sin b \"#$WF\"%$ÑÀHFOMFSJOEFTJOÑTUFPSFNJOJZB[BSTBL 22 sin a 9 3 AC x AC == sin b 6 2 32 = ve = sin a sin 30° sin 45° 1s4in4^41820°4-4a43h ÖRNEK 23 sin a #JS\"#$пHFOJOEFB C DпHFOJOLFOBSV[VOMVLMBSES x 32 j x =CS sin WA + sin WB = 3 sin XC WFB-D=-C = >sin 30° >sin 45° 1 2 2 PMEVôVOBHÌSF DLBÀCJSJNEJS 2 abc a+b = c ==& sin XA sin XB sin XC sin XA + sin XB sin XC ÖRNEK 21 a +C=D+JTF A \"#$пHFO 2c + 4 = c 30° m ( B%AD ) = 30° 3 sin XC sin XC 4 % D+ 4 =Dj 4 =DEJS m ( DAC ) 6 = i | |AB =CS ÖRNEK 24 | |AC =CS | | | |#JS\"#$пHFOJOEF AB =CS CB =CSWF B DC m ( A%CB ) - m ( B%AC ) = 90°EJS |BD| = |DC| :VLBSEBLJWFSJMFSFHÌSF UBOiOOQP[JUJGEFôFSJLBÀ- #VOBHÌSF cot ( % ) EFôFSJLBÀUS US BAC \" \"#% =\" \"%$ PMEVôVJÀJO m ( B%AC ) = a PMTVOm ( A%CB ) = 90° + a PMVS 11 . sin i TJOÑTUFPSFNJOEFO ·4. AD .>sin 30° = ·6. AD B 2 2 1 8 6 2 68 90°+a = C sin a 1s4in4^4c9o20s°a4+4a4 3h 1 3k k = sin i cos a 8 i 3 j= 22k 2 sin a 6 a tan i = 4 4 A jDPUa = 3 20. 3 2 3 4 21. 22. 23. 4 24. 23 4
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ÖRNEK 25 ÖRNEK 26 #JSDFQUFMFGPOVõJSLFUJ NÐõUFSJMFSJOFEBIBJZJIJ[NFUWF- ¶¿HFOõFLMJOEFLJCJSQJTUJO\"WF#LËõFMFSJOEFCVMVOBO SFCJMNFLJ¿JOWFSJDJTBZTOBSUUSNBZQMBOMZPS:BQMBO JLJBSB¿$LËõFTJOFHJEFDFLMFSEJS IFTBQMBNBMBSTPOVDVWFSJNJOFOZÐLTFLEÐ[FZEFPMNB- T J¿JO WFSJDJMFSJO пHFO õFLMJOEFLJ BSB[JOJO LËõFMFSJOF A ZFSMFõUJSJMNFTJHFSFLUJóJCVMVOVZPS BC | | | |AC =LN m % =LN BC ( BAC ) = a WF A % B m ( ABC ) = 90° + a PMEVôVOB HÌSF UBOa EFôFSJ LBÀ- US A 16 18 a = sin a sin^ 90° + a h 8 44 C 90°+a 89 B6 sin a = cos a 8 UBna = C 9 7FSJDJMFSõFLJMEFHËSÐMEÐóÐHJCJ\"#$пHFOJOJOLËõFMFSJ- ÖRNEK 27 OFZFSMFõUJSJMJZPS | | | |\"# = 40 7LN m % 45°PMEV- 2 LN \"$ = ( BAC ) = :BOEBCJSLFOBSNPMBO | |ôVOBHÌSF #$ FOÀPLLBÀLNEJS A LÐQ õFLMJOEF CJS BTBOTËS LBCJOJ HËSÐMNFLUFEJS ,B- A 5 CJOJO ÐTU LËõFTJOEF CVMV- 45° 52 OBO ËSÐNDFL ZFSEFLJ ZJ- 40 2 ZFDFóJ GBSL FEJZPS WF IB- 1 SFLFUF HF¿JZPS ¶TU LËõF- 70 C EFOËODFLBCJOJOBMUBZS- B 1 2 x 1B C UOOPSUBOPLUBTOB EBIB TPOSB ZJZFDFóJ BMQ EJóFS \"#$ÑÀHFOJOEFLPTJOÑTUFPSFNJZB[MSTB BZSUO PSUB OPLUBTOB VMBõZPS %BIB TPOSB ÐTU LËõFZF x2 =702+ ^ 40 2 h2 - 2.70 . 40 2 >cos 45° HFSJEËOÐZPS x =LNEJS 2 ²SÑNDFôJO J[MFEJôJ SPUB \"#$ ÑÀHFOJ öFLMJOEF PMEV- 2 % ôVOBHÌSF cos ( BAC )EFôFSJLBÀUS ,PTJOÑTUFPSFNJOEFO ^ 2 h2 = ^ 5 h2 + ^ 5 h2 - 2. 5. % 5 . cos ( BAC ) 2 = 10 - 10 % cos ( BAC ) cos ( B%AC ) = 4 5 8 4 27. 95
11. SINIF .0%·- 53÷(0/0.&53÷ www.aydinyayinlari.com.tr )(1/m6(/(5m1(<q1(/m. ÖRNEK 30 ABC üçgen 4JOÑT5FPSFNJOJO¦FWSFM¦FNCFSJMF÷MJöLJTJ A % = 68° 60° m ( ABC ) %m/*m % = 52° Bir ABC üçgeninde m ( ACB ) A | |52° BC = 12 br 68° 12 B C c b O C :VLBSEBLJ WFSJMFSF HÌSF \"#$ ÑÀHFOJOJO ÀFWSFM R ÀFNCFSJOJOTOSMBEôBMBOLBÀÖCSEJS Ba a = b = c = 2R dir. a sin (WA) sin (WB) sin (XC) = 2R 3¥FWSFM¿FNCFSJOZBS¿BQ sin A 12 = 2R j3= 4 3 CS 3 2 Ö3=Ö^ 4 3 h2 =Ö ÖRNEK 28 ÖRNEK 31 A | |\"#$ÑÀHFOJOEFm (XB)=WF \"$ =DNPMEV- 12 46 C a ôVOBHÌSF \"#$ÑÀHFOJOJOÀFWSFMÀFNCFSJOJOZBS- ÀBQLBÀDNEJS HB b = 2R sin (XB) 4 = 2R & R = 4 >sin 30° 1 2 ÖRNEK 29 ôFLJMEF\"#$пHFOJWF¿FWSFM¿FNCFSJWFSJMNJõUJS ,FOBSV[VOMVLMBSCS CSWFCSPMBOÑÀHFOJO | | | | | |[AH] m [BC], AB = 6 br, AH = 4 br ve AC = 12 br ÀFWSFMÀFNCFSJOJOZBSÀBQLBÀCJSJNEJS dir. :VLBSEBLJWFSJMFSFHÌSF ÀFNCFSJOZBSÀBQLBÀCJ- SJNEJS 4 A 4 12 = 2R = 2R a 4 B1 15 sin a sin a 4 D 1C = 2R 12 15 = 2R 4 4 8 15 6 R= 3=CS 15 8 15 54 4 15
,PTJOÑT5FPSFNJ4JOÑT5FPSFNJ TEST - 16 1. A D ôFLJMEF 4. ôFLJMEFCJSLFOBSCSPMBOLÐQWFSJMNJõUJS,OPLUB- 34 | |AB =CS T\"%)&LBSFTJOJOBóSMLNFSLF[JEJS HG 4C | |x CE =CS | |4 BC =CS EF 6 | |B AC =CS | |E CD =CS K C D L [ BD ] a [ AE ] = { C }EJS AB | |:VLBSEBLJWFSJMFSFHÌSF %& =YLBÀUS % FKL A) 30 B) 34 $ =| | | |BLLCPMEVôVOB HÌSF cos ( ) EFôFSJ LBÀUS D) 2 15 E) 2 17 A) 15 B) 7 C) 30 10 5 10 D) 15 2. 5 3 30 E) 5 Dx 2 C A 60° 6 6 B | | | |\"#$%LJSJõMFSEËSUHFOJ AB = BC =DN #JS\"#$ÑÀHFOJOJOLFOBSV[VOMVLMBSBSBTOEB | |AD =DNWF m ( % ) =EJS a =C +D + CD BCD CBôOUTPMEVôVOBHÌSF m (XA)LBÀEFSFDFEJS | |:VLBSEBLJWFSJMFSFHÌSF %$ =YLBÀDNEJS \" # $ % & \" # $ % & 3. A \"#$пHFO ,FOBSV[VOMVLMBSCS CS CSPMBOÑÀHFOJOJÀ 2 [ DE ] m [ BC ] BÀMBSOEBO LPTJOÑT EFôFSJ FO CÑZÑL PMBO BÀ- D OOTJOÑTEFôFSJLBÀUS | |AD =DN x 3 B4E | |DC =DN | |2 C BE =DN | | | |&$ =DNPMEVôVOBHÌSF \"# =YLBÀDN 7 7 5 A) B) C) EJS 4 3 4 A) 21 B) 22 C) 23 5 5 D) E) 3 7 D) 2 6 E) 5 1. B 2. & 3. \" 4. $ D \"
TEST - 17 ,PTJOÑT5FPSFNJ4JOÑT5FPSFNJ A 4. A 1. 5 5 2 120° B4 C D | | | |\"#$пHFO AB =DN BC =DN B3 C % = ôFLJMEF [ BC ] m [ AC ], [ AB ] m [ AD ] m ( ABC ) :VLBSEBLJWFSJMFSFHÌSF A^ & h LBÀDN2EJS | | | | | |BC =DNWF AB = AD =DNEJS ABC & A^ ACD hLBÀDN2EJS :VLBSEBLJWFSJMFSFHÌSF \" # $ 2 3 $ 15 2 \" # D) 4 3 E) 6 2 D) 25 E) 25 4 2 2. D 3 C [DC] m [AD] A \"#$пHFO A 5 [AB] m [AD] | |AB =CS | |DC =DN 6 23 | |B BC =DN | |AC = 2 3 CS sin (WB) = 4 5 BC A^ & h FOCÑZÑLEFôFSJOJBMEôOEBsin ( % ) ABC BCA EFôFSJLBÀPMVS :VLBSEBLJWFSJMFSFHÌSF \"#$%EJLZBNVôVOVO A) 1 B) 2 3 BMBOLBÀDN2 EJS 2 C) D) & \" # $ % & 5 2 | |3. #JS\"#$пHFOJOEFm (WA) = 60°, BC = a, #JS\"#$пHFOJOJOJ¿B¿MBS XA, XB, XC CVB¿MBSO | | | |AC =C AB =DEJS HËSEÐLMFSJ LFOBSMBSO V[VOMVLMBS TSBTZMB B C D D- + C- = EJS sin (WB) = 3 sin (WA) - 2 sin (XC) C+D= 5a - PMEVôVOBHÌSF A^ & h LBÀCS2EJS PMEVôVOBHÌSF BLFOBSLBÀCJSJNEJS ABC \" # 6 3 C) 10 3 D & \" # $ % & 1. $ 2. D 3. B 4. \" D $
,PTJOÑT5FPSFNJ4JOÑT5FPSFNJ )(1/m6(6m7(67m 1. #JS\"#$ÑÀHFOJOEF 4. ,FOBSV[VOMVLMBSCS CSWFCSPMBOÑÀ- m (WA) = 120°WF BC = 6 3 CS HFOJOÀFWSFMÀFNCFSJOJOÀBQOOJÀUFôFUÀFN- CFSJOJOZBSÀBQOBPSBOLBÀUS PMEVôVOBHÌSF ÑÀHFOJOÀFWSFMÀFNCFSJOJOZB- A) 25 B) 17 C) 17 D) 25 E) 22 SÀBQLBÀCSEJS 6 6 3 33 \" # $ % & 2. #JS\"#$ÑÀHFOJOJOJÀBÀMBS XA , XB , XC CVBÀMB- ôFLJMEF 0 NFSLF[MJ ¿FNCFS WF \"#$ пHFOJ WFSJM- SO HÌSEÑLMFSJ LFOBSMBSO V[VOMVLMBS TSBTZMB NJõUJS B C DPMNBLÑ[FSF A TJOA -TJOB = 2 WF 3 O 4 B -C = B6 PMEVôVOBHÌSF \"#$ÑÀHFOJOJOÀFWSFMÀFNCFSJ- OJOZBSÀBQLBÀCSEJS C A) 1 # $ 3 % & 2 2 | | | |BC =CSWF OB =CSEJS :VLBSEBLJWFSJMFSFHÌSF sin ( B%AC )EFôFSJLBÀ- US A) 1 B) 2 3 D) 2 E) 1 3 3 C) 5 4 4 3. A 12 10 a 2a A ôFLJMEFLJ\"#$пHF- B C OJOJO ¿FWSFM ¿FNCF- O SJOJO ZBS¿BQ DN %% 6 EJS \"#$пHFO m ( ABC ) = a, m ( ACB ) = 2a B C | | | |AB =CSWF AC =CS EJS :VLBSEBLJWFSJMFSFHÌSF \"#$ÑÀHFOJOJOÀFWSFM ÀFNCFSJOJOÀBQLBÀCSEJS (sin2a = 2sina.cosa) \"#$ ÑÀHFOJOJO ÀFWSFTJ DN PMEVôVOB HÌSF sin (XA) + sin (XB) + sin (XC)EFôFSJLBÀUS A) 25 B) 25 C) 13 D) 17 E) 15 4 2 2 22 # 2 4 5 7 3 C) D) E) \" 3 4 4 1. B 2. $ 3. B 4. D $ D
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr 53÷(0/0.&53÷,'0/,4÷:0/-\"3*/(3\"'÷,-&3÷ ÷MJöLJMJ,B[BONMBS 11.1.2.4 : 5SJHPOPNFUSJLGPOLTJZPOMBSOHSBGJLMFSJOJ¿J[FS 1FSJZPUWF1FSJZPEJL'POLTJZPO ÖRNEK 2 7$1,0%m/*m G Y GPOLTJZPOVOVOQFSJZPEVPMEVôVOBHÌSF g ( x ) =G Y+ 1 ) +GPOLTJZPOVOVOQFSJZPEVOVCV- G\"Z B, rx `\"J¿JOG Y+ T ) =G Y FõJUMJóJ- MVOV[ OJTBóMBZBOQP[JUJG5SFFMTBZMBSOEBOFOLпÐ- óÐOFG Y GPOLTJZPOVOVOFTBTQFSJZPEVEFOJS Tf^ x h 4 G Y GPOLTJZPOVOB JTF QFSJZPEJL GPOLTJZPO Tg^ x h = = =2 EFOJSrx `3WFL`;J¿JO 2 2 sin (x + k.2r) = sin x 4 oldu€undan, T = 2r cos (x + k.2r) = cos x tan (x + k.r) = tan x 4 oldu€undan, T = r cot (x + k.r) = cot x L B C`3WFLáWFBáPMNBLÐ[FSF ÖRNEK 3 % G Y =LTJOn ( ax +C G Y GPOLTJZPOVOVO QFSJZPEV PMEVôVOB HÌSF H Y =LDPTn ( ax +C GPOLTJZPOMBSOEB ff 2x - 3 pGPOLTJZPOVOVOQFSJZPEVOVCVMVOV[ OUFLEPóBMTBZJTF T = 2r 3 a O¿JGUEPóBMTBZJTF T = r G Y JOQFSJZPEVOB51 a 2x - 3 % G Y LUBOn ( ax +C fd 3 n OJOQFSJZPEVOB52EJZFMJN H Y LDPUn ( ax +C GPOLTJZPOMBSOEB T= T 6 &T = =9 OEPóBMTBZJTF T = r 1 a 22 22 33 %m/*m ÖRNEK 4 G Y GPOLTJZPOVOVO QFSJZPEV 5 JTF G BY + C G Y GPOLTJZPOVOVO QFSJZPEV H Y GPOLTJZPOV- nun QFSJZPEV GPOLTJZPOVOVOQFSJZPEV T ES a G =WFH = ÖRNEK 1 PMEVôVOB HÌSF H - G JGBEFTJOJO FöJUJOJ CVMVOV[ 3Z3 G Y =Y+ GPOLTJZPOVOVOQFSJZPEJLPMVQPMNBEôOCVMVOV[ F10 6 4 758 4 8 G Y =G Y+5 PMNBMES f^ 6 h = f (6 + 4) = . . . f (6 + 4.13) = 12 G Y+ T ) = 2(x + T) + 1 = 2x + 2T + 1 F20 g^ 13 h = g (1 3 + 7) = . . .g (11434+274.1483) = 4 >2x + 1 = 124x4+22T4+413 & T = 0 f^ x h f^ x + T h 139 T ` R+ PMEVôVOEBOG Y QFSJZPEJLGPOLTJZPOEFôJMEJS 2 . 4 - 12 = -4 1. QFSJZPEJLGPOLTJZPOEFôJM 2. 2 3. 4. m
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ÖRNEK 5 ÖRNEK 7 \"öBôEBLJGPOLTJZPOMBSOQFSJZPUMBSOCVMVOV[ G Y =TJO Y+ +DPT Y- a) Z=TJO Y+ GPOLTJZPOVOVOQFSJZPEVOVCVMVOV[ C Z=DPT Y- D Z= 1 DPTd x + π n G Y =TJO Y+ 1) +DPT Y- 3) [[ 2 34 E Z =TJO f 2x - 1 + π p 2π 2π T= T= 3 13 22 2π π n = 2π (T1 52)PLFL = d 3 okek 2π 2π a) T = = 33 2π C T = = 2π 1 D T = π = 3π 1 ÖRNEK 8 3 f^ x h = 3 cos2 c x + 50° m + tan3^ x - 20° h 2 E T = π 3π = GPOLTJZPOVOVOFTBTQFSJZPEVOVCVMVOV[ 22 3 3 2 d x + 50° n + 1ta4n434^ 2xπ-42404° 3h 44444 3 cos T = =π 1 4 4 4 442π2 T= = 2π 21 11 2 ÖRNEK 6 TG Y =0,&, 51 52) =Ö \"öBôEBLJGPOLTJZPOMBSOQFSJZPUMBSOCVMVOV[ ÖRNEK 9 a) Z=UBO5 Y C Z=DPUd - x + π n L` R+PMNBLÑ[FSF G Y =DPT LY+ - 1 35 D Z=DPU f 3 - x p + 4 2 2 GPOLTJZPOVOVO CJS LÌLÑ PMEVôVOB HÌSF CV LÌL- E Z =UBO f x - 1 p + 2 UFOEBIBCÑZÑLPMBOFOLÑÀÑLLÌLÑOÑCVMVOV[ 3 ππ a) T = = 22 C T = π = 3π 1 - 3 D T = π = 2π 1 π - 'POLTJZPOVOQFSJZPEV= 2 k E) T = π = 3π %PMBZTZMBCJSTPOSBLJLÌLÑ 1 π 2k + π 3 2° + = EJS kk 2π 3π π 7. Ö 2k + π a) C ÖD ÖE a) C ÖD ÖE Ö 8. Ö 3 22 k
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr 4JOÑTWF,PTJOÑT'POLTJZPOMBSOO(SBGJLMFSJ D Z=+DPTY 7$1,0%m/*m y =TJOYGPOLTJZPOVOVOHSBGJôJ D y 2 Õ Õ x 0 2 Õ 2 Õ mÕ Õ OÕ Õ Õ x y = sinx 0 1 0 –1 0 2 2 2 – Õ – y 2 1 mÕ – Õ O Õ Õ Õ Õ x 2 22 –1 E Z=DPT Y+Õ y =DPTYGPOLTJZPOVOVOHSBGJôJ E y Õ Õ 1 x 0 2 Õ 2 2Õ OÕ y = cosx 1 0 –1 0 1 2 y – 3Õ –1 Õ 1 2 –Õ – Õ Õ 3Õ 2 2 Õ OÕ Õ x 2 2 2 mÕ – Õ 2Õ –1 ÖRNEK 10 F y = - 3 cosd x n + 2 2 \"öBôEBLJGPOLTJZPOMBSOHSBGJLMFSJOJÀJ[JOJ[ a) Z= -DPTY F a) y mÕ y 2 –2Õ 5 Õ O Õ Õ Õ O Õ – Õ 22 –1 2 –2 x mÕ – Õ 2 C Z=TJOY | |G Z=TJO x C y F y 1 mÕ 1 – Õ Õ mÕ O Õ 4 4 Õx –1 – Õ – Õ OÕ Õ x 4 2 4 2 Õ –1
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF 5BOKBOUWF,PUBOKBOU'POLTJZPOMBSOO(SBGJLMFSJ ÖRNEK 12 7$1,0%m/*m y y =UBOYGPOLTJZPOVOVOHSBGJôJ 5 2 O Õ Õx Õ 6 3 –1 12 O :VLBSEBLJHSBGJLG Y =BTJO CY +DGPOLTJZPOVOB BJUPMEVôVOBHÌSF B+C+DEFôFSJLBÀUS y =DPUYGPOLTJZPOVOVOHSBGJôJ mÕ Õ G =PMEVôVOEBO x =JÀJOG =DjD= 2 O Õ Õ G Y =BTJO CY +PMVS mÕ -ãG Y ã -ãBTJO CY +ã -ãBTJO CY ã a =ZBEBB= -PMVS (SBGJLUFOBOOPMEVôVBOMBöMS G Y =TJO CY + 2 fd π n = 5 jC=PMVS 12 a =C= D= 2 3 ++ 2 = 11 ÖRNEK 11 y ÖRNEK 13 y – Õ Õ 1 Õ x 3 4 4 4 2 O – Õ 4 1 x O Õ Õ Õ Õ 42 4 :VLBSEBLJHSBGJLZ= a +UBO CY GPOLTJZPOVOBBJU (SBGJLUF WFSJMFO GPOLTJZPO BöBôEBLJMFSEFO IBOHJTJ- PMEVôVOBHÌSF B+CEFôFSJLBÀUS OFBJUUJS ππ \" DPTY+ # TJOY+ $ TJOY+ T = = & b = 2 jC= HSBGJôJOöFLMJOEFO % TJOY+ & DPTY+ b2 x =JÀJOZ= a j a = 1 D a +C= 3 11. 3 12. 11 13. D
TEST - 18 5SJHPOPNFUSJL'POLTJZPOMBSO(SBGJLMFSJ 1. y =G Y GPOLTJZPOVOVOQFSJZPEVJTF G Y =+UBO Y+ 5 ) g_ x i = ff 2x - 3 p GPOLTJZPOVOVO QFSJZPEV BöBôEBLJMFSEFO IBO- 3 HJTJEJS GPOLTJZPOVOVQFSJZPEVLBÀUS \" Õ # π C) π D) π E) π 2 4 6 7 \" # $ % & 2. G Y GPOLTJZPOVOVO QFSJZPEV H Y GPOLTJZPOV- G Y =TJO Y+ +UBO Y- OVOQFSJZPEVEJS G 3 ) =WFH =PMEVôVOBHÌSF GPOLTJZPOVOVO QFSJZPEV BöBôEBLJMFSEFO IBO- f_ 28 i - g_ 34 i HJTJEJS _ fog i_ 10 i \" Õ # 3π $ Õ % π E) π 2 2 4 JGBEFTJOJOFöJUJLBÀUS A) - 4 B) - 2 C % 2 E) 4 9 9 9 9 3. ôFLJMEFG Y =TJO BY GPOLTJZPOVOVOHSBGJóJWFSJM- 7. f_ x i = sin_ 2x + 5 i + cos2f x – 2π p + tan32x NJõUJS 2 GPOLTJZPOVOVO QFSJZPEV BöBôEBLJMFSEFO IBO- y 1 HJTJEJS O Õ x A) π # Õ $ Õ % Õ & Õ 2 –1 #VOBHÌSF GGPOLTJZPOVOVOQFSJZPEVBöBôEB- LJMFSEFOIBOHJTJEJS A) π B) π C) 2π % Õ & Õ 3 2 3 4. f_ x i = sin9_ 6x + 3 i 8. f^ x h = sin5 f x + 2 p + 7 tan6 x + cot3 ^ 2x + 7 h GPOLTJZPOVOVO QFSJZPEV BöBôEBLJMFSEFO IBO- π HJTJEJS GPOLTJZPOVOVO QFSJZPEV BöBôEBLJMFSEFO IBO- HJTJEJS A) π B) π C) π D) 2π & Õ π π C) Õ % Õ & Õ 6 3 2 3 A) B) 4 2 1. B 2. \" 3. D 4. B $ $ 7. $ 8. D
5SJHPOPNFUSJL'POLTJZPOMBSO(SBGJLMFSJ TEST - 19 1. G Y =+DPTY 3. G Y = -+TJOY GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- GPOLTJZPOVOVOHSBGJôJBöBôEBLJMFSEFOIBOHJTJ- EJS EJS A) y B) y A) y B) y 3 3 1 Õ Õ Õ 2 2 O Õ Õ 2 Õ x O 2 Õ 2 Õ x 1 1 –1 2 –1 O Õ Õ Õ Õ O Õ Õ Õ Õ x –2 x C) y D) y C) y D) y 1 Õ 1 Õ 2 Õ x O Õ Õ 2 Õ x 3 3 O ÕÕ –1 2 2 2 1 1 –1 2 O Õ Õ Õ Õ O Õ Õ Õ Õ –3 x x E) y E) y Õ Õ 2 Õ 2 x 3 O 2 Õ 1 –1 2 O Õ Õ Õ Õ x y 4. y 2 2. Õ Õ 24 Õ 2 OÕ 4 x O Õ Õ Õ x –2 Õ f(x) –2 õFLJMEF [ Ö] BSBMôOEB HSBGJôJ WFSJMFO G Y õFLJMEFHSBGJôJWFSJMFOG Y GPOLTJZPOVBöBôEB- LJMFSEFOIBOHJTJEJS GPOLTJZPOVBöBôEBLJMFSEFOIBOHJTJEJS \" G Y =TJOY # G Y =TJOY+ \" G Y = --DPTY # G Y =+DPTY $ G Y = -+DPTY % G Y = 2 sin x $ G Y = -DPTY+ % G Y =TJOY- 2 E) G Y =DPTY- & G Y =DPTY 1. \" 2. D 3. D 4. \"
TEST - 20 5SJHPOPNFUSJL'POLTJZPOMBSO(SBGJLMFSJ 1. y 3. y 2 3 mÕ Õ Õ Õ x O f(x) 1 –1 Õ 4 x O Õ Õ Õ –4 24 õFLJMEF[ -Ö Ö]BSBMôOEBLJHSBGJôJWFSJMFOG Y –1 GPOLTJZPOVBöBôEBLJMFSEFOIBOHJTJEJS \" G Y =DPTY # G Y = -+DPTY õFLJMEFLJHSBGJLBöBôEBLJGPOLTJZPOMBSEBOIBO- HJTJOFBJUPMBCJMJS C) f^ x h = 3 cos x D) f^ x h = - 1 + 3 sin x \" +TJOY # DPTY 2 2 $ -TJOY % -DPTY E) f^ x h = - 1 + 3 cos x 2 & -TJOY 2. y 4. sin x = x 12 3 EFOLMFNJOJOLBÀLÌLÑWBSES Õ \" # $ % & O 4Õ x ÕÕ 42 f(x) –3 õFLJMEF [ Ö] BSBMôOEB HSBGJôJ WFSJMFO G Y GPOLTJZPOVBöBôEBLJMFSEFOIBOHJTJEJS \" G Y =TJOY # G Y =+TJOY $ G Y =TJOY % G Y =TJOY & G Y =+DPTY 1. & 2. D 3. & 4. $
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF 5&3453÷(0/0.&53÷,'0/,4÷:0/-\"3 ÷MJöLJMJ,B[BONMBS 11.1.2.5 : 5SJHPOPNFUSJLGPOLTJZPOMBSOUFSTGPOLTJZPOMBSOB¿LMBS %m/*m ÖRNEK 1 A Z#UBONMCJSGPOLTJZPOVOVOUFSTGPOLTJZP- \"öBôEBLJJGBEFMFSJOEFôFSMFSJOJCVMVOV[ OVOVOPMBCJMNFTJJ¿JOCVGPOLTJZPOVOVO–WF ËSUFO PMNBT HFSFLJS 4JOÐT LPTJOÐT UBOKBOU a) BSDTJO C BSDTJO 1 WFLPUBOKBOUGPOLTJZPOMBSOONFWDVUUBONLÐ- 2 NFMFSJOEF–PMNBELMBSJ¿JOUFSTGPOLTJZPOMB- SZPLUVS D BSDTJO - 3 E BSDTJOf - p #VGPOLTJZPOMBSOUBONLÐNFMFSJOJO–WFËS- UFO PMBO BMU LÐNFMFSJOEFO CJSJ UBON LÐNFTJ 2 PMBSBL TF¿JMEJóJOEF GPOLTJZPOMBSO CV LÐNFEF UFSTGPOLTJZPOMBSWBSES B BSDTJO= a =TJOa = 0 l a = 0 1 1π C BSDTJO = a jTJOa = j a = 2 26 π D BSDTJO -1) = a jTJOa = -1 j a = - 2 E BSDTJOf - 3 p = a jTJOa = - 3 π ja= - 2 23 G Y =TJOY'POLTJZPOVOVO5FSTJ ÖRNEK 2 TANIM \"öBôEBLJJGBEFMFSJOEFôFSMFSJOJCVMVOV[ y π C sinf arcsin 5 p a) arcsind sin n 7 7 1 D cosf arcsin 7 p 25 mÕ –Õ Õx 2 O Õ 2 –1 π B BSDTJO (sin ) = a >a 7 :VLBSEB CJS LTN WFSJMFO TJOY HSBGJóJOEF HË- ππ SÐMEÐóÐ HJCJ <- π , π F BSBMóOEB GPOLTJZPO TJOa = sin j a = 7 7 22 6 447a 448 –WFËSUFOEJS 5 C sin (arcsin ) = x PMTVO G <- π , π F Z [ - ] 7 22 55 5 arcsin = a & sin a = j x= G Y =TJOYPMBSBLUBONMBOEóOEB 7 7 4 7b 7 G-[- ] Z <- π , π F 6 4 4 4 8 22 7 G-( x ) =BSDTJOYGPOLTJZPOVOBTJOÑTGPOLTJZP- D cos (arcsin ) = x 25 OVOVOUFSTGPOLTJZPOVEFOJS 77 Z=BSDTJOYl x =TJOZEJS arcsin = a & sin a = 25 25 A 24 cos a = 25 25k 7k a 24k C B ππ π π 5 24 1. B C D - E - 2. a) C D 62 3 7 7 25
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr G Y =DPTY'POLTJZPOVOVO5FSTJ ÖRNEK 4 TANIM arccos 2 - t JGBEFTJUBONMCJSJGBEFPMEVóVOBHËSF 3 y U OJO BMBCJMFDFôJ FO CÑZÑL UBN TBZ EFôFSJOJ CVMV- 1 OV[ Õ Õ -1 ≤ 2-t ≤1 2 Õ 2x –Õ 2 O 3 –1 -ã-Uã -ã-Uã -ãUãjUJS :VLBSEBHSBGJóJOJOCJSLTNWFSJMFO G Y = DPTY HSBGJóJOEF HËSÐMEÐóÐ HJCJ [ Õ] BSBMóOEBGPOLTJZPO–WFËSUFOEJS G[ Õ] Z [ - ] G Y =DPTYPMBSBLUBONMBOEóOEB G-[- ] Z [ Õ] G-( x ) =BSDDPTYGPOLTJZPOVOBLPTJOÑTGPOL- TJZPOVOVOUFSTGPOLTJZPOVEFOJS Z=BSDDPTYl x =DPTZPMVS ÖRNEK 5 \"öBôEBLJJGBEFMFSJOEFôFSMFSJOJCVMVOV[ a) BSDDPTd cos π n C sinf arccos 3 p 8 5 D UBO BSDTJOY ÖRNEK 3 \"öBôEBLJJGBEFMFSJOEFôFSMFSJOJCVMVOV[ π a) arccos (cos ) = a a) BSDDPT C BSDDPT 8 D BSDDPT - E BSDDPTf - 1 p ππ 2 DPTa =DPT j a = 88 π 3 B BSDDPT= a jDPTa = 0 j a = C sin (arccos ) = sin a 14 442a 4 4543 2 C BSDDPT= a jDPTa = 1 j a = 0 3 cos a = D BSDDPT -1 ) = a jDPTa = -1 j a =Ö E arccosd - 1 n = a & cos a = - 1 j a = 2π 5 2 23 5k 4k 4 TJOa = a 5 3k 6 4 7a 4 8 xk D tan (arcsin x) = tan a tana = BSDTJOY= a jTJOa = x k 1 - x2 k x = a LmæY2 xk 1 - 2 x π4 x 4. a) C D π 2π 85 3. a) C D ÖE 1-x 2 23
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ÖRNEK 6 ÖRNEK 7 arccosf sin 11π p \"öBôEBLJJGBEFMFSJOEFôFSMFSJOJCVMVOV[ 3 a) BSDUBO C BSDUBO_ i JGBEFTJOJOEFôFSJOJCVMVOV[ D BSDUBO - E BSDUBOf - sind 11π n = sin 5π = - 3 p 3 32 arccosf - 3 p = a & cos a = - 3 B BSDUBOj aUBOa = 0 j a = 0 22 C BSDUBO^ 3 h = a jUBOa = π 5π 3 ja = a= 3 6 π D BSDUBO -1) = a jUBOa = -1 j a = - 4 E BSDUBOf - 3 p = a & tan a = - 3π & a =- 3 36 G Y =UBOY'POLTJZPOVOVO5FSTJ ÖRNEK 8 TANIM sinf arctanf - 15 p pJGBEFTJOJOEFôFSJOJCVMVOV[ 8 y 64 4 447a 4 4 448 –Õ O Õ Õ x sin (arctand - 15 n) = sin a 2 2 2 8 arctand - 15 n = a & tan a = - 15 88 a ! d - π , 0 nPMVS 2 - 15 sin a = 17k 17 15k a 8k :VLBSEBHSBGJóJOJOCJSLTNWFSJMFOG Y =UBOY ÖRNEK 9 HSBGJóJOEFHËSÐMEÐóÐHJCJd - π , π nBSBMóO- tanf arcsin 3 + 3π pJGBEFTJOJOEFôFSJOJCVMVOV[ 22 52 EBGPOLTJZPO–WFËSUFOEJS Gd - π , π n $ R 6 447a 448 tan (arcsin 3 + 3π ) = tand 3π + a n = - cot a 22 52 2 G Y =UBOYPMBSBLUBONMBOEóOEB G-3$ d - π , π n 33 arcsin = a & sin a = 22 G- ( x ) =BSDUBOY 55 GPOLTJZPOVOB UBOKBOU GPOLTJZPOVOVO UFST 4 -DPUa = - PMVS GPOLTJZPOVEFOJSZ=BSDUBOYl x =UBOZPMVS 3 5k 3k a 4k 5π πππ 15 4 7. B C D - E - 8. - - 6 346 17 3
11. SINIF 1. MODÜL 53÷(0/0.&53÷ www.aydinyayinlari.com.tr ÖRNEK 10 ÖRNEK 11 sinf arctanf 2 p + 3π p \"öBôEBLJJGBEFMFSJOFEFôFSMFSJOJCVMVOV[ 32 a) BSDDPU_ i C BSDDPU JGBEFTJOJOEFôFSJOJCVMVOV[ A 2 D BSDDPUf - p E BSDDPU - arctan = a 3 13 2 π jUBOa = 3&a= 23 a) arc cot^ 3 h = a & c ota = 6 a sind 3π + a n = - cos a 2 π C BSDDPU = a jDPUa = 0 j a = B3C 2 D arc cotf - 3 p = a & cot a = - 3 2π j a= 2 23 E arc cot^ - 1 h = a & cot x = - 1 & a = 3π 4 G Y =DPUY'POLTJZPOVOVO5FSTJ ÖRNEK 12 TANIM arccotf cotd - π n p 5 y JGBEFTJOJOEFôFSJOJCVMVOV[ mÕ O Õ x arc cotd cotd - π n n = a 5 cot a = cotd - π n 5 4π BSDDPUY GPOLTJZPOVOVO UBONOEBO EPMBZ a = 5 PMVS :VLBSEBHSBGJóJOJOCJSLTNWFSJMFO ÖRNEK 13 G Y = DPUY HSBGJóJOEF HËSÐMEÐóÐ HJCJ Õ TJO BSDUBO+BSDDPU BSBMóOEBGPOLTJZPO–WFËSUFOEJS JGBEFTJOJOEFôFSJOJCVMVOV[ G Õ Z3 BSDUBO= a BSDDPU= b G Y =DPUYPMBSBLUBONMBOEóOEB UBOa = DPUb = 4 G-3- Õ G-( x ) =BSDDPUYGPOLTJZPOVOBDPUYGPOLTJZP- π UBOa =DPUb j a + b = PMVS OVOVOUFSTGPOLTJZPOVEFOJS 2 Z=BSDDPUYl x =DPUZPMVS π sin = 1ES 2 -3 π π 2π 3π 4π 10. 11. a) C D E 12. 13. 1 62 3 4 5 13
www.aydinyayinlari.com.tr 53÷(0/0.&53÷ 1. MODÜL 11. SINIF ÖRNEK 14 ÖRNEK 18 BSDDPU Y- -BSDUBO Y+ = TJO BSDDPTY JGBEFTJOJOEFôFSJOJYDJOTJOEFOCVMVOV[ PMEVôVOB HÌSF EFOLMFNJ TBôMBZBO Y HFSÀFL TBZT- OCVMVOV[ 6 447a 448 sin (arccos 2x) = sin a BSDDPU Y- 1) =BSDUBO Y+ 2) BSDDPU Y- 1) = a jDPUa = x - 1 DPTa = 2x WFUBO Y+ 2) = a jUBOa = x + 2 1 mæY2 2 a sin a = 1 - 4x 2x UBOa DPUa = 1 (x - 1) (x + 2 ) = 1 13 - 1 x = - 13 - 1 x2 + x - 3 = 0 j x = 2 22 1 5BONLÑNFTJOFEFOJZMF 13 - 1 PMVS 2 ÖRNEK 15 BSDDPTY=BSDTJOY PMEVôVOBHÌSF YEFôFSJOJCVMVOV[ BSDDPTY= a jDPTa = x BSDTJOY= a jTJOa = 2x ÖRNEK 19 TJO2a +DPT2a = 1 ôFLJMEFLJ¿PDVLMBSLVMBLUBOLVMBóBPZVOVPZOBNBLUBES 4x2 + x2 = 1 = 1 &x= 1 WFZB x = - 1 x 55 5 1 x = PMVS 5 ÖRNEK 16 1. çocuk 2. çocuk 3. çocuk 4. çocuk G Y =BSDDPU Y- 0ZVOEB ¿PDVL CJS HFS¿FL TBZZ ¿PDVóB TËZMÐZPS PMEVôVOBHÌSF G-1 Y GPOLTJZPOVOVCVMVOV[ ¿PDVLCVHFS¿FLTBZOOBSDUBOKBOUOCVMVQ ¿PDV- óB ¿PDVL CV EFóFSJO TJOÐTÐOÐ CVMVQ ¿PDVóB y =BSDDPU Y- 1) ¿PDVLCVEFóFSJOBSDDPTJOÐTÐOÐCVMVQTPOVDVTËZMÐZPS DPUZ= 2x - 1 ÀPDVLHFSÀFLTBZPMBSBL 3 EFôFSJOJTFÀFSTF ÀPDVôVOTÌZMFEJôJEFôFSOFPMVS cot y + 1 = x & –1 ^ x h = cot x + 1 f 22 ÖRNEK 17 π tanf arcsin 1 + 2 arcsin 3 p JGBEFTJOJOEFôFSJOJCVMVOV[ 6 4 4473 4 448 2 Arccos(1s4in4(4a4r2cta4n4433)) 3 2 3 7π 3 3 3π 6 3 Arc cos = a j cos a = j a= tan (1a44rc2sπin413 + 2 arcsin ) j tand n= 2 26 1 4 42π 424 3 2 3 13 - 1 1 cot x + 1 3 2 π 14. 17. 18. 1 - 4x 6 2 5 2 3
TEST - 21 5FST5SJHPOPNFUSJL'POLTJZPOMBS 1. \"öBôEBLJMFSEFOIBOHJTJ ZBOMöUS 4. f^ x h = arccosf 1– 2x p A) arcsinf 1 p = r B) arccosf 2 p = r 5 26 24 GPOLTJZPOVOVOUBONLÑNFTJBöBôEBLJMFSEFO IBOHJTJEJS C) arctan ( 3) = r D) arcsin (- 1) = 3r A) [ - > # <- > $ <- > 3 2 E) arccosf - 1 p = 2r % <m > & <- > 23 2. \"öBôEBLJMFSEFOLBÀUBOFTJEPôSVEVS G Y =BSDDPU Y+ * cosf arccos 3 p = 3 PMEVôVOBHÌSF G-1 Y GPOLTJZPOVBöBôEBLJMFS- 44 EFOIBOHJTJEJS A) G-( x ) = -+DPUY ** arcsinc sinc - r m m = - r B) G-( x ) =+DPUY 44 C) f–1^ x h = 5 cot x *** arccos c cosc - r m m = r 3 44 D) f–1^ x h = - 3 + cot x *7 arccosf sinc - r m p = 5r 5 36 E) f–1^ x h = 3 + cot x 7 arcsinf tan 3r p = r 5 42 \" # $ % & 3. arcsin 4 + arccot 4 tanf 3π + arcsin 8 p 53 2 17 EFôFSJBöBôEBLJMFSEFOIBOHJTJEJS JGBEFTJOJOEFôFSJBöBôEBLJMFSEFOIBOHJTJEJS A) r B) r C) r D) r E) r 15 8 C) - 1 5 A) - B) - 2 8 15 8 15 D) E) 15 8 1. D 2. B 3. \" 70 4. & D \"
5FST5SJHPOPNFUSJL'POLTJZPOMBS TEST - 22 1. arcsinf cosf arctan 3 p p 4. arctan_ - 3 i + arccosf - 3 p 3 2 JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS A) π B) π C) π D) π E) π π π C) π % Õ & 5π 2 3 4 6 8 A) B) 22 6 3 sinf arctan - 3 pp 3 + arccotf 3 2. BSDUBO 1 =YPMEVôVOBHÌSF JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS 2 +UBOY DPTY A) - # C) 1 4 JGBEFTJOJOEFôFSJLBÀUS D) 1 2 & 5 $ 1 3 A) # 2 D) E) 2 2 f_ x i = arcsinf 3 - 2x p 4 GPOLTJZPOVOVOFOHFOJöUBONBSBMôBöBôEB- LJMFSEFOIBOHJTJEJS 3. BSDTJOZ=BSDDPTY A) > 1 , 7 H B) >- 3 , 7 H 22 22 PMEVôVOBHÌSF x2 + y2LBÀUS 1 3 $ % 3 & C) >- 1 , 7 H D) > 3 , 7 H A) B) 2 22 22 2 2 E) >- 3 , 5 H 22 1. B 2. \" 3. $ 71 4. $ B $
KARMA TEST - 1 Trigonometri 1. x + N-O Z = n - 4. ²MÀÑTÑ - 59π SBEZBOPMBOBÀOOFTBTÌMÀÑTÑ EFOLMFNJCJSJNÀFNCFSCFMJSUUJôJOFHÌSF NO 5 LBÀUS LBÀEFSFDFEJS \" # $ % & \" # $ % & 2. ôFLJMEFLJ\"OPLUBTCJSJN¿FNCFSÐ[FSJOEFEJS y y A 1 a 1 x D CO x A B –1 O –1 \"OPLUBTOOZFLTFOJOFV[BLMô 1 CJSJNPM- ôFLJMEFLJCJSJN¿FNCFSEF[AC] m [BD], 2 | | | |% EVôVOB HÌSF Y FLTFOJOF PMBO V[BLMô LBÀ CJ- m ( BAC ) SJNEJS = a WF OC = CD EJS :VLBSEBLJWFSJMFSFHÌSF UBOaEFôFSJLBÀUS 3 B) 1 D) & A) 1 B) 1 $ % 5 E) 7 A) 2 C) 7 5 2 3. a =hhh b =hhh a =hhhPMNBLÐ[FSF PMEVôVOBHÌSF a + bUPQMBNBöBôEBLJMFSEFO 4a EFôFSJBöBôEBLJMFSEFOIBOHJTJEJS IBOHJTJEJS 3 \" hhh # hhh A) 7hhh # hhh $ hhh % hhh $ hhh % hhh & hhh & hhh 1. D 2. \" 3. $ 72 4. D & B
Trigonometri KARMA TEST - 2 1. DPTY+UBOYTJOY-TFDY tan π cot 5π JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS 4. 4+ 4 \" TFDY # DPTY $ 1 + cot2 π 1 + tan2 π 33 % TJOY & DPTFDY JöMFNJOJOTPOVDVLBÀUS 1 B) $ 1 D) & A) 3 2 DPTx +TJOYDPTx +TJOx JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS 2. π < a < 3π PMNBLÑ[FSF A) - # - $ % & 2 1 + cos a - 1 - cos a 1 - cos a 1 + cos a JGBEFTJOJO FöJUJ BöBôEBLJMFSEFO IBOHJTJ PMBCJ- MJS A) -DPUa # mUBOa $ UBOa % & ôFLJMEFCJSJN¿FNCFSWFSJMNJõUJS y C y=1 D x A aB O 3. 5BONMPMEVôVBSBMLUB \" B1 B2 # C1 C2 $ D1 D2 WF% E1 E2 PM- NBLÑ[FSF f tan x - sin x p.f cos2x + sin2x + cos x p C +D E - a) sin2x tan x JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS sin3 a cos3 a sin2 a A) B) C) \" DPTY # TFDY $ cos a sin a cos a D) - & TJOY cos2 a E) cos a D) sin2 a sin a 1. $ 2. \" 3. $ 73 4. B & B
KARMA TEST - 3 Trigonometri 1. A = 13 - 2 sin x 4. GGPOLTJZPOVOVOQFSJZPEVWFHGPOLTJZPOVOVOQF- 3 SJZPEVUÐS JGBEFTJOJOBMBCJMFDFôJUBNTBZEFôFSMFSJOJOUPQ- G =WFH =PMEVôVOBHÌSF GPH + HPG MBNLBÀUS UPQMBNOOTPOVDVLBÀUS \" # $ % & \" # $ % & 2. A DF C 7 7 a C BD | | | |\"#$JLJ[LFOBSпHFO AB = AC =CS A EB | | | |BD = 15 - 2 CS DC = 15 + 2 CSWF | | | |\"#$%LBSFTJOJOCJSLFOBSCS DF = EB , % m ( ADC ) = aES m ( B%EF ) = aWFUBOa = -UÐS :VLBSEBLJWFSJMFSFHÌSF DPTaEFôFSJLBÀUS | |:VLBSEBLJWFSJMFSFHÌSF \"& LBÀCJSJNEJS A) 1 B) 1 2 2 \" # $ % & 2 3 C) D) E) 6 3 3. D C y F a k O x –k Õ AB E | | | |ôFLJMEF\"#$%LBSF FB = CF WF ôFLJMEFLJHSBGJLG Y =LDPTNYGPOLTJZPOVOBBJUUJS % #VOBHÌSF GGPOLTJZPOVOVOQFSJZPEVLBÀUS m ( EDC ) = aES :VLBSEBLJWFSJMFSFHÌSF UBOaEFôFSJLBÀUS A) 9π B) 8π C) 4π D) 9π & Õ 8 9 9 4 A) 3 B) 4 C) 3 D) 4 E) 2 4 3 5 5 5 1. & 2. $ 3. $ 74 4. & $ B
Trigonometri KARMA TEST - 4 1. A \"#$EJLпHFO 4. A \"#$пHFO [AB] m [BC] 8 | |AB =CS 5 H [BH] m [AC] | |x BC =CS | |BH =YCS a % B m ( ABC ) = a % C m ( CBH ) a = a | |a <PMEVôVOBHÌSF \"$ =YJOLBÀGBSLM B C UBNTBZEFôFSJWBSES | | :VLBSEBLJWFSJMFSFHÌSF \"$ OJOaWFYUÑSÑO- \" # $ % & EFOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS \" YTJOaDPTa # YTFDaDPTFDa x x C) D) cosec a sec a & YTFDaDPTa 2. H G \"#$%&'()LÐQ \"#$пHFOJOJOB¿MBSBSBTOEB E a | NF | = | BF | sin2 WA - sin2 WB = sin2 XC D F% CBóOUTWBSES m ( AGN ) = a % = 27°PMEVôVOBHÌSF m ( % ) LBÀEF- CN m ( BCA ) ABC SFDFEJS AB \" # $ % & :VLBSEBLJWFSJMFSFHÌSF DPUaEFôFSJLBÀUS 6 6 6 D) & A) B) C) 3 2 5 3. BSDUBO_ - i BSDTJOf 1 p BSDDPTf - 3 G Y =BSDDPTf 2x - 1 p p 5 22 GPOLTJZPOVOVO FO HFOJö UBON LÑNFTJOEF LBÀ UBOFUBNTBZEFôFSJWBSES JöMFNJOJOTPOVDVLBÀUS π B) 2π $ Õ % 4π 5π \" # $ % & A) 3 E) 3 33 1. B 2. B 3. B 4. \" D $
KARMA TEST - 5 Trigonometri 1. y 4. y 2 1 O Õ OÕ x Õ Õ Õ 3 Õ Õ Õ 3 63 x 4 24 –3 –1 –2 õFLJMEFLJHSBGJLBöBôEBLJGPOLTJZPOMBSEBOIBO- õFLJMEFLJ HSBGJL Z = BDPTCY + D GPOLTJZPOVOB BJUPMEVôVOBHÌSF B+C+DLBÀPMBCJMJS HJTJOFBJUPMBCJMJS \" m # $ % & A) - 2 cosc x - π m B) 2 sinc x - π m 6 6 C) 2 cosc x - π m D) 2 sinc x - π m 6 3 E) –2 sinc x - π m 3 2. y BSDDPT 1 +BSDDPT 2 3 55 JGBEFTJOJOEFôFSJBöBôEBLJMFSEFOIBOHJTJEJS OÕ A) π B) π C) π D) π & Õ 3 6 2 4 x Õ –1 õFLJMEFLJHSBGJLBöBôEBLJGPOLTJZPOMBSEBOIBO- HJTJOFBJUPMBCJMJS \" Z=+TJOY # Z=DPTY+ $ Z=TJOY+ % Z=DPTY+ & Z=DPTYm Õ< a < b < 3π 3. cos^ 2 arctan 3 h 2 PMEVôVOBHÌSF BöBôEBLJMFSEFOIBOHJTJLFTJO- JGBEFTJOJOEFôFSJBöBôEBLJMFSEFOIBOHJTJEJS MJLMFEPôSVEVS -3 B) - 1 C) - D) 1 3 A) 2 E) \" TJOa <TJOb # DPUa <TFDb 2 $ TJOa <DPTa % UBOa <UBOb & DPUa <DPUb 22 1. D 2. D 3. B 4. $ $ D
Trigonometri YAZILI SORULARI 1. 5BONMPMEVôVBSBMLUB 4. 1 + 1 = 8 sin3x - cos3x 1 - cos x 1 + cos x tan x. cot x + sin x. cos x PMEVôVOBHÌSF UBOYJOQP[JUJGEFôFSJLBÀUS JGBEFTJOJOFOTBEFI»MJOJCVMVOV[ 1 1 1 + cos x + 1 - cos x =8 += 1 - cos x 1 + cos x ^ 1 + cos x h^ 1 - cos x h ^ sin x - cos x h.^ sin 2 x + sin x. cos x + 2 h ^ 1 + cos x h ^ 1 - cos x h cos x 1ta4n4x2. c4o4t x3 + sin x. cos x 2 = =8 1 1 - cosx ^ sin x - cos x h.a 1 + sin x . cos x k = sin x - cos x 1 a 1 + sin x . cos x k 2 = 8 & 2 = sin2x & sin x = ± 1 114-4c2os442x3 8 2 sin 2 x 4 2. AöBôEB WFSJMFO BÀMBSO FTBT ÌMÀÑMFSJOJ CVMV- A 13 tan x = = OV[ 33 a) - C 163π 2k k D - 72π 7 x B 5 E C 3k a) –3060° 360 b) 163 14 Ö –3240 –9 14 11 7 180° 23 14 9 c) –72 10 d) 2870 360 \"õBóEB0WF0NFSLF[MJEõUBOUFóFU¿FNCFSMF- –80 –8 2520 7 350° SJOPSUBLUFóFUJWFSJMNJõUJS 8 A 43 B Ö 3 53 3 5 33 a O2 O1 3. a =hhh | | | |\"01 = 4 BO2 =4 3 CS WF m ( % ) = a AO1O2 b =hhh PMNBLÑ[FSF a + b JöMFNJOJOTPOVDVOVCVMV- PMEVôVOBHÌSF TJOaEFôFSJLBÀUS 23 43 4 OV[ sin a = = EJS 53 5 44° 112' 34' ' a = & 22° 56' 17' ' 22 2 18° 45' 174' ' b = & 6° 15' 58' ' 33 3 22° 56' 17'' + 6° 15' 58'' 29° 12' 15'' 9π 8π 77 3 4 1. TJOYmDPTY 2. B C D E 4. 75 3 5 3. hhh
YAZILI SORULARI Trigonometri #JS\"#$ÑÀHFOJOEF π < a < b < 3π PMNBLÑ[FSF cos2_ WB + XC i + sin2WA 2 tan_ WA + XC i. cot WB *TJOb <TJOa **TJOa <DPTb JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS ***UBOa <UBOb *7DPUa <DPUb 7UBOa <DPTFDb \"#$ÑÀHFOPMEVôVOEBO JGBEFMFSJOEFOIBOHJMFSJLFTJOMJLMFEPôSVEVS DPT #+$ =DPT -\" = -DPT\" UBO \"+$ =UBO - B) = -UBO#EJS y cota 2 64 4 4471 4 4 448 2 2 cotb ^ - cos A h2 + sin A cos + sin A A cosa = = - 1EJS cosb ^ - tan B h. cot B 1-4ta44n 2B. c4o4t4B3 sinsainb tanb –1 tana x 7. f_ x i = 3 sind x + π n *%PôSV **#JMFNFZJ[ ***%PôSV *7:BOMö 7UBOa WFDPTFDb PMEVôVJÀJOZBOMöUS 6 10. ôFLJMEFLJ \" OPLUBTOEB CVMVOBO LVSCBóB [QMBZB- GPOLTJZPOVOVOHSBGJôJOJ[ Ö]BSBMôOEBÀJ[J- SBL # OPLUBTOEBLJ CËDFóJ ZBLBMBZQ $ OPLUBTOB OJ[ EÐõNFLUFEJS y B 3 3 2 Ö 3x O Ö Ö Ö Ö 36 6 –3 8. ,FOBSV[VOMVLMBSBCS CCSWFDCSPMBOCJSÑÀ- AC HFOJOLFOBSMBSBSBTOEB #ÌDFôJO ZFSEFO ZÑLTFLMJôJ NFUSF LVSCBôB- OOBMEôZPMNFUSFPMEVôVOBHÌSF m ( A%BC ) (a -C B+C -D B-D = LBÀUS (sin2a = 2sinacosa) CBôOUTPMEVôVOBHÌSF m (XA) + m (XC)LBÀEF- B TJOa TJOaDPTa SFDFEJS aa 3 4 (a -C B+C -D B-D = 0 TJOa = 2· · a2-C2 -BD+D2 = 0 5 5 55 C2 = a2+D2 -BDPMVS \"#$ÑÀHFOJOEFLPTJOÑTUFPSFNJOJZB[BSTBL 4 24 C2= a2+D2-BDDPT WB EJS TJOa = 25 a2 +D2 -BD= a2 +D2 -BDDPT XB A3 3C m ( A%BC ) = arcsind 24 n cos XB = 1 & m (XB) = 60° 25 2 m (XA) + m (XC) = 120°EJS m 8. 120° 78 * *** 10. arcsind 24 n 25
Trigonometri <(1m1(6m/6258/$5 1. ôFLJMEFCJSCJSJOFQBSBMFMLBMESNMBSEBCVMVOBOCBO- 3. ôFLJMEFLJ.OPLUBTOEBOGSMBUMBOCJSQBS¿BDLTSB- LBNBUJL NBSLFU WF :Bó[hO FWJ HËTUFSJMNJõUJS :B- TZMB WFFOHFMMFSJOF¿BSQQEPóSVTBMZPMMBS ó[NBSLFUFHJEJQCJSõFZMFSBMNBLJTUJZPS\"NBQB- J[MFZFSFLUFLSBS.OPLUBTOBHFMNFLUFEJS SBTOOPMNBEóOGBSLFEJZPS 1. 2. #BOLBNBUJL M a 3. Market :Bó[OFWJ .BSLFU 1BS¿BDóOFOHFMJMFFOHFMBSBTOEBBMEóZPM FOHFMJMF.OPLUBTBSBTOEBBMEóZPMBQBSBMFM- :Bó[hO NBSLFUUFO ËODF CBOLBNBUJóF HJEJQ QBSB EJS ¿FLJQ EBIB TPOSB NBSLFUF HJUNFTJ HFSFLJZPS :B- ó[hOFWJJMFCBOLBNBUJLBSBTOEBLJFOLTBV[BLML 4BOJZFEF LN ZPM BMBO CV QBSÀBDL . OPLUB- NFUSF CBOLBNBUJLJMFNBSLFUBSBTFOLTBV[BL- TOEBO FOHFMFTOEF FOHFMFTOEF ML NFUSF :Bó[hO FWJ JMF NBSLFU BSBT FO LTB FOHFMFTOEFWFUFLSBS.OPLUBTOBTOEF V[BLMLNEJS VMBöUôOBHÌSF DPTaLBÀUS :Bô[FOLTBZPMMBSLVMMBOBSBLÌODFCBOLBNB- A) 1 3 C) 3 D) 4 E) 5 2 B) 5 5 6 UJôF PSEBO NBSLFUUF NBSLFUUFO EF FWJOF HFSJ 2 HFMEJôJOEF J[MFEJôJ SPUBMBS CJSMFöUJSFSFL CJS ÑÀ- HFOPMVöUVSVSTBLCVÑÀHFOJOBMBOLBÀN2PMVS \" # 10 2 C) 10 3 % & 2. ôFLJMEFLJ CJMBSEP NBTBTOEB UPQMBN EFMJL CVMVO- 4. ôFLJMEFLJHJCJCJSBUDEPóSVTBMCJSõFLJMEFZFSMFõUJ- NBLUBES%JLEËSUHFOõFLMJOEFLJNBTBJLJFõLBSFOJO SJMNJõPMBO\" #WF$IFEFGMFSJOFBUõZBQBDBLUS CJSMFõUJSJMNFTJZMF PMVõUVSVMNVõUVS %FMJLMFS EJLEËSU- HFO WF LBSFOJO LËõFMFSJOF LPONVõUVS OVNBSBM EFMJóJOËOÐOEFOUPQBWVSBOPZVODVLBSõCBOEOPS- UBTOB UPQV ¿BSQUSQ OVNBSBM EFMJóF UPQV TPLV- ZPS5PQEPóSVTBMIBSFLFUFUNFLUFEJS b 123 2a a ab A BC 6 a 4 \"IFEFGJOFZBUBZMB #IFEFGJOFa $IFEFGJOF 5 EFaB¿ZBQBDBLõFLJMEFBUõOZBQZPS õFLJMEFWFSJMFOMFSFHÌSF DPUa UBObEFôFSJLBÀ- DPTa PMEVôVOBHÌSF DPUaLBÀUS US (b > a) A) 1 # $ 5 % & 7 2 2 2 \" # $ % & 1. $ 2. & 3. $ 4. B
<(1m1(6m/6258/$5 Trigonometri 1. \"õBóEBLJõFLJMEFHÐOÐOJMLTBBUMFSJOEFLJBQBSUNB- 3. \"õBóEBCJSPEBOOUBWBOOBõFLJMEFLJHJCJBTMBDBL OOHËMHFTJJMFJMFSMFZFOTBBUMFSEFLJHËMHFTJHËTUFSJM- MBNCBMBSHËTUFSJMNJõUJS NJõUJS A 53° 37° B 53° 37° 53° 37° ab N 2.gölgenin ,BCMPMBSBõBóEBLJHJCJLVMMBOMBDBLUS boyu | |r AB =DNEJS 1.gölgenin boyu r ,BCMPMBSJLJ¿FõJUUJS (ÐOFõZÐLTFMEJL¿FBQBSUNBOOHËMHFTJOJOCPZVL- r :BUBZMBMJLB¿ZBQBOLBCMPMBSLFOEJJ¿JOEF TBMNõUS\"QBSUNBOZBUBZMBMJLB¿PMVõUVSBDBL õFLJMEFEÐ[CJS[FNJOFZBQMNõUS FõJUV[VOMVLUBWFNBWJSFOLUFEJS r :BUBZB MJL B¿ ZBQBO LBCMPMBS LFOEJ J¿JOEF HÌMHFOJO CPZV HÌMHFOJO CPZVOVO ÑÀ LB- FõJUV[VOMVLUBWFTBSSFOLUFEJS U BQBSUNBOOCPZVNPMEVôVOBHÌSF sin a r ,BCMPV[VOMVLMBSIFTBQMBOSLFO sin b TJO=DPT= DPT=TJO= PSBOLBÀUS BMOBDBLUS A) 2 # $ 3 5 E) 7 #VOBHÌSF LVMMBOMBOTBSSFOLMJLBCMPMBSNBWJ 3 2 D) 4 SFOLMJLBCMPMBSEBOUPQMBNEBLBÀDNGB[MBES \" # $ % & 3 4. \"õBóEBLJõFLJMEFHËTUFSJV¿VõVZBQBOEËSUV¿BóO 2. 4BBUUFLJI[MBSLNWFLNPMBOJLJCJTJLMFUMJBSB- PMVõUVSEVóVпHFOTFMCËMHFMFSHËSÐMNFLUFEJS TOEBLJB¿PMBDBLõFLJMEFBZOOPLUBEBOBZO BOEBEPóSVTBMCJSZPMJ[MFZFSFLJMFSMJZPS A 60° BC D TBBUTPOSBBSBMBSOEBLJV[BLMLYLNPMVZPS%B- && IBTPOSBI[MPMBO I[O 1 ünFEÐõÐSÐZPS EJóF- #VпHFOTFMCËMHFMFSEFOA^ ABC h = 3A^ ADC hEJS 4 SJ BZO I[EB EFWBN FEFSFL TBBU EBIB JMFSMJZPS | | | |AB =LN AC =LN m(B%AD) = 30°, WFBSBMBSOEBLJV[BLMLZLNPMVZPS % = a PMEVôVOB HÌSF TJOa EFôFSJ LBÀ- #VOBHÌSF Z-YLBÀLNEJS m ( CAD ) \" # $ % & US A) 1 B) 1 C) 1 D) 3 E) 2 2 3 2 1. $ 2. $ 80 3. & 4. B
CEVAP ANAHTARI 75m*2120(75m r Sayfa 34, Örnek 30 e) y 5 a) cosx b) –sinx c) cos2x d) sin3x mÕ –2Õ O Õ e) cot4x f) –tan5x g) –sina h) –cos2a –1 Õ x j) tan3a k) –cot4i n) –tan5x o) cosx l) - sin x m) - cos i 2 2 p) cot2i r) –tan2i r Sayfa 60, Örnek 10 a) y f) y 2 mÕ 1 – Õ x x 2 O Õ Õ Õ mÕ O Õ Õ 22 –1 mÕ – Õ 2 –2 r Sayfa 78, Soru 7 b) y 1 y – Õ Õ 4 4 Õx 3 – Õ – Õ OÕ Õ 3 4 2 4 2 2 Õ –1 3 O Õ Õ Õ Õ x 36 6 –3 c) y 2 mÕ Õ OÕ Õ Õ x 2 2 2 – Õ – 2 d) y – 3Õ 1 Õ x 2 OÕ –Õ – Õ 2 Õ 3Õ 2 2 –1
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