Home ADVANCED LEVEL TOPIC 7: TRIGONOMETRY(II) ~ ADV MATHEMATICS FORM 5

TOPIC 7: TRIGONOMETRY(II) ~ ADV MATHEMATICS FORM 5

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TRIGONOMETRY
iii) Consider Nr Ykw6N4Iwjg3It9Pdbfbdh6Hgyvhcnkdwoc6Kk8Njbasmwndzcgchfehlamaiuvmpcvujjktheo51Gvgnueopqyimaf1Ifgiqszbcqxyd0Zczllegp1Brsjz
Nr Ykw6N4Iwjg3It9Pdbfbdh6Hgyvhcnkdwoc6Kk8Njbasmwndzcgchfehlamaiuvmpcvujjktheo51Gvgnueopqyimaf1Ifgiqszbcqxyd0Zczllegp1Brsjz=Qv432Qs2Clyrxmqkuzi58Ufj4Vakjxaghonito9W Dl Bve7Ajxr21Qo2Y49Fl5Pdvtspn Xun5Ff8Fdwi1Eotytemcwbrl64Iuzpvpo06Ucrxogxo09Nhb88 0Fhyod Camwgu
Qefw
But = Ux789Mmkqnunhlsw3Cwkoox0Xngdkjipnopjuytizupddjrtrr7Lionic8M5Dxiec2Jympm5Miuxjvlaeq7E9Gjkyn6Pdnefupqa8Hm78X50Ujmj 6Cwqoiyamtokbucyo2Khay
2Exk61Iotdubpaad7 Bvyi6Te7Arhyjfbtgcwt770H6Eivqoy4C9Viixpmsiur Opdkljjhxou5Fyvyo5Fwlfr7Dmggfhayf1Fijtj8R99I744O1 Tevjtqwtqkcyd Xb3Tk6 0
=1Mnas8Gbg8Ptmkguau Vboa2Cenas Dl4Gm6I4Gusqgzy5 Kxcihfph Ddhhgfua2Ugm88Sbmog2Vu7Do2Rt39N0Orlxermarsy60Itrzptvo1Pnd9Jewgbcvyupbuyh6Aj2E
7Mrtobucabzhpsl9Bdfhoxnjhnaor7Mj9X 4Nzhdxdmkizbhn0Z Auvogk9Ftkck9Xexxjxxw69Jze1Uezup0Ih4Pen4
=Hkxspfsfptorqtpoq6Uwnl3Moqc9Mnjm5Kfuakyona9Pw X4I4W N2Gsvwwngkcgihno9Dc0E3Jr Qydhbhzhxx
Dyvsngjwes S67Dmfpoo4Tx1Eaedxjb4El7Zdov90Fzhhbuhqn4Ypvuf5Nbxjxcejnavkrigfjs8Ivohp5 P1Zhhdbptgowuzrzb20L2Ledrsckvmor5Guopyqifqly1Hmwrchq
Alternative: Using Dve2X Gccddqqy0Boboscpfrqmapz1Uyzqpkzwyu3Uzai0Pokdceo8Yy=Nsgqahnkhplhpjre7Bo6O1Pu33Gbvh Knfuijs9Ysru5Rbzy
Bel4Uoiud Ugaitivuphtfzv23Hidlmheayafczvfij5Jeiypfd3Jmdccoumfqboqfnkcehyv5Sqd5Ui7Gk5Cxpqprj4Shvnkufrhukbd5O9L7O1C3
Dividing by cos3θ numerator and denominator
3Awvdt07Xdbayks5R Irqjzjpor1Tvtk Kweq Qwbjcc6Mmymj
Aqr4Uetzhgzqq93Mtac9Kw58T Gfor1Ysln2 Wtohud0K1Aktre7Qs2Rtnngs1Mwyngbkzs8J3Oazy9Mmq1Fcpbzllvnnhwt8Mzoakvvdrjv2Gutrivjnjn T2J Fapkzexfjac
Applications of the double and triple formulae
A. Proving Identities
Examples: Prove the following identities
(i) + Xemz80Btoqmjoqqawztblqxn
(ii) Hjv6Rxo A8Hl81Qq6Bua 6Po Zcucnmwflt5Uagwx858Onlowd2Stpgfottxydy70V1Nn8V8R4Twahso0J9Fneyh3Xcinz9Txai7Hmnzqbsq3Ohdivdnha0Y Wasylo8D C2N8= Xbs1Cqtpw2Rzegkayborl3We9Wzkkblo3Cmrcrol4Vzt8Iq25Wsk7Thxzb0Mqiilf0Wdsc62K4B4Rxudntsyrv9Zyw Szza60Pmrc
(iii) Ec Jrqm6 Qo 9Dfnmolnsa5Ey6A3Ad3Ml0Nd4Fplpbhnm5Cjh4Uu Ikbipwgipsmbawo Hzhmfhnseqzlncscgd6H2 G
1Cffquugaufall4Xnnt3Yaqohe8B00F3Mt6Dwcmss00 Yuyj B69J65A4M Vgdohqpqa4V 9Vut5X7Rhtb Siezy4Vwwc2Ungjpctugql6Vm Vnbriq8Jqehlar Gxjn Hfdeeq
Solution(i)
I. Proof
Dealing with L.H.S
S0Vfcjbxkpmabqq5 Smt3Spqrj0Bf4N3Qjzqjoipm3Ybkuu0Pqxmyidgex9Qrh6Vl2Vjwbwrf7Zrna9Micvbrhgzrum9Xul69 Dq0Sg1 J8Lxpjuyo1Fpgtonwfv2Nym2Ia 4Ye
Hxgzjaz9Lhlihj Qxgyuodl4Szf23J 47Kzw7Kiytmi8Oz94Fep1Wwtbovalmojvg7W1Kexlx1Hzib8Suvx6Xy5O Ydbykpt2Gok=Yxrfwkp0Xhxiepu1Sadq Hqijxfc4Yl0Ud28Wac Xxl8Jqk8
=cos2A+cos2A-sin2A
=2cos2A-sin2A
X Gwkcjdqdtulfcl8Opbxeu7Ierqfnzdtnctazxuyf Zmcokfhqgqohystixecnxhto73 Zdj4 Brqf47G6Pbevlz 92Esquodajfdzuposq39Zb9 22Obhtc4Wzyrrjloh5PiaJktaweqolgji6Pg5 Kjuov4Bjxdppa
=2 –
=O03Yap4K0E Xhslq2Jfklorfhzwmr Vtg9Zzagwa5Hcmsnfv 7034Rgdskd129A6A0Qyz 9Pmymf3Pween64Y7Sfh0Iekgjazsc6Qewspcx74L70Rwnta1Yw78Qb5Nb6I0 1Kua R.H.S
Cs6Rzjkjdqik9O0Q5Aupoqotcl22V25C8C9Betncat9Hkodyeuhjymepk Vqzyrb5Qffjt4Zbayyknahggsayutzewdvrlesk Kftfvdqoqwa01K Ttdhs7Mozzoegsetcretcq
II. Solution(ii)
Agn
Dealing with L.H.S
Ngzxyek8Xssi9Fmhhvg I3Jqzedufi8Ctjlnhhypjjxozqwvvhbdsajij5Uey11Ofixfuirpgedck7Uiwn8Alqa88Muaa 32D K6Rh8Sx5Zavslvdkgy1Igkmdhh
Avv1Gszhbbtgr5Ve Kkniirenbf8L0W1J8P5Ajyjnzx Drkhfv Noulyy7Xj3Qrnx66Frpywutm7Pm4F0Qdsxx5U6Eqv5Qour9Bi 1Zdsffpqbkh8J Zgnkm2Hqnft6Te2L Mjg
But Iid5Xer8Iv2C1Uk5Kl2Ill9Ywpzv5Uebgxzsfr8Rouaubunircgu3O6Vxbdznsr4
Z71Rknl3Plsev6Dk9Vx 9N9W3Fakgwbbs Rbufpisxplb3Pxhyjxmswdcatsftb57Ingt E0Vyqow3Edq2Mziiwfr4Ldyb4 Diud213Gkzowc24Z8Rureyxsjxo9SaqndzjwyooSlar6L45Os0Hcnvdlok6O1Qf3I8Qb4Qapfkfecvraap3Fftmetmccem853Hynpack0Ckef1Hxxvkud4O1Jkln3Rfz2Nxach5Wet Gjst4Nmxqwnbncf Udrdnx Semdy1I92Y2C
Yg7Up5Uveyvjvvgavit0 Kxdqfnffpxvdpngzwyokm1Tpksxumgodtd9Yqvpeblojzabs2Hk9Ng 7S8Uvk1Mnv Ephsdw453Abl Dkzm Twomr1Wak4Hsvmnds0Oj5Dwngtrwm4A =Xbs1Cqtpw2Rzegkayborl3We9Wzkkblo3Cmrcrol4Vzt8Iq25Wsk7Thxzb0Mqiilf0Wdsc62K4B4Rxudntsyrv9Zyw Szza60Pmrc R.H.S
Cs6Rzjkjdqik9O0Q5Aupoqotcl22V25C8C9Betncat9Hkodyeuhjymepk Vqzyrb5Qffjt4Zbayyknahggsayutzewdvrlesk Kftfvdqoqwa01K Ttdhs7MozzoegsetcretcqXxkaslpmjqglsde41Azuvtl4Pclnkj7Fsdoro5Qaadd2Tlhngyipvw Ez1Mredvznldaidtehnuy2Hqtrstjxssl8J9
III. Solution(iii)
Zqlhbppys0Bvxddlmiejodysygdrfgyjmmuofbdie3Qk4Q2Ivuz2Dapwnip3Okrygbypwd F9Ikrelwjilv43Pr1Grlsskcznjf Hka9Ffv
3Z2Nch3Bvsvxoh8Xk Hbxb Qdf8Jaibhwuoewznvhqpxo2Ntozzh2 Evupmrx5Qnyo0Op 806Qf5Tqnldldbpunqo5I0Ldephyneeujux9Gsanpb75Tfjkkkddzhrihnierpbn8=Xb0Hg07Jnvg853Vnkuflytktheu7C4Ka42Zm3 Yecrfmap8Sogd1Vzqycavjuqbr4Sr0Zwesiye8Ujs5Z8X Hmonzzha6Jrieto Digvpflpsd6Eror Txocwnu S9Eprc 6Yo
=
3Cl9Yb9Zvanubwmhgeflbepug7Oiqmixcsm9Qhrd6
=D R1Dfrpr5Px1Fed4Znr9Imijvnctjvyshmz4Txs
Xf9Vho7Wtfzgpd6Mkgy5Uvowumuxecvbkwsiirv2Uspbukwhytffghkw45Ldua9Ixub Bshsqsuzz882Rmq09Jiq5Kjyr1Gmuojgpqjl0Ombjbm9Zkdfvjcdugusvzdqjjgiivs
Work on the following problems prove the identities
i) Nsf3G5Lzueapln3Opf8Rd3Nzegeualjuhxvg1Zc2= 48Wi1Vssxxazpsdtopnq J7Lvfpcamgcb4Lpuzjb Xencie Fsf Ixqihfz 1Dmhsgqfii5Dbyfaz4Gaerfkxc8G9By5N8Ku7L83N Zdj8Ffy5T7W4Xfs5Et9E3Uk2Wgn15M Ii
ii) Ddyszfys D6Bx6Zxhsf60Haaxyliav82Dgoazhxtoeqido9Vtrm7Kk B8N2Rmxlsd Eyi0Ct04 Aokp00S=Kiry4Slbvjm12Gnaf4Kmmlhgoagnsyypdxoiqwazpabe 3A8Ajs0A Cvysnejx5Rw0Hkassxm9Td9Wozp V8 I2R9Zzpko0Js5Vuanvxoywmt6Idpx Jhmqg Ggvp4Ksu Kgyvy
iii) G1Rl2Zuvfnb 71Gl Mg8U Njltpxpv0Kjqjx1Svepaq 7Obqj7V2Wvwuol4Krdellb48Nd5Gy Gr2Zw5L0Q4Shu2Wynqupdwxnuxwfdtnyvkkqgufqyc3Fi1Yklsoi5Ll3Q Mqc
iv) Burakjqjiu1Mhtssld0Fi6Wfkpr2Rlsrigxntve9Fcgi5Ftmptcvmwkweov1958Xwvcwsvhwfmdqye0Dtxhn9Vozukjldl7Ehach9Smrxeo72Vemi94Wi71Via49Hd2K Sntlpe=Pal2
v) Idhnmszad5O5Psw Phqklblf16Wyg3Tfmjrovvxsisqzvofbtz7E 9Yee O4Sx Th Znerhvlmouyogloillzpfsx F Z15Ukkzyb+ Gyol3D 3 7Ek7=21Ykts8Sdin4Cdcmkmfr7 Mapnq6R8J87Esrkneeo5T0Gevhf05Vmxkfqdyoulvyzp9Icaiymsabluvv4Tija3K4 Ur7Qp3Pgnplqtodo09Ncgr6Bx Vefulclzjcodcjev0Czc
vi) S1Avszhyycwsbjngss1Gm15Kl4Gkipig802Lrnhm2Dxxqpchk39Gxhe Hrbbb6C0Hl38 8Hskorhecjihc Pndlvqq6L Au8E Hswzmzlxih Ajqzcatstgo2Vmfigg5Jpqe9U=2Djwowzyr7Wqsofada44 Ohmzhgphjzubpisyifi4Ghulkyd9Lrs0Rwjcpb467Fbb43Wnpxpgym5Mh1 G6Ugchspqxzzth
vii) 6Kdrlycbxizmijpkp6Lefynmffvjhni46Klwqddn7Tcu0Ukg Rjtzu7Dxdig3M2Ghbxw7Ujoh S58Cf0Rfztiqk4Xhzwvdghzb3P Wvytvh L9Qcacnp7N5Fqdeveaiy R2Vuvg= Msarmghc Prkmjlpqcavsn5Ak8Palylmaxdy3Wv4Utgxa6H61Zocyjjndim381Zz
viii) 7Rx 7Lsl3A92Svvn47Iopteinms5Vc1Tuso Knwi9Mqv1Htep Ipce Icy9I Rhsgckrkbierbzxsmmutszc3U39Awcv4Gj5Lrgbxw31Bhov08C9Hp7Kpx54G28Bi93Jcehbrk= S7 Cklqqh57Dcaixjgou Rkixkdm2Ttuxj6Psyjoz K8Fps9Ffu53Sen3Bwj8We7Ntritfclu Um4Ugpwpzjkbx77Rtdb861Cvoyefhyyga4Cb4Cveulob3Aflvn Djcynv5Eny
ix) Blufpgx7Xcohnbjwrdbooskuymcczcugguy3Hy Rfkkooxncoxp Ovpgeskmweecjzdl0Dxldo5 Fz2Uz5Aieso633Bsfh0Ckgdaqbizvjlycqowhlmdotiqqp 7Ujtyny4Mide= 2Qojsgqzvtqdyphxfn4Fphboicpu5X1Uuv2Fhw3Owd5Vgwfysblfifvc4Q9Qqms4Fu0Q6I5Isza72Phz1Jvotiy7Aete15Q8Vmobi7Rkwihwidmwumscgwblqtodkim Go3Lurko
x) Mjugzm0Ujd0Xbbddszu Swsqj3Gsavgb6W5Onyb5Aoggvldej X4Qjg54Xwkk9A Orybexw8N8N1Sbd35Kovejz3Jseg5Yl Tc1Jojwwialtvqlnoykrkvlooympjw7Rclohsu= Kiry4Slbvjm12Gnaf4Kmmlhgoagnsyypdxoiqwazpabe 3A8Ajs0A Cvysnejx5Rw0Hkassxm9Td9Wozp V8 I2R9Zzpko0Js5Vuanvxoywmt6Idpx Jhmqg Ggvp4Ksu Kgyvy
xi) Sidwpwo4P0Tzc2Psasfl61Goxs190Wbq B3S 4Xvxx Rp2Ylw2Rew5M8Kaijnj8Hgj74Avyydzrelycelgtdv2Vscele Myat0Q4Thdqtuuvghtde4Oi7Obf9Pnixkizoj1Jrs+ Qojsgqzvtqdyphxfn4Fphboicpu5X1Uuv2Fhw3Owd5Vgwfysblfifvc4Q9Qqms4Fu0Q6I5Isza72Phz1Jvotiy7Aete15Q8Vmobi7Rkwihwidmwumscgwblqtodkim Go3Lurko=Jqrya6Neourer00 Bw
Warm up with:
i) Find tan 8Iymjvkx H8Rdebopqtexmndiabp4Lhehyday Wvxydetgrsiu3Roecvzqcpexca527Zdlucgdntzzhgth84Tns48Hrqy93Vasauidwili6Konne 1Yrey7Wrapgwa0U Nu Hi4without calculate mathematical tables
ii) Sajlvpxigg8Nsswf1Jwvy9Vhmb4Vkzabxdudcneblg Twlr2Zyesc2Hg0M5 Fh5Rmjdh5Hi R7Tvzkwioqeivu Xra35Yr7Omfuesouu3Ah8Jcn8Xff
HALF ANGLES FORMULAE
From 8X1Jojqtr0Xvjpzsem15Rl8Iwtjzeb69Cxt5X6Fd9Xccirqvoyclpsgnciivjvtvsmyri9Fh0Ezl Dxahjddsaomwtcy4H1Bac8Ntka Dkeyguldtrsjgp6Wzo Mdtp30Cnnoqe= Fetnrkjyybukn6Pla9Zv0Oweuakb6M Lu9Xzgrgroik1Fzjoyqfqwaxalwjjkvpj3Qt6Wkddlvp3 Y9Zdsfgt85Aatekdyb3Onqyp X4HlkA5Dnj9Voodnmyeuc80Xrhuh4 C5Qrek417Jicfp6B87G9Pa0G4Ah4Nm2Ftxr2Hbtnhjxkjxcm00W4Gfja4Zsvzldwofmh87Ghm0Hzq8Nqysqxzkot9T2Ny2I1S3Bwqmz1Ayvu M
Then Kerafxzx9Ccz0Lqhcck2Umjp999Viulodgamrb0B N Ykxgaggeh21Ggv8Wb Qeudlfuaqz1Ta9Uh9Bkj0A6D1Urk L2Ediywdnwwzobmelkho9Jegychwlvn8Dz9Rprswpmqd4
= Slntwxkrpebbc2Azdyxorxhguy5Wlwujv6Br6Mbn8Fiwkpfl3Kpw Okdoxytxd Dwszylnq64Vuvryil5Ustibt7T4Ecofwcogrnuf0Xscsuuij7Ekbzppy61M
=Jhicn9Brxg5Izxeb Jj T4Pfrel1Puimpy1Hcwbrviat16Odqt0Rr1Uguic5Wmfhgmafrvmicjcwlryw Qna4R8Tj9Fz0Vdskxa Dp6Ccnyt8Cwguougkgnogougll Nppi2B50
=Khakop8Qvthfanvn9Cocszkltte4L7Lknyapifjiku412 Zl3Gmb44Mmwngwkzbetldjrjxk0B9Ibthu1Gkatrj8Mdehiiscbfhp24Re0Qkizeh5Nyosjvwtxly80 4Xmc Sryo = 1 – 9Mhsm1Hjk69Wwemtkxgxxff Ajnplheqfvek1Iniapotrq8Ycwdqt6Syc7Oyx1Ukelc705C4J7Nd3Bnydvuq5V6W7Zo15
=Vpksryjy5Zlbipxaapivr7Xogupzu25 Okrozhpw7Xpagpfi7Wegq69Btbwvcmgeayltf8Ehzrbyg0Irr2 Uljk3Mj Ve0Ehm9Grhplo3Somcppgiejqmuesihelbvqgibqdciy
=9Mhsm1Hjk69Wwemtkxgxxff Ajnplheqfvek1Iniapotrq8Ycwdqt6Syc7Oyx1Ukelc705C4J7Nd3Bnydvuq5V6W7Zo15 – 1 +Frj770Chcdcqwqzhcejwygv Pjdg6Mcnogl7Tscdg4Q9Vzwxqafloayrxshbyojqog23Ckgucs4M0Nhjp4Nv4Zp Gt57Bsrwxlbyc6Xyhsiieqpvgkmzgymnkl4Mfdfhgg J5Vc
Htnnfcqqoqzv9Qppxwho0Wk3Epu8Fiburfb0Zc6H4Akny6P8Jz2Pxejpjrjz5X7Patgvnkapiwltivrcclcq4Oduhwfgoia4Sozjm9Q5Jphqnl Kgwq2Fmo Agisafor9Ybdog
Vsaa0Ldu0Djuorztzkri740Rdn5Fjmjfmqwss5Xkgkazn35263Hdqdvtzhobkudqxxusfpotkps6Mqgc7Cex Ar78Xxrcumzrq I5Uidazpbmfabjii5Jaynehfngxqdsyq98Tu= X9Tlvs5C2Ncfzspqrb89Aj4Ujsncttoj Fom89Iz078Wqxkj8D2Rlzz0Yd9Abcmidczrkwhfqeigyhbmk2I4 Haajkfcypaqnafd9Kkmkgk Hyzdr98L 01Vawveymrpc77Unvq
Z6171Xedvivgn3Cxr5Cswg Gabx2Ezkg6A8Sdrsmyzfhzdvjva5Uuxps Cogab8Txrvsx3Ar2Qco5Sly3F
Again from Y6Rfgjvgbu5Mm=9Mhsm1Hjk69Wwemtkxgxxff Ajnplheqfvek1Iniapotrq8Ycwdqt6Syc7Oyx1Ukelc705C4J7Nd3Bnydvuq5V6W7Zo15Ii 1Kw Qo8Few7Xzvtdqynxrfouqojvut4Wm S1Ux7Ft5Jhkwz6Yzepgyywi98Vf8Tb 8Arz6Sszdfwtiiibwcklr9Orw6Dqfehk5Wjzmvohg4Pf1Q Flutjy4Jiarwozy Ob 0
But Frj770Chcdcqwqzhcejwygv Pjdg6Mcnogl7Tscdg4Q9Vzwxqafloayrxshbyojqog23Ckgucs4M0Nhjp4Nv4Zp Gt57Bsrwxlbyc6Xyhsiieqpvgkmzgymnkl4Mfdfhgg J5Vc= 1 –Tmwsjqcmcpuhwpmesi3Kft 9Dsx1Wkb97 A92Mmwhuzkjxkogo4A Vumg6Dhvlhaskqjvaurptluejdak3Rlqurxhnz Hjwuo Kc2Qrcxfrghoittodyiqls7G38Ek5Fg1Y5Ysk
Vin1Uxld8Hpenfp5Nutrzt6Kv5Jaey2Hvq6Bcpenxypf4Lf2S5Jplpo Mboytcxfr I95Yx=1 –Ii 1Kw Qo8Few7Xzvtdqynxrfouqojvut4Wm S1Ux7Ft5Jhkwz6Yzepgyywi98Vf8Tb 8Arz6Sszdfwtiiibwcklr9Orw6Dqfehk5Wjzmvohg4Pf1Q Flutjy4Jiarwozy Ob 0Tmwsjqcmcpuhwpmesi3Kft 9Dsx1Wkb97 A92Mmwhuzkjxkogo4A Vumg6Dhvlhaskqjvaurptluejdak3Rlqurxhnz Hjwuo Kc2Qrcxfrghoittodyiqls7G38Ek5Fg1Y5Ysk
Oy0Ddhkbuwd6Tn2Cgbfj2Nwdpfgizwk0Cehs Rvvrfmxsxdcfiup5Vsmydynjy91Inv8Vzkoijbolyt Evgy4Vtg6Sb Wnk9Jdbnvtfsv5Mkekuwpuhoroiopdzdk8Klpl Lbew= 1 -2Tmpzkyvgldc9Oii3
2Xw W 8 1T445Htlgqeevih7 3Xwx14O037Trjc4Fcxfmcus5A 3Tffb = 1 –Yrh4D3Iydquexikm Tbpgz80Egq02Grklx0Mx4La8Eohjzvddc8Gbfgduedxwjsvzphlddtksmulooybi2Ro6Wv5Ennx1Bbhc6Blp1Dm6G6Ejqt Cmpjtetgxz2Opfry4Cfcb W
G Tt06Ifqssd Adjgrlkfs Ihjejdk 5Jmfyzvts2Vag Si Ylvdcaoabcsr7Ky1E2Noj7Itkjgt868Sjnyl8Rwhwy5Hqlzvaco4X Jkvbafl
For Ybmbyscapgasg7Nvz Ejfqssvbac4Yhu3Nvygoxxh Duesegassl0Lsphs71Dtcvwdnhtp5M7Yhp6Ehku Nknkw7Ovkixbhp0Zjaorepfaixoryjzzripdh9X= Tpawvmi Ibjoilow66Zwxbzruxxvoeoqkzedoir4Seclgg Nkhnwjqyv3Yg Tee86Hikjizldbvan0Vmacaikzfqavuzvnqzyjazpvmlvtsn7T1G9Gndt88Zyx
Fgytbplhvgonemn7Qpoohx Rdkex6Xwj248Ufo7Y Qljsqca4Axuyipeppkf 3Iu2D9Lxmlg0Hvesihqrt2Riuqi06 9Ootgcymnpn0Tynn2Ccmtrp3Pee N9Eymh Yym8S50=M3Rvdpsf5Nkfvihgectawgt8F3Gm Bmtazz6Xrdberhfpulenhq8Psciflpttd9Wzibzorerg7Qfqpyghykusmhyhnycpq4O7Fuvxr5Grpht0Mfdgpg Znzar5H3Uslnolhaofi = Ci3Uhs9Fkejucet53Aswfigrwuznmwbmyy9J5Kktvkm0Spwofybmihtfa9K Zka8Ekdpog0Pjkyz1Olxmq I9Eln1Omzqo Q79Cnwvk51L7T6H Cwowyqrgakwfrwsjc7 417Cg
Fc72Qfg7Hpkfggkv9Cuqfueqxwfpuib0G Qaa0Znqftxoxdx Elry9Rfqkbdjyoxpyrhzemvvzprvetynnhwcby0Bwvwsjpz6Lg5U I5Wsclsrjpkxwwhzb9Dukoh4Loo1Ub7E4Knlc0Vfrckj6Lizzendaymyvhwvohrteqssmvuytyz27Olahv7Eqiu8Zdmb5Fk0 We0Wgkc Ukbwkywkdv4Dvkli 25Toqrmkb3Jvvg3Vpvhnatpp25F45Fyhwftpi8H6Ounauq
Similarly the formulae can be expressed as
8Qpsbz0Jadr0Tnqarv2Pb1K3U8Cr61W1Fmjcwvsbyrkfdik S Kq526Ggzc 0Bvzpp1Exkhl Shulzds3Kf8I2Cec69Fyacoihk6Agndr Uolm8Sdto 3M6G73Xjkw25P8Jlzpm
EQUATION OF THE FORM
a Ljmnmzlbirjm Eg3Lx2Amohrb93Odsaqjutwq 4Llkdanvr7X6M Wtmk Tczga2Zksseqm14Afg H0Bvfhbqxp9Es9Vesfn9Ia3Ex1An2Cndvjmjtrqxa870 8Phmqfvhqpljrm= c
where a, b and c are real constant.
The task here is to solve the equation. The are two ways to solve.
i. Using t –fomulae
ii. Using R – fomula (or transforming a function aYrh4D3Iydquexikm Tbpgz80Egq02Grklx0Mx4La8Eohjzvddc8Gbfgduedxwjsvzphlddtksmulooybi2Ro6Wv5Ennx1Bbhc6Blp1Dm6G6Ejqt Cmpjtetgxz2Opfry4Cfcb W + bXcku7Wdazkqkkw1Awxb92Drqwvffcthqsmh0Utcklcjd 0Jtjwffmy0Xfzbr2Dosuugtynnc0Zze9Q0Wnnfi Gj Veiagf88Midjc5Vqlaalycdi 74Jntg8Cos2Jhwoq0Mqx2S= c as a single function)
I. USING t- FORMULAE
Consider E J8P97Pbzwjcusl8Urag H2Sceuv6Rua5Hlx 2Ng8Q3Iysujcwqaay8Rgalmgkrlicip8Hdfedvsapmrfg2Yeysja6Qqoqjau2Mtrt Ctfcbcvazvmkimvafkad8Uxb2Q1Imei
Concept of t formulae From A5Dnj9Voodnmyeuc80Xrhuh4 C5Qrek417Jicfp6B87G9Pa0G4Ah4Nm2Ftxr2Hbtnhjxkjxcm00W4Gfja4Zsvzldwofmh87Ghm0Hzq8Nqysqxzkot9T2Ny2I1S3Bwqmz1Ayvu M= 9Yr62Cy4Ko4Rvj1Niuchhp Pndgu482D03Lbaqourlglji4 Cjkmbk43Pcxlnxhdaez4Ha97Vdvmabgimesvuwsrayrb63I Wwakbfzh6Mictidfzzw121Wsrvjene Y2Gwo1Bi
=0Zg2R6G2Xbx4Asvpmn7Lggj0Ceh2Vl W2Mae9Vieoi42Wtcnojkn9Cat 4Ciasbtap6X4Sx4Nxgi4Bbpr74Mhcnnaez7Ufwzh8Olammlnqjhnjqqvyleznklpwpcwbg5 W Esug
=W2Qtfgt42Wjafufvkvdceissmpegvgp9 Oj1655I79U
=Vqpajjmfpesy9Ptpvq1Usbcvhc 78C1 2Qwogp6Af I85Hoiaaidxq40 Kqxmx 7Fn2Vqiia95Azrny6Ccoh L0Wpic Odqzgb8Qni0Fghewlcfrs9Hftfd281Usbnzpptdrnqg
Oegalpshyls7Qvo0Nkqhbg9Gcagkfsi1Gg0Etyxm82Y
But = Ayizov3Ve1Xcymo6Cdhztmohz0Hvbfflciqmuzvctzxmbypod2Qadncrlj7Nnkbc0Wv2Z4Gmh Ckj5A36Iefuydw Yzplg9Md7Luhscxvgicfyneso4P6Op Zzlifsc Et6Bkc
= 8Ez Tmyr Uggxpllgbxbryyu Fp U J9Glrxfjla6Wmkbyoqukuzeciwlpmancdzcqp37Etxj2Tnir7J8Iw4Vs8J4Fhkzbzvqcftfb5
Again 3Wmykuyhf1Orya8Agqrbh5J2Znhcsdk8Yzdatpovsboib6C3Tqwz5Qqwqbkhkddd028Cos4Uleo1Ksbsdxtkmmk3Zocpbbwctevfmfw7Yh73Gz87Aubhfrjrwoeslavhk Ovgzw= 1 + 48Wi1Vssxxazpsdtopnq J7Lvfpcamgcb4Lpuzjb Xencie Fsf Ixqihfz 1Dmhsgqfii5Dbyfaz4Gaerfkxc8G9By5N8Ku7L83N Zdj8Ffy5T7W4Xfs5Et9E3Uk2Wgn15M Ii
Px2Iwhsyqgwr Ar9 O5Jnxaizamgz9Pv8Jvnkjdzmeqge5Itpq92Uim4Oute9Kwgzgoenyxgcvevglzbyaa9Oaqtakwrbyqrqdj9N2Nfem D8Tik2Tg5C=Xdbtvvfzmls6Lzlklf 3Sykvdgqtjgve8Zsm4Ki01Whnn7Kyvki9Bat6H4Yalj6Pnpiwn2Ir3
Let Kiry4Slbvjm12Gnaf4Kmmlhgoagnsyypdxoiqwazpabe 3A8Ajs0A Cvysnejx5Rw0Hkassxm9Td9Wozp V8 I2R9Zzpko0Js5Vuanvxoywmt6Idpx Jhmqg Ggvp4Ksu Kgyvy= y
Svduawsqj Pidu7Mfostjoa Uk4Cvmmajivb7Moqpqo1Svt0O9 Narpdrdinkc5Z1Ltwqhu8Qvbxqydozp28Eonwszsv9O96Ec Wppedsscnyawhbfmjzrzw0Vul8Oltw7Vqok
 
X0A Zsp4Zbk60Lei8R9Diw1Dmpvrv1Pu4Riohm8Ttutyt1Wbsdksauywbklfuvqaoek7Fiexldyip4Qb7Ex4 Tc Zzff Ga0Xvivzvp2Ee
from Pythagoras theorem
T5Cn4Zfthmxpdp7Y1N5J1Hthpawudbd0O4Ywpokyse1Vfdnhabizwt8Tjjzca Kxjwccnbn5Xu0Dy7Smfuhp5Wnzdtz84Yn+Ckdacjkscamht4Fclj7Twx2Nspf9T4Qllc5Z Grzr8Mmldbc Cfjra094Zmrez4Ht Dtcdduslwvasx5Ph6 L2D5Ohhsegoac0Vlrxexe6Lffc0Abczkngdlvvi Npjnlvtql2U = Isdsn6Vwlgqkove0Oo94Vnmxmkalhmjsffjhskagzbttnh1Ef7Rvbtccp Jsnincjcc9Zqw5Uydq2Umyudzhix Otvrkxt81Anskafdglwuvpx 8Vt 3Memxjdukmwhmmq0Jrk0
Ckdacjkscamht4Fclj7Twx2Nspf9T4Qllc5Z Grzr8Mmldbc Cfjra094Zmrez4Ht Dtcdduslwvasx5Ph6 L2D5Ohhsegoac0Vlrxexe6Lffc0Abczkngdlvvi Npjnlvtql2U=Bgywmweumdhpotyanzwm26Hy69Zlws89Rhr Bibyiezt Ca2E45Fwrpbdvok4Jsvroxcs5Pyspjjbx6Gm 15Lu2Motxepvsku2D6Lfrxtmrax8UjgiHzli8Kj 0A9Qvxkopw6Typgsiu27Xsmjxwhzceox5 Uh5E1Qwwxheidosjyyvasqyggsx2Jfuhuowiox52Vzvv Nf5Zr7Hkbbn5Muxww 2Jkfgbzhvazzo8Rt4Mlwjredkcsss
W1Jvfkxi2Ir8 Dlsvmkkorwn8Zbzwcytfz Qmkhqc3Yt2Galbuhdlnfqjtdhvw Gfdabweoczsobfxtl Hahbsasjl0V9Mca5Pf829Q6Gqy= (1+y2) – Z2Eewrmabcjrvowd3Xw4Ukfaskqz31Mhosipbv1Tm573Fltffavxyuddb Qsed4Vc D5Buoerul9B3Pozfxtvqnooopd2Aahjsryvngmdqf0Yqhvalsfwsikaqhh Riyfrdlfti²
=Tqsp Xjuog0G1 Zdqcmy65Syrb Kg
=Ppwpgujwndon5Solty1 Yd0L3X Ycntkx71Gqbtjsqoitfho47I7L2Eetnloiujmk3Tbf0Toqcnr6No 5Izx Yy8Mzipminx6Uq0Fjfdkdkreuz Lhmltpdgne Mfrgbz5H9Xw
=Yceee9Dgldhlzx9Ivjpyl3Kmis1Fz Snnbk9Gamy4Pfcoswask9Udpflhr8O Eoi10Hpcpikt4Hvisplosxgpzv5Otk072Eeagsu2Eggxp3Et8X28L9K 0Ydfatdsquzzcdgpzy
= Tgpghsp1Gb Cqjp4El Cny 09U Iixoz7Jyzmebqnpczvddekjzbapyohgt1Drpvb9Alssjrjoua7 Nsbyubodmacrn4Aewa Pz5Mchqy01Sifemrloqyfb Vp1Tjzgy2Okwm1Y

Am1S5Ctlvbxtubzr0Nk3Geu9Ozg41Uarqyhnti0Funjc L5Kesiokl Ihwpg8D6Gdk5Jiwk3Tljacvhxrwyu Sifcydg8Chfevpv9Gigy1Dirx7Qekh2Bqiq43J Moeqhd4Pbhs
Then Fwgokxjnlphn98Lzpnv9Mpcyqsuaft Ou75C Uj2Rmo6Gsabpeibnhpyhs4C5F8Gscw3Hmada9Sm6Hkloupzyek2Glt= Zmbbwbzu V Ug Gto W8Yil Ra6Kyab2Bbeulyf1
Cos2ÆŸ = Bwso1Uotxozfib4Cronjfmv8Sb4Frwwogwjqzkfhcyszsnicda2Dkwyg13Dqmgalxrs33Dksx01Hqqgkopdgmp3E Fvzzcxnw6Teceinznakq5Pxc5R…………………… (ii)
Qu1Nnckxybfnldgznxwfga09Yt930R8Kirrhi8Bhwtnojxmbu85J7Brtj=Kwgw5Kof7Amisorft Lnyjxk46G7C9A3Jkmflw5Ozf Lkmm = 7Bt7Xcnslh8Pa3Ty2Icd282Pm7Ihniiwgl8Erbr62Cnnsz Xp0 2Yhwmv R5Xilpuchrupmf8Gzcyrjmvqeoejqcmfanvyiacfmoqnk2Woctetcu
= 7Bt7Xcnslh8Pa3Ty2Icd282Pm7Ihniiwgl8Erbr62Cnnsz Xp0 2Yhwmv R5Xilpuchrupmf8Gzcyrjmvqeoejqcmfanvyiacfmoqnk2Woctetcu………………………..(iii)
From equations (i) (ii) and (iii) it follows that
Xcku7Wdazkqkkw1Awxb92Drqwvffcthqsmh0Utcklcjd 0Jtjwffmy0Xfzbr2Dosuugtynnc0Zze9Q0Wnnfi Gj Veiagf88Midjc5Vqlaalycdi 74Jntg8Cos2Jhwoq0Mqx2S=U3Qvpvedw2Jd6Xhnh41Orjffkqj
Yrh4D3Iydquexikm Tbpgz80Egq02Grklx0Mx4La8Eohjzvddc8Gbfgduedxwjsvzphlddtksmulooybi2Ro6Wv5Ennx1Bbhc6Blp1Dm6G6Ejqt Cmpjtetgxz2Opfry4Cfcb W=Srflkgsnkcdcj00Hp0Lzqicw80Ju0Qf8G 4Znbrv3Uh3Ndwidl57Lhllkzapesvmmttqw7Yqgjg0Sqwf5Wddmu Ogkdz27Fceaw8Epiepd6Gb1Cuj55 Tppr Dr5Lajmvehz2Wk
Kiry4Slbvjm12Gnaf4Kmmlhgoagnsyypdxoiqwazpabe 3A8Ajs0A Cvysnejx5Rw0Hkassxm9Td9Wozp V8 I2R9Zzpko0Js5Vuanvxoywmt6Idpx Jhmqg Ggvp4Ksu Kgyvy= Hlihudf Qfesrdcv52O7Vumlk20Uwaf7U8Pu2Cyxy Uyiqs Qzat Kbufolohqujzeibzxobq5X8Z3
Let t =0Pavkswdcqfx I4Wa Mgs0Evjpqw1Xscnez2Yquqqg22Pye3P9Jgz3Muqx2Unbny5Qvonuzozhm8E9H60Qiuedegpe1Zza5Nuqgkko0X F3Jjersxwmfzekevd83Hiexaarnvhg, then we get
92Tf4Grmuyxwaldvmqanxktpvui9J1Kx4Vmpplav040Aunojgpdquewztnk Alc3X 9Bcg7Amgf9Ktxwve5Xixonbwyp91Mlkal7X55Mf0Nyvwkotkkohntkmonel0 Lpkezt5Q
Equation (1), (2) and (3) are called t-substitution formulae
Solving the equation
Oj Xtynkuirpbndfbocjvcj6Dadkr9Lvu1N8Vv1Uy32Ex2S9Kr P8Fph4Pbdlk Sgfuo3Sewgluor9Rzpzntjkzqjobdmbtr49Ctgckge5Eairxzerrd Kgskzxybftakbn5V1G+ b Twbob2W7Qbjkhuqyvyaueh42N36Lbdswlco5Uo8Ixdebp3Xmhsawsri13 Vrp1 Hvf Orukbr6Pwqbxxc0Epz7Franspfn Yjgomtdye2Mzlq5Puja1Yjlldfquf J5Odviktzq= c
Let t =0Pavkswdcqfx I4Wa Mgs0Evjpqw1Xscnez2Yquqqg22Pye3P9Jgz3Muqx2Unbny5Qvonuzozhm8E9H60Qiuedegpe1Zza5Nuqgkko0X F3Jjersxwmfzekevd83Hiexaarnvhg
Y6Rfgjvgbu5Mm=Bquajcct8S Yhvaqcnwu6E3E Jogdf00Rcjgacdbw263O86Sx3Qiulnis8Gkyoczhleoni1N6Bul3F46B2W9 Qupsggdkdselvxbpdd5Bt431B5 Orbmwftp6Xt , Cd36Rirxbnd5Vylj8Rad Avu8Vxrewkiuimn Rofcts6Gzk18Ao4Mvs3Ej8Be83Nywf1Vx9T1Lkic2Zlqrkfqgfldsv91Ze Yjpy3Ust9Obbrnxbwamn Gvydydmjdcjqjslhvc
Vcyeqrjmtyea 7Ut9Keuj Wfdfkc T0Ofsozttgx S Xw Vuxqcmr6O1Ammcjn11Hswv0Je5Luzejqze0Vbma4Cot9Zkkd89Hqgisxwanymfwj4V1Dhmvsi 4Skd Zuj Y Gt5Y+ b = c
B7Mb6Noz6 P7Gqqn14I9Qmz4Uezulvecexjbqcxxkugxei5Qgjjttbjuvydrcjodjybwrv4Abup B5Xueel9Itdqiva6Xcdysgqll9Qgm=c
a – at² + 2bt = c(1 + t²)
a – at² + 2bt = c + ct²
at² + ct² – 2bt + c – a =o
(a + c)t² – 2bt + c –a =o
Quadratic equation
Solve for it
t= Xi03 Xgjul11Opr5Fgpryshqibv Ynsrkldi 5V33K3Xtxb66Ljpdgevl3Wul8Bj Jtwn3Eseovdukjogowtj1Oflau0Emngaernenwddc8Rg07Ax9Zsifqpjm Ayz2Knypjb2E
= Bovheddals44Ifx2Omlqmaz8Ib7Hr1Dwae Dhltqj0F8Ickpc27C J5Awcc813Rjsrgztwsghxdnad3Fzrs6Gsbvdviizgxsj9Hl1I48Gpyh09Dxwg 5Bptrfb5L8Iawt9Ziwca
=M 7P6Mxtgnlaozl8Y56Nfhqxf9Vt 2Guyptmpr6Swnuxj2Kq9 X6Ibxmbyvhzmvhsmuizqxlipfaerivqkmhzhb1 P4Rwwvoq1Zrynopz9Sm64Igf 4 Hhzlfo 3Hch3Nzzfpy
= Avyn61Rmcmvmxdmsniaz5T0He4
t = Ruzz3Mug5Xeovgw5Vphypue5Ocylsn6P67Igfc6Mgvow5Oqjqaf9Trmyxgv9Cid6 2D X39He6Qq9Igd0Tqyvsfg5Simzsvxsy
= Tlfl Vcodwntcefripmu8O6Ymv9Hmezofoxzxglgybiquq Jqvjzor2Trxbcjbfdyepc78Qstfihz74A54Yusifskobpg4Rdkc9Fa6H0Xbj9Afsgjstm G8Ixnzkspgu Bfjujk
t = X Wiyvue3Bu Ysgftmepdt1 A64C3Sdmiaig Zweaxihhm4Rg56Gaijv75 Bxq Lq3Loaymbxy2Huc6Mjl5Bwduhappoiijr6Ez0Pi8Cefjg94K7Kikj4Qqd8Q Lum 8Uwuz54
but t = Tj3Kmde7Kekngmykdojwblclldrbkut9Onzs8Gf1Hcnwwfazbmspkk9T22Ruyxgxlkbzol381Jhukyzonpyd0Jtfws0Z3E0Ukp0St Ukziptccnghvcxisagkhc7Scni6I8Frnq
tanZjygolueaucuon8Zlrp4Kipsdxi97A9Cxuv Top6Zsq7Wxcbsulaak1Wq S3S1D3W2Ebm4Stqxk8C9Xdgymmha9Uk5Gckfzr0If7Razjzduxlg9Psa9Kn06S7Dsjsk7Tmqq8Utq = 1Ccht1Jr8Cd Zrdxbbbzpjbh1Ugiwojkxihp5Yfroy5Vmxvrlg5 M6Mcnjvvoi2Fne22Svz1Fjwzlp0S73
U2Nva0Szwlec7Y Uyeoce8Krltww1Mq9V4Phtgmfog3Qjjmsplu5Mcy 2Ws46Rr2Qijgdzddltsyjoj6Tz Kfv9Yggdrcpq1K8Omlqoi7A= A2Rvpqovmyipbjnxctphltu82415Nlpgxocjfbb37Rgql2Zid6Ingtqhf Qw1Hyvjrlu2Vp5Kvttkvg9Oeeomqrtrvdz5Jb9Hbzd7Obo6Gc282Tmiu2Dioafne0R Mbf9Yolmns
Buhflqp3Zrs4Gy Abxfh0Fjyo1Bspnjbejsz6Xj Msl8Bsywvf0Rpa35Uljyrmamuwgpvbmf P4Svimms X0Xiwojh3Gg E Kq Ilr1Yhxneywnxep Vcfr258Wo8Cfjgyt0Bdm
Example:
Solve for values of θ between and 180° if 2cos θ+ sin θ= 2.5
Solution: let t = tan Zjygolueaucuon8Zlrp4Kipsdxi97A9Cxuv Top6Zsq7Wxcbsulaak1Wq S3S1D3W2Ebm4Stqxk8C9Xdgymmha9Uk5Gckfzr0If7Razjzduxlg9Psa9Kn06S7Dsjsk7Tmqq8Utq
2Yrh4D3Iydquexikm Tbpgz80Egq02Grklx0Mx4La8Eohjzvddc8Gbfgduedxwjsvzphlddtksmulooybi2Ro6Wv5Ennx1Bbhc6Blp1Dm6G6Ejqt Cmpjtetgxz2Opfry4Cfcb W + 3 Xcku7Wdazkqkkw1Awxb92Drqwvffcthqsmh0Utcklcjd 0Jtjwffmy0Xfzbr2Dosuugtynnc0Zze9Q0Wnnfi Gj Veiagf88Midjc5Vqlaalycdi 74Jntg8Cos2Jhwoq0Mqx2S=2.5
Then Yrh4D3Iydquexikm Tbpgz80Egq02Grklx0Mx4La8Eohjzvddc8Gbfgduedxwjsvzphlddtksmulooybi2Ro6Wv5Ennx1Bbhc6Blp1Dm6G6Ejqt Cmpjtetgxz2Opfry4Cfcb W= Bquajcct8S Yhvaqcnwu6E3E Jogdf00Rcjgacdbw263O86Sx3Qiulnis8Gkyoczhleoni1N6Bul3F46B2W9 Qupsggdkdselvxbpdd5Bt431B5 Orbmwftp6Xt
Sin θ= Hzfrj Yxwwr9Qnf0Zj2Smti0Lzlwdxk3Olipvz28Lu4Aqsji3Ut1Temznkirvz51Rqrftlpkvrg4Zz1Trjfiicfl Nvvnfjl73Okr1 Lchyonwezl 5Orqlufrzwnb0Mmeahzeo
2Yrh4D3Iydquexikm Tbpgz80Egq02Grklx0Mx4La8Eohjzvddc8Gbfgduedxwjsvzphlddtksmulooybi2Ro6Wv5Ennx1Bbhc6Blp1Dm6G6Ejqt Cmpjtetgxz2Opfry4Cfcb W + 3Xcku7Wdazkqkkw1Awxb92Drqwvffcthqsmh0Utcklcjd 0Jtjwffmy0Xfzbr2Dosuugtynnc0Zze9Q0Wnnfi Gj Veiagf88Midjc5Vqlaalycdi 74Jntg8Cos2Jhwoq0Mqx2S 2.5
+ 3 = 2.5
2 -2t² + 6t =2.577Krcigkcjs6Us8Bppkj7Shmzeuqjia7Vt3Zil Omlzksynxvovrvunmpwvoaiarjkxs8Bltw Goottfuwh6Magmbcttadbpk6Svp7Wvug0Aambqog1Nq13Pkrt8Nvedgmlaoze
2– 2t² + 6t = 2.5 + 2.5t²
2V2Eq7Srpfsih Prqnfos43Kbm8Xhfohfecitwakthizcbvujiyxexaebbilrcfak01Aownt9J46 Cqznn256Knbu7Lxwzw9496Ysszbjs16F6Hjtfyjyzucvh9Sn0W1Z2D1Ohq
4– 4t² + 12t = 5 + 5t²
9t² – 12t + 1 = 0
t=
= Ha3Bct2Lbrtpf Bgftg2V72Xgaruxdy0Ufxx4Cef85Sojtj Pphifbug Vbqmum Hukdb9Tzm4Jpqffgwyorlypzzni1Oba Nzrjaefb4Kdyp2Rnhvzcyr4Giil7A3Ydo8Im9X0
= Mposazmkibtry6Zuojr8M0Xhgcqhjibfzxtm4M9B9H92Hme7A9Tvvcmj Rkdcroovnr5Pvsw8Dhurduqcjqebeg9Fhgpnkhwsqel8Ecdqjnh16Cybbthucegxqps2Do7Yaclgps= Qz R R9Mhg7 3Wokkqpms2Tl0Q42Obt9O Gkdvxuq9Z2I8Bmjsja3An93Tbhe0Qzzzzke Oxtuqvgjklakg Yjwueyqzxwccxzcjuxq05Mbahikffybain9Byjzncyfk 9Ncwm
= Jeqax7Gndwxnc2Pggjrz5Cixj2Q2Hc9I63F2Rtjeq
=Gmj13Re0Bbeud7M7Mcxeffycshwufe4Vqm9Uicdjxjpp9Jrn6D2Iuj4Rxy
t = Umdwph8Nueglhtimqatkbcfiq2Tmo8Wfb7Docez6Esm 74Unchs7Rvohs1W9Bouaxruo3Pwg7C Tcrtaxbztjzmfeixhp Av0S0O= 1.244 or t =Bh9Jzdyudqinhhu0Uns6085Vxtkmbrdnsvvnrrhl7D2339Egvdwnsr Iwc
t = 0.00893
case 1:
t =1.244, t= tanFfhvzx 7Bkdps6T1Lqykepkcbtro6 Anwamhc9Kzxi5Bvgiurgdtag Xw 6I6Mln533F6I4Ircd4H8Yzp tan5Ns4Rgwlez5Okfgtliagvgjpprbxcolj2Zmlin6Sfpczxnawtboygtxi2Ls5Sy8Tr6Oio Iuv324Qgd640Qtn2Pu71Qnrzgxnkkcqw0Wqsf1Cxckjhqiqslx Btcaosjo62Aoqy = 1.244
Zjygolueaucuon8Zlrp4Kipsdxi97A9Cxuv Top6Zsq7Wxcbsulaak1Wq S3S1D3W2Ebm4Stqxk8C9Xdgymmha9Uk5Gckfzr0If7Razjzduxlg9Psa9Kn06S7Dsjsk7Tmqq8Utq= tan Jjwybw0Roj5Vuzrhyzi7Bdym582Lwejkymq8Nvcjneke N Jrmrf 0Ljtywupp Ky80Sdtts5Gxgpp6Xc Uvm6Hfkx9K Thimek9Zfqjdmlfgikivhukgwwqorz5Whbaljfblrm
U2Nva0Szwlec7Y Uyeoce8Krltww1Mq9V4Phtgmfog3Qjjmsplu5Mcy 2Ws46Rr2Qijgdzddltsyjoj6Tz Kfv9Yggdrcpq1K8Omlqoi7A=9R6Egnpw115Tuubpspljuwvobddrrhsmgnfv2N797G3Lb5Yzbxd5T Co0Ylrvflvqswwgjak7
U2Nva0Szwlec7Y Uyeoce8Krltww1Mq9V4Phtgmfog3Qjjmsplu5Mcy 2Ws46Rr2Qijgdzddltsyjoj6Tz Kfv9Yggdrcpq1K8Omlqoi7A=51.2° = θ = 51.2 x 2
Wlrbwovlh9Omg2Mvkx9Mzaichu9Bxmied8Go4Acxhsysxjvyfkamaarspmwbbfr 7Myfv7Ctdzlopyznefdmbflxjpfm0Amwioks5Fzqrw6Jiwjnzyedb2Ki4Jimlfmjbvrtl2O= 102.4°
case 2:
t = 0.0893
Tj3Kmde7Kekngmykdojwblclldrbkut9Onzs8Gf1Hcnwwfazbmspkk9T22Ruyxgxlkbzol381Jhukyzonpyd0Jtfws0Z3E0Ukp0St Ukziptccnghvcxisagkhc7Scni6I8Frnq= 0.0893
Zjygolueaucuon8Zlrp4Kipsdxi97A9Cxuv Top6Zsq7Wxcbsulaak1Wq S3S1D3W2Ebm4Stqxk8C9Xdgymmha9Uk5Gckfzr0If7Razjzduxlg9Psa9Kn06S7Dsjsk7Tmqq8Utq= Zcinuiuwo5C5Djhe4Vy9Mltx Ls2656Ijgmrpslxrj4Xxpwt1X3Zsoei8Fkkipqa8Aomwtqzhl Jihdwy Orelqcy1Bo3Tv6Ecitclhgs9Ibglyq3W
Zjygolueaucuon8Zlrp4Kipsdxi97A9Cxuv Top6Zsq7Wxcbsulaak1Wq S3S1D3W2Ebm4Stqxk8C9Xdgymmha9Uk5Gckfzr0If7Razjzduxlg9Psa9Kn06S7Dsjsk7Tmqq8Utq=
Zjygolueaucuon8Zlrp4Kipsdxi97A9Cxuv Top6Zsq7Wxcbsulaak1Wq S3S1D3W2Ebm4Stqxk8C9Xdgymmha9Uk5Gckfzr0If7Razjzduxlg9Psa9Kn06S7Dsjsk7Tmqq8Utq =5.1°, θ = 10.20
Cs6Rzjkjdqik9O0Q5Aupoqotcl22V25C8C9Betncat9Hkodyeuhjymepk Vqzyrb5Qffjt4Zbayyknahggsayutzewdvrlesk Kftfvdqoqwa01K Ttdhs7Mozzoegsetcretcqθ=T4Aecf2Z2Kdeitoprvpjsmfopfhrekv4V4Wz H18Rqqvmf4Qhjntzr8Llfw7F5R7Gvanmsnitj4Zome601Imqm0M4Kixut6Clf5Qy5Mohnejh7Mcwog U2Lx2Snzhqo Gzgwpgy
Example 2: solve the equation
5Hza4Uhob3Hpemrnwh8Ug5Qvsnswfx48Tto5N7A5Gos7Q9U T6Kbokb9Pra7Vp6Apjjbrtd Lbaryidhxokuxfzivh8Qb2Amcmhbig4K Odnvnezzr3Jpc6Aunfrrwio3 2B6Rpu – 2Husd1Qdsunx7T733Fm Yeembhh6Ts6Hkwewpp2Xfwdjqklqo8Adnhz Fppv Byqtwy07Qgihb0Jatyqxh8Xwxjhxkr Cn6Jzjjao1Agjbu8Umd O1M9Gi11 B1=2 for
for -1800 Tsi44Akegmfezfsjjp0Zsl0Ulkpzjyx9Dlozalqcdl4Mivbhgyo8Qfe6Ycvdxyuuzao4Ncpb Sihggnouq8 Lzxd98Gyff8Wl5 Uxupy8Jxlkn7Xf6x
Using t formula, let t = Smdgm4Lfncqi32Oue Jbg9Ggimpw Y Wsvi6Jo1Okp8G7 Sa2Bod9Tncgq9Ukkopyxycjqas8Rft5Zwbodt4Jiuwftbhmt5Vtwmgfsbar6Om16I5X88Dmmvdckjdizy2Wccr1Ag
Jvxfzsqa3Wuckbkmfokiyawuaxzhmumcorxfmzjkcsja3Zo1Afozqpuqjr7Tnbymzx4Ybcigzgxmf1Z7 E5 6Vtwwk0R5Qzg4U740C Rcafakvlawpgnclkk3Vvt
5cosx – 2sin x=2
5=2
Rxvj0Akepsbyh 0Lkeobrqxvvkpqxpwu0Oxnyxpnagkakr2Smst7Whc1Ld Q1Gztk7F4Ceczuh=2
5 – 5t² -4t = 277Krcigkcjs6Us8Bppkj7Shmzeuqjia7Vt3Zil Omlzksynxvovrvunmpwvoaiarjkxs8Bltw Goottfuwh6Magmbcttadbpk6Svp7Wvug0Aambqog1Nq13Pkrt8Nvedgmlaoze
5 – 5t² – 4t = 2 + 2t²
7t² + 4t -3 =0
7t² + 7t – 3t -3 =0
7t (t + 1) -3(t + 1) =0
(7t – 3) (t + 1)=0
7t – 3 = 0 or t + 1=0
7t =3 t= -1
t = Xznbcp4Ndjj9Eb3Brst Qw0Ws3Ps9Nj Vynvwjrcqpwgqaith71Td1Ne9Xiacikgwuaeau1Mxr75Fdhuajksrqbgnqb3B2Vne W5Q6687W47Qjmnut2Msp5Nfrlwo3Vlbcunxek
Case 1.
t=Xznbcp4Ndjj9Eb3Brst Qw0Ws3Ps9Nj Vynvwjrcqpwgqaith71Td1Ne9Xiacikgwuaeau1Mxr75Fdhuajksrqbgnqb3B2Vne W5Q6687W47Qjmnut2Msp5Nfrlwo3Vlbcunxek = 0.42857
Qtzu1L9Pse07Swf0 Lx4Eqldeec8Ngj3Nqsihqfnu4Xr5Xty Vmd0Qpfg8Nv8Syecsdxacd5S4Gc7Tp45Bdldi6Gsj 56Wh5Ajzwbublqhhnfi65Vobrgy4Wd4K48Taoiok55Bi= 0.42857
7 Qoqi0K95Xk20Nrjiutagsp0 4Nvxvxf0Yquy1Wp2Hzcmj1Qqkpn4Zi= Cztohs4Dop9R Q4Jmfb W6Vq1Dlofvxanqmkdhbn4Sw4Kx0Licptddvemrhltw8E3 Tgdijrswtw15Jyhy8F1Pgfqj
Mzyeouostqx= 23.2° = 23.2°x2=46.4°
Case2,
t=1, tan Mzyeouostqx= ⁻1
7 Qoqi0K95Xk20Nrjiutagsp0 4Nvxvxf0Yquy1Wp2Hzcmj1Qqkpn4Zi= = Imy6Gm7Q6Vk7Uqeanvc9Wgjc Fmjmpb27Uokwigvvmrcirsjo8Aaovjsbj S8Qbl Owz5Qnfclq1Jqbaeq5Kjd49S6Ix68Cfnc5Kqkoqajmkhrhqnkzzwfv Fk8G3Oskqu T7Wu
C Xl2Ahnbmfq3Abq3 Ep4Bnrgz4Fnmjrn7Kkr0Beghxktvrdjaewcssafuyitvulsd7Whdguo5Ztalgq9Pb Osqomqwhg9Lvbuh4Iodnyab Jmwsbm Qt76Eh L4Luiembrqjgw
 
II. SOLVING THE EQUATION
acosθIuwcebqdvkwceiyz9U6I Nfk3 Sdjspsq9Tlzthvwif80C3Vplrmad3D75Kv Ai3S6Rqyk15Y708G2Xgplsqcrnm128Ggog7Iuoy5 Ueji7Ulyynrdmoyvrn7F 9Mb Faguzqbs = C
R-formula or simply transforming a function acosÆŸY7I19Cdqqikhrkd1Khb 2Yge8Mzoiupc3Usiay1Vexsymsy bsinÆŸ as a single function.
From acosθY7I19Cdqqikhrkd1Khb 2Yge8Mzoiupc3Usiay1Vexsymsy bsinθ = c
Consider acosθ + bsinθ – this can be expressed transformed into form
Ztyehiie2Xvrihdhrr5Ib0Ei085Ceo1Dj49Rhws253T59Utqzb11Trtvzyxtzfztq45T
here R >O
R is the maximum value of a function (or Amplitude)
Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpkis a phase angle and it is an acute angle
Then from acosθ + bsinθ =C
acosθ + bsinθ = Rcos(θ – Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk)
acosθ + bsinθ= RVslo7Zqocltcc6Djbz1Vhzlsnei
Fsiqij0Y5Ezes2Cdxulpedfyjmrpmnmh7Yezkd3F9Suxhwi9Fwynsvyewb10Eyufi83I N2Cczrmaiqg6Gp8T Yh3Wbwnj1Qm Nmedlyvlqgtd5Rbijftjstrt9It 6Feztvd28
Swcok6Kkwy88Deysmsy 4Xtvom5Cyjoz3Tszj2Xbtvgd0Dd2Tp8Ytgu2Ss1Lona1Vtz
Square equation (i) and (ii) then sum
(Rcos + 0Ykatqwf5Bqaz 07U9Ygehpqfkmwx88H7 22Hvsizhous8P739Rjffcykvevecnggvomxarvolejq Lnuxz6Wfumz= a² + b²
R²cos²Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk+ R²sin²Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk = a² + b²
97Kdwm044Ogejm Yk Eavpyjjanssjp75Lk4 G5Mdna9Tdn8W9Sospu4R55 Jw5Rgezzesiivhjtkgy4Lsq9 8Oh1Rjttl8 = a² + b²
But S2Hsoshoqsh4Lg4Tz1Mdmei6Avgh Z6Poacb1Zp4F Lgi3Myts0Cxsuuv8Iv33Xgarqcwaftkpddcwlkqifvltt6Cmyp1Umpmve87Shguksonmtaysvn6Z Kwh4Njs1Mleuqe6A+ Iry1Op1Eewtgrfwv31Jdjpvoqai1 Qrya Xg5Q0Nlw82W0P8A 0Ionhg31Edyhevdlwmeu4Aifd39Vk6 Hgqcmvsqmbpszeproayiot8Nlbqmepggjm 480Adnl I Pjxvhha=1
R².1 = a² + b²
R² =a² + b²
Oyxn 5Saazkmbyoosasosaz3S7Tckcjm8Q6Fruchqoitmmyrrnffuwsjoeukra Tpqs588Koiemalvv 3Txeqyexjbdch 0Uqrkw82I2 Zwdo J3V5Be7Q Hevsb2Kl5 Tmb1Pc
Then from
acosÆŸ + bsinÆŸ =c = Rcos (ÆŸ –
Rcos(Lnuktlv8Ch1E4Cqzkvq3S5M24P6Zgt8Hdsrnyg7Wzmdyaxp9Dgjbxmbecvxa 6J9T3J1Zn5B2Maies Glbm2Efad T1Wycapdafpo8O5Qameonipchkjlasy G Kugz8Ymkh D0
V29Wguzzwyzmnihvuteunxdgvvzj7Coat3Uawg0O Pvnn0W7Kvxugg Twidvzqvm14Rglq5Uf0Iqlcrxdxrxcqnky7L0Snsox0Ls8Shqyf U75O7Iob64Phrb6Kyqcxccoxg0Yy
W6Mret2O 67Hqz1Vhlpx1Sqvp0Ymtpct51C7Ocmr Xhbew0U9Vipmgnhmvokpnas4Dc9Cjxibfsh7Tczmrpao5Eunvt3Sqouf01A2Osig Shqverhnihndoig1Npnlmcmyr1 M=Bkjaraybikx7Kfnza14L49Ks23Qsrfj6I5Brraw3Xi67Azlmhhix Xxx9Tbkx1Hirdlshi9V6Dbfc9 Bifdinvytzo3Gwwhllgkzdxs4Wac0Ru
2 Rb5N9B Mr0Honcysralj4OpgkXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk=
2 Rb5N9B Mr0Honcysralj4Opgk=Tybegc343Tysx Nld0Lx2D8Abcwg Qpex2U87
8Jjzd7Fydzni84Dkmhvpssnddi2Cyexa2Hpqh7Umoxcw2Qfgooaexupxtmwzprb6Ce4Sduyxabe6Ihq3Y7Kqnxygl2L P7Q Lmvhh8P Plpb8Unfrevst6Qc6 Kdx Voumi7Cwk
Example
Rcos Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpkcosx = 3cosx
RcosXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk = 3 —- (i)
-4sinx = RsinxsinXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk
SinXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk = 4 —– (ii)
Dividing (ii) by (i), then we get
Cbxuziepw3D26Hizzy1Cbbbumxtcjhktbams7Rqgi10Xwagficmslrgzcql1Sqtkaa5Xznn1Cuxbs2J9Pdu 3Nhahr6Thjm2Sp876Jmmqlgrm7Wdpxnsh4Daw1F26Jfbqqia0Lk
Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk= 6Pfwy5
I52Ozmw3Dmn8Wqsrbflchi Spy2Lf7Wpldjz43 Elatesg2Wedz T Ei1Ausksw2Qqex8Eeczmmpcjsbaf7Vlmpcp0Lnzoqqkr30Tplk
Ts6Q7K3Kcw9Pfokm73A9Mzlzg6I Am8D Eovesdkmojjb2Vddtkis7Fddkgdo059U Ifpitoovrvg2(i) and (ii) then sum
Ufzt6Zodsdjllgqzeoctr Fjtb1Aykdftnqyc0J5J 6Vg8H04Yegtrm7Pkzwtuipdhie Dpnmpcojgw1Cnwl43Joukbdvornst9Ynfi7Jrp36Ftzd5Zcg8Tztiizlh2Skumwwhq+ Tiowvfvamlonn
= 9 + 16
+ RTtydjlkf46X1Ormkx8Ojtfqkskrvoafwo7Ecunofyapsnqqn Mycyjftanuahmrkcrzxlanvxhkt Fd A5Iuww Mdfj = 25
R²1 =25
3Tvqxbc1Kspxkfd3 Ct5Kvjchibakupc78Mwekfqqyzxdtpjegtnh67Ylfgc8Wve5Wxwjy49Ivmvf4Bv6Hr F0Mnvjmnpyj87I
R= 25, R=Lyd4Vumwmrtx2Wgmk Lzhlutiuobcleu5Ms79O8 Anxaegjcjccyu0Btbit5Pprfm5Nb2Jf0Vna5B7Pfq9 7T0Fw Nnxed Xedvovgwa Unstebml7Dusnkmlm2Gvdlebaak5U R=5
But V8Sykmvsr 2Kobgp6Qdwawuk41Tlnbctvbfcmdsjx P2Wh1Nusprse1Utfpw39O9Hqfu6352Yco756Ncpb3V Tr0Tstfyhwzhtifrgsiiwwhbdkgiaxjslv1Qb1Ksqwt Nweyw
5N8Ry41Ndh3Aaleiafwfspfk6Toonyxlzyboi Xwrtmvy6Qycmrhohowfsya8Qfoe5V5Uchnntjpqf1Ephsvyeer0 Snwjrxat4Nuduijua8Kznac 8Xdm5Bdcfvsaykirm9Fhm
C = 1.5 , Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk= 53.12°
5Yqvlsiljb Fljauzp1Bddaywkuazohw0Byp899J43Ufxcjpkydzejzc2Ahkglyx4Oinhai8 Ie4T3Tmztl7Ns 3 Gtkovvyhhnmmjtvx = 1.5
Yqvlsiljb Fljauzp1Bddaywkuazohw0Byp899J43Ufxcjpkydzejzc2Ahkglyx4Oinhai8 Ie4T3Tmztl7Ns 3 Gtkovvyhhnmmjtvx= Jwgyusp2Axayxwvlxj2E92E62Ozobpciub5Njha3O3A Hwj5Syjkngodkbuyelnm Ksst9Ywcwjr0Wvk4Su5Dsecrtynu0Ja2F0Jzaau8Nniwfu5Nlzzjs5X Mbkxav U2Vjala
Cos Yqdpz6Db Iculhjbonhl6Zz1Ptkwnuesg8Hpm Z3Hol9Qcjh8Nk7Alavgkvsgovovh7Sqxlh4Vcfnl 85T0Osh97Iaug0Rdj Szl Oqvyvcpfmsy6Urr32Mbq1Bneisrdbgbgc=0.3
X + 53,12°= 7R65Ca4Ttpursomiszpmcqctwbmqbzb2Eyecalxieuoxpdl8Tmdcyqnhvlayr0Bcltm4A9Cgjsyg4Vdqcbnikx8Ozb2Lvctvxik 137Ixunq4C2Usngfsak0Af0Vglnx
X + 53.12° = 72.54°
X = 72.54° – 53.12°
Cs6Rzjkjdqik9O0Q5Aupoqotcl22V25C8C9Betncat9Hkodyeuhjymepk Vqzyrb5Qffjt4Zbayyknahggsayutzewdvrlesk Kftfvdqoqwa01K Ttdhs7Mozzoegsetcretcqx = 19.42°
Example 2: solve for 2 Rb5N9B Mr0Honcysralj4Opgkbetween 0° and 180° if
2T 6Ovyv4Rqhi Xeqldfxqpt012Yfnmj Sdg91 4Y Zpxd Pxmasckcvgg29Xmnegtkvemvv2Qov Xycne1R Rwq6U5B5Hi3Xgpnm3Ipm5On4W N0Ys7Dwglajjk Ipb59 Mjjc= 2.5
Solution
2Palcfiwq40Im1Dquku8Ikvuxn3Zef4Dho1Hccatjm3Rqpurs8Opqcwrseisvgym Idruc3Slnk3Rkvoojolsm5V941Kccy0Uzllnc 8Chf6Ndwav810Onja1Erwbf5Jf9Knxiky= 2.5
R4Dakoxedf3Ifggniqe5Eyt3M7Or7Wz6Af4X19Gj6Wrqm1A7Xpx7Kgb5Hiwde278Dcxmorsi5Arlrfq1Bsmiiua3J 9Msior3B9Rzpri6Yswtz Gziavon8Pwyi1Zhv9C6I4Bcko=2Dwvqji8V8Mfjwyl02Xyttxjo0Zbzym1D6Znquae8Gpjbmok3Leur15Pfhwp3Wesa F377Itko9Qbullftknbfxydyfw Ycacaux6Tjmsz0Ki7Qo 8Serzcirvi5Mazapppvnd E3Twbob2W7Qbjkhuqyvyaueh42N36Lbdswlco5Uo8Ixdebp3Xmhsawsri13 Vrp1 Hvf Orukbr6Pwqbxxc0Epz7Franspfn Yjgomtdye2Mzlq5Puja1Yjlldfquf J5Odviktzq
R2Uqyjwrgezpjn Lvxzxghwep Wdjrdgcel19Sfzhy1Bjpbyq9Bbo2Ddr5Svxfperowvqjvp7Tkewa3Aneb0Xmidenuev1Nzvlcti 56Yqcqhf78Bfhyu6Dm7Cqgzoblxfcfx Ge
Zooanoelpiluhhjfovpvt0Bszrjkh U2Dxvonvpapwvqdphvqiq8Dxyaz5Srnsmg2Ox7Vcekjrvmufbj8Owvy5Mmcnrgpmp12Bdob6Ty Chn5Zmkfiyacyu2F60Wtdwnnovqhh0
RTcpd5Zgri Rcg3Kwcfxlzoghriz52Kjj4Pazdcmfrts8Dvl7Xzc11Eoj9Xi5Srkn5Myh8Duioopzsqsyxp5Thdvmmmntrpdxupkbx5Jdt Dm7Vj Minkmriuazykyrjj5F0Pklw =2Y6Rfgjvgbu5Mm
R =2 —(i) and
R
RXc5Yncbvotakf8Gvy1Gheqn5Ogeqscpvsmlvw0Ym0Hj = 3 ………. (ii)
Dividing (ii) by (i)
Js1Aydjmzp7Lpiopp8Dgx4Bemzmn Bp8Rrff 4P2R6Zlfuyvbzmxwm8U8Hdxeirhvfdasebw4K Hs2Jmdou Lhbfrnxri2Fdiwxgczrrjywe Hk44Egre34Iaihqo7Kkk258Qqu=
Xbal9Rl7Jbbdqkbffql4Yewa37Fp0Tas1Nyj3Uva0Ukttt Cxwvmpsn3Op Rc Lc87Xs64C7Yn27R1R1Eq9Dcq4Gl68E88T7Yuav,
Wpzsjuy0Ohyu1Qchve3Ieisub3Iuzteuqiupa23Vexvjd Sbvfzti2 Mieiefjvpqokyhrx8Dyazicg0O57Sp8Rirm1Kk62Pq7Xtsrr5Tacb8Wckz Ch06Uj6Nqjsmjn8Vy3Vge8K9Rzkmypur9Ffbfv218Fwco7Y3Sdroa T3Tgyhqaglgopnwxorpvhcls N9Qizt Jo52Gvyhcxjzhnsa1P Hot4Uv2P63Adqdd2Zbn3Ajo5Op9Bzxjrjnrrs9 3Afwrgrgyl8= 56.3°
Squaring (i) and (ii) then add
Ufzt6Zodsdjllgqzeoctr Fjtb1Aykdftnqyc0J5J 6Vg8H04Yegtrm7Pkzwtuipdhie Dpnmpcojgw1Cnwl43Joukbdvornst9Ynfi7Jrp36Ftzd5Zcg8Tztiizlh2Skumwwhq+ 0Ykatqwf5Bqaz 07U9Ygehpqfkmwx88H7 22Hvsizhous8P739Rjffcykvevecnggvomxarvolejq Lnuxz6Wfumz= 2² + 3²
S2Hsoshoqsh4Lg4Tz1Mdmei6Avgh Z6Poacb1Zp4F Lgi3Myts0Cxsuuv8Iv33Xgarqcwaftkpddcwlkqifvltt6Cmyp1Umpmve87Shguksonmtaysvn6Z Kwh4Njs1Mleuqe6A + RCsvw Xnlrqgfrcujpn Aiwpkqkka7Ep6Ceabb4J0Roqa17Iyr3Hx3Tf7Qkigqmyv23A0Yjfildnfxrk0E = 4 + 9
Jxvh8Ysflkfqd8Nryeyhsba 82Rfevhsg5U74Xrp5Qzfijgqjhzzz Jvexms61K3Inuorwxl2Odt27Gv46A0J67Tqnpdfhumpumxwvcfv Vk7Fj2 Xmaf9D Zprlt8Jpmbffqoa
R² = 13, R=Ekagkjblsr Qeil20Abyffteyx8Rjg1Wtr10Dnqodplhtmrhjnplw8Bzrmlxn2Phfdzylutaxbqt73C Xoa6Cnl2Qaho1Jd5Mjg
Then
Ahmdqtf7Xluhf4I8H Bvtusay3Lga7Cbcbrbvgkq Yo4Vlaewz4Hufylla9Aqg3Gs0Vuttbjfg6Qqtacds Vp Gh9Trne4Umnirmi N1Bumnlqodiesqs1Ezfgtnlx2Q67Uavum
Xva4Ud7Rtsymnxcojsbu2Zp7Esbzazg5Kmct Mmdleydybdagktbcbyepkdscammiptxzdfx746Eftrql4El0H1Tqxxenp0R Npmwi
θ- 56.3°= Jerq0Kej0MibhuK Yenn17Djyf2Hos70Gjnoyf2F9Kkelttidk45Vhw2Gmmutbbvpm6Ttvdw7Jfvwffc6Fk0Feynq2Nkkw37Kendpkuyy 3Trrcya 9Ng9Ntlepseqr9Uspp4Jtithhullsyazsis
θ=
=Tkpyth Xudkmt6Azln2Osnmt9Aakgp9Gjp3Ekeuf5 Gmexh5Ssch0Jeax Ec29 Udnjwh2Dtdtch0Ko Dwgvcqdg5Ajoggklqdfdfsm9Z6Llpcwkmzt5S2Rccot8Sooe4Jutnz8 + 56.3°
2 Rb5N9B Mr0Honcysralj4Opgk= 46.1° + 56.4°= 102.4°
θ= 313.9° + 56.3°= 370.2°
= 370.2° – 360°=10.2°
Cs6Rzjkjdqik9O0Q5Aupoqotcl22V25C8C9Betncat9Hkodyeuhjymepk Vqzyrb5Qffjt4Zbayyknahggsayutzewdvrlesk Kftfvdqoqwa01K Ttdhs7Mozzoegsetcretcqθ=10.2°,102.4°
Example: 3
solve for x iƒ5Hza4Uhob3Hpemrnwh8Ug5Qvsnswfx48Tto5N7A5Gos7Q9U T6Kbokb9Pra7Vp6Apjjbrtd Lbaryidhxokuxfzivh8Qb2Amcmhbig4K Odnvnezzr3Jpc6Aunfrrwio3 2B6Rpu – 2sinx =RGoryl4Kqja3Cho0Adhhqn1Dns=2
5Hza4Uhob3Hpemrnwh8Ug5Qvsnswfx48Tto5N7A5Gos7Q9U T6Kbokb9Pra7Vp6Apjjbrtd Lbaryidhxokuxfzivh8Qb2Amcmhbig4K Odnvnezzr3Jpc6Aunfrrwio3 2B6Rpu – 2Husd1Qdsunx7T733Fm Yeembhh6Ts6Hkwewpp2Xfwdjqklqo8Adnhz Fppv Byqtwy07Qgihb0Jatyqxh8Xwxjhxkr Cn6Jzjjao1Agjbu8Umd O1M9Gi11 B1= RDftn21S0Totuildsu2Exmtz8B Ev16Ob3B5Alxyhxcpypolalsqqhothc2 V7Dh Uvverrcgbnei8G6Ewc5Xnifbdxnusdhcvc5Ds6R Kp4O7Dhbcyed1Jsczlqyuwaauk9Yqd0
Ip M3Ym4Q3Vbuuoakxbwb5Ibiifcrums1Mrtxywpinouxicoeitrlkksw8Stnskht0Nk1Mkrj Rjona2Ciauupo7Yhncnt 5Lamgpk7Nnzwuzz U Ourn69 Wyl9I9Czrdcp9Hc
5Hza4Uhob3Hpemrnwh8Ug5Qvsnswfx48Tto5N7A5Gos7Q9U T6Kbokb9Pra7Vp6Apjjbrtd Lbaryidhxokuxfzivh8Qb2Amcmhbig4K Odnvnezzr3Jpc6Aunfrrwio3 2B6Rpu = RE
R = 5 ……………………. (i)
2Husd1Qdsunx7T733Fm Yeembhh6Ts6Hkwewpp2Xfwdjqklqo8Adnhz Fppv Byqtwy07Qgihb0Jatyqxh8Xwxjhxkr Cn6Jzjjao1Agjbu8Umd O1M9Gi11 B1 = RXc5Yncbvotakf8Gvy1Gheqn5Ogeqscpvsmlvw0Ym0HjHusd1Qdsunx7T733Fm Yeembhh6Ts6Hkwewpp2Xfwdjqklqo8Adnhz Fppv Byqtwy07Qgihb0Jatyqxh8Xwxjhxkr Cn6Jzjjao1Agjbu8Umd O1M9Gi11 B1
RXc5Yncbvotakf8Gvy1Gheqn5Ogeqscpvsmlvw0Ym0Hj = 2 ……………………..(ii)
Dividing (ii) by (i)
Nkowe5Ilqfn4Lhpbo58B= Wsbngjtvgxbvrl6Uhvl8Ba Plowvcjmnl7 Qggzptzmzvfczushxjqykzf Ywy1Rzbjkfvdsmv03B1Ozzzw75Kgifcusz, R7Ypnyzjswwvccida0I78= 7Yzqjc6Kf3Ehd8Ztullx7Rqyunsrbswjfe6 Ylz0Tihqeoekdidtuqrunbycb0Ieemvmrvlefo= Wsbngjtvgxbvrl6Uhvl8Ba Plowvcjmnl7 Qggzptzmzvfczushxjqykzf Ywy1Rzbjkfvdsmv03B1Ozzzw75Kgifcusz
Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk= Plydsgrxieobfy3Yk2Yjqywvkbybe6 Qcekzvh Gugguiohvmd0Cwjmsqm6R3C9Fg Ylmuk Yqn2Kvbdz4Jaimstseydvrhyofao6Bioma9Gk Ng7Qvd1Xeejbfkrni6Rkbh7S= Nhdi4Sgnwbjnehvln Tw11Qi0Xw L 9Nkmcmvlgkcw1Jqiyxhfjnrmv Z 5Msni1 Bhc1Ljbajhrg2Aaosiphe2 Yixbou Xb3L4Qek1Wk6Wxgoczl7G4Edexai 6Dmdxpj5Yo=21.8°
Squaring equations (i) and (ii) the add
V6Lh4Krhfcf0C0 Wzca6I8F5Xzcqll Uueux9M4Pmkv5Knv Iekdoec2R54Hjzpmi3Ojesr Rvi Syynlfzdhkc Rck4Iz380Fqcckmgryylyr Zqs1Jbdpr Yc1Cqygppihluy2² + 5²
Hsdr1Onyskox3Aqr27Bue2Z4G Dira24Cpqfqtgdlhb7Ea Cfuhq2O 6Koeiao8Dvmhupg9Ilkfyeeplh4Ln2Eugvksvg Dlpo2 2Bqgahjd1Fs5Yqc Kdzczlsamjay4F Goue = 29
Qzlex1Yyeixisxpagiuqp9Eqjumjc9B0Mm9Mithghrtfklq A3Bfgdeeyylqtgi Mim66 Boh3Whqlypdd Mfncaldwnhfkxlp70Nhetqd0Zdeaysq27 Crqqd8Ngb5Xob InliXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk+Iry1Op1Eewtgrfwv31Jdjpvoqai1 Qrya Xg5Q0Nlw82W0P8A 0Ionhg31Edyhevdlwmeu4Aifd39Vk6 Hgqcmvsqmbpszeproayiot8Nlbqmepggjm 480Adnl I Pjxvhha = 1
R²x1 =29, R²=29, R = Gadtlxsel7Ntx4Mcm5R3Dorydxfv9Bv Sloakdnu7N5Ffzc7Pnuiq 9Lhasix F1Wtwb1Lw3Jjnl84Wfotny4Wikulisfrz4Ctbrdevl9N9Icgtwbpvwb4Wcsmljzdfxbh Saw
From RGoryl4Kqja3Cho0Adhhqn1Dns = 2
Gadtlxsel7Ntx4Mcm5R3Dorydxfv9Bv Sloakdnu7N5Ffzc7Pnuiq 9Lhasix F1Wtwb1Lw3Jjnl84Wfotny4Wikulisfrz4Ctbrdevl9N9Icgtwbpvwb4Wcsmljzdfxbh SawRbjb2 Laqike 73Pphcktwgivclekgrf4L6Uchsbl5Vzuif06Hxyqo0Ornjzopha Pu5Ujfzoueddgthpg 607Dhujzhcgbq Lej= 2
Rygtl6Cife 7Znwm8 P2Prsit 7U3Gpyv5Yecqrk1Ddjweiyw Ql9Vf8Dnmvtplivthrkn5Oo2J5Pwfutnpekeyxh9We3Ermo3Orska8U 4Gj913Nacev6C3R9L4Pu Ezs7Yzbc= Cr V223Ylh6Hcihgylw1Nswakiv8M Iuynqs9Nmo4Kz44G8Ksauzhocihhx7Ruuc3Ibsni45Cppysas5Ad5Qh8Nbhtztwzv7Lybdv3Zovmdbxilrj Unoxszmd73Qaneyewhmx0
X + 21.8 = Oxw9Jrd7Nomqzydcl62Xonsw9Apkmkfkeuhlstdisi4Zbrfsky 84Bzwnuvx Eagcgbou6Uvs9Vj7Kz610Cq1Cik1Hpmlm0Ws8Apdqtm8Gstmjkmi Epaplxfqm8890Kv7M9Sje
X + 21.8° = 68.2° , -68.2°
X= 68.2° – 21.8° = 46.40°
Also x + 21.8° = ⁻68.2°
X = ⁻68.2° -21.80 =-90
Cs6Rzjkjdqik9O0Q5Aupoqotcl22V25C8C9Betncat9Hkodyeuhjymepk Vqzyrb5Qffjt4Zbayyknahggsayutzewdvrlesk Kftfvdqoqwa01K Ttdhs7Mozzoegsetcretcqx = Zxbhqfelgpdvnsmhturvd2Kki1Gcqv5Jgkjyjhtqecufm1Zafmqogy5Zhsqnb17Em7Lh6Hqhlnfcasnv6Isjwp0K 2Awgqul6Aiwtko1 680Ri6Phl Vaxk26Rbmbh75Lce89Vy
Clnv2Vsibbdwh6Yxlfmydbimnnt4Lxya6Lcmhyi
NB: The R- formula ( Transformation) can also be done using an auxiliary angle approach; where we substitute constants a and b as functions of sine or cosine.
Thus considering the same problem solving 5Hza4Uhob3Hpemrnwh8Ug5Qvsnswfx48Tto5N7A5Gos7Q9U T6Kbokb9Pra7Vp6Apjjbrtd Lbaryidhxokuxfzivh8Qb2Amcmhbig4K Odnvnezzr3Jpc6Aunfrrwio3 2B6Rpu – 2Husd1Qdsunx7T733Fm Yeembhh6Ts6Hkwewpp2Xfwdjqklqo8Adnhz Fppv Byqtwy07Qgihb0Jatyqxh8Xwxjhxkr Cn6Jzjjao1Agjbu8Umd O1M9Gi11 B1 =2
Imagine a triangle
Mqgauadowl1Yrflkvweyebzwb0D7Sdgvfktco4Cpbhbbrfrrvzw5Kimifi Fil2 9Jfoiqyjfja5Bt Flgzmdath4Gvgrfzw Xuf8254 1 Cosky5Fusjwbv628Biamcfvztorq
Using Pythagoras theorem
= Ckdacjkscamht4Fclj7Twx2Nspf9T4Qllc5Z Grzr8Mmldbc Cfjra094Zmrez4Ht Dtcdduslwvasx5Ph6 L2D5Ohhsegoac0Vlrxexe6Lffc0Abczkngdlvvi Npjnlvtql2U+ 8Ccora6Ngzr2Sxn9Iaybkhcy7Aaemikewu7Ifvmke12Juy²
= 5² + 2² = 25 + 4 = 29
Jm6Betcmso8O2Gzncg Bh3Dm 2Eaysz0Lbu1Secobvg7Kjndbbmuzuryufu7 E8Pcmnaqsph9Y Ejwt18Lbmc4Ygdumbkpssftcthsu8Wgnjamv5L7N7N1Qcbznxa6Wk0Nh Ois= Gadtlxsel7Ntx4Mcm5R3Dorydxfv9Bv Sloakdnu7N5Ffzc7Pnuiq 9Lhasix F1Wtwb1Lw3Jjnl84Wfotny4Wikulisfrz4Ctbrdevl9N9Icgtwbpvwb4Wcsmljzdfxbh Saw
From the figure above, it follows that
Cyiyet0 Ntzr0Czveybdtjdhlnlmlbpneywkgp78Yajjm5 3Orzoxjj339Ffgmcjgphcqzkbqw6Tl0Vrz31Pfdxj2L4Rkznwamaiq Ve5M Lmo1Ydcz5Km6Atmbirz2Feoayi7S
= Cr V223Ylh6Hcihgylw1Nswakiv8M Iuynqs9Nmo4Kz44G8Ksauzhocihhx7Ruuc3Ibsni45Cppysas5Ad5Qh8Nbhtztwzv7Lybdv3Zovmdbxilrj Unoxszmd73Qaneyewhmx0, 2 = 8Dumq Detrazqejkvewcyv0Nsuu9Jp1Wtw4M0Q6Zjwqr Bgkb 2Bmgxy1Zyyxwczrf3B Obajfd7Iztza5Pk8Zrvmz6Ze45Lacygafw0Unjjgitmff Qpjaq78Dz9Du3K4Nbk Ecos0Rf67K3Dz6Sqedxt1Rzazemo5Uacessexia Oycao6Kjiajzi09Pyzgmqisktnp7Kxhigdfl1Wktpw3Lpvhqnxusduklhu Dd82Mola7Nitfnvgg Stlwky Cuv6Xg4Yr04Upvw
Then from 5cos x – 2sin x = 2
Gadtlxsel7Ntx4Mcm5R3Dorydxfv9Bv Sloakdnu7N5Ffzc7Pnuiq 9Lhasix F1Wtwb1Lw3Jjnl84Wfotny4Wikulisfrz4Ctbrdevl9N9Icgtwbpvwb4Wcsmljzdfxbh SawFlp834Q Bvyjwlrktzngxsp Khypb Za2Ezbjjcpqoff7Jbufbergpedl4Yfq91Rg0B1T 5Fekw6Ky69Zmpkig3Eyq1Dxvv82Jabq Wqgasy2E1Czaijrn4Tyqsxhis0Ycpa0WgGadtlxsel7Ntx4Mcm5R3Dorydxfv9Bv Sloakdnu7N5Ffzc7Pnuiq 9Lhasix F1Wtwb1Lw3Jjnl84Wfotny4Wikulisfrz4Ctbrdevl9N9Icgtwbpvwb4Wcsmljzdfxbh SawG5Y 73V2Gptmglg1Iuajaybko0Exwlpn2Jkwo9Kvakx24Xofzmklmyjgqlmpyv5Uzrz0Ys7Inhkvkncr Vv37Jbe Zsdn4Kda5Al1L6Jni6Sln2Iqggiyqiych33Tpli3O Yva4 = 2
Gadtlxsel7Ntx4Mcm5R3Dorydxfv9Bv Sloakdnu7N5Ffzc7Pnuiq 9Lhasix F1Wtwb1Lw3Jjnl84Wfotny4Wikulisfrz4Ctbrdevl9N9Icgtwbpvwb4Wcsmljzdfxbh SawNad0Hyhidc8Fcyi3Opb5Ooh55Otndcqt1Ewhlzyb= 2
Gadtlxsel7Ntx4Mcm5R3Dorydxfv9Bv Sloakdnu7N5Ffzc7Pnuiq 9Lhasix F1Wtwb1Lw3Jjnl84Wfotny4Wikulisfrz4Ctbrdevl9N9Icgtwbpvwb4Wcsmljzdfxbh SawUpinuyyt990Allipmpdto 2Dasjnky6Bpffn Kxiheck 2=2
Obh1Vngwupsv7Pt= Cr V223Ylh6Hcihgylw1Nswakiv8M Iuynqs9Nmo4Kz44G8Ksauzhocihhx7Ruuc3Ibsni45Cppysas5Ad5Qh8Nbhtztwzv7Lybdv3Zovmdbxilrj Unoxszmd73Qaneyewhmx0
Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk– x =
Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk-x = 21.8°
So, the principle angle = 21.8°
Using the general solution of sin
Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk– x = 21.8°, thus 180°n + Qcly2Nwmr6Jcotddgmutbldfkpjzgcwoiniewmijywp4I7Xgahsbyhyfds1V4Woc3Kmupx Etgwp44Nkttwsgwb55674Ukm Qbprs4Mh5Miihpkzzjhxzhyr1S4Willkn6Vx20nFsisftz Ftyky Sdxmqmz Rbkmuy0Sogikthrneqgscfcjfsyvybkljrb32A9O Dch8Qhgrjivby2Inqo54Sdxo
Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk= 68.2°
X = Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk– 21.8°
X = Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2GpkLgdf9Yxmyfyiucvbgig
X= 68.2° – Lgdf9Yxmyfyiucvbgig
n=
find x values according to the limits given in the question
OR imagine a triangle
Mqgauadowl1Yrflkvweyebzwb0D7Sdgvfktco4Cpbhbbrfrrvzw5Kimifi Fil2 9Jfoiqyjfja5Bt Flgzmdath4Gvgrfzw Xuf8254 1 Cosky5Fusjwbv628Biamcfvztorq
Then sinAk759Pg8 Txvbwumq4Wndsizpqlwb2Qw9Is2Vrrnwxnhglfmm93Nimgnsqc 348Cghqkptwzj Zpo6Wn2Dqn4Kru Does1Evkgf Tcwgsm8K3Kydyiff6Mecbpl9Wslhsp1Ay, 2=Gadtlxsel7Ntx4Mcm5R3Dorydxfv9Bv Sloakdnu7N5Ffzc7Pnuiq 9Lhasix F1Wtwb1Lw3Jjnl84Wfotny4Wikulisfrz4Ctbrdevl9N9Icgtwbpvwb4Wcsmljzdfxbh Saw sinXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk
cosXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk = Uv1Kp6Q6Fckyg9Hpr967Gqcp9K9L Cqvnkujan7Gds09Xmk Mod5Nps5Eqkdozd1Bo5Fcrwmqjyq Emtflgzmkn00Lkyit Hqd99Vx5S Thtldy9Wsaxxpnidw, 5= Gadtlxsel7Ntx4Mcm5R3Dorydxfv9Bv Sloakdnu7N5Ffzc7Pnuiq 9Lhasix F1Wtwb1Lw3Jjnl84Wfotny4Wikulisfrz4Ctbrdevl9N9Icgtwbpvwb4Wcsmljzdfxbh Sawcos Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk
from 5cosx – 2Mhucegunmhadwngrkyvj0Nxwibaqmsu3Fqio2Klkt2Wlxaygiuwjhucjnjt3Phixe 4Xgywbhdwdqn5Vscq
Gadtlxsel7Ntx4Mcm5R3Dorydxfv9Bv Sloakdnu7N5Ffzc7Pnuiq 9Lhasix F1Wtwb1Lw3Jjnl84Wfotny4Wikulisfrz4Ctbrdevl9N9Icgtwbpvwb4Wcsmljzdfxbh SawHpo2Ujv63Qmlk Vjjougv1K5Vhnvdoetd8Orth9Rpxqesarjqeardwsv5Rfkjmcmlyrpeh7S7Gdv I E1F8Wbhf9MeseqiretZyxsmcev4Wm Uf Pmn25Rfjiact3Ls0Agcp33Dkhmzbhrsi Gss0M39Cdkf6= 2
Gadtlxsel7Ntx4Mcm5R3Dorydxfv9Bv Sloakdnu7N5Ffzc7Pnuiq 9Lhasix F1Wtwb1Lw3Jjnl84Wfotny4Wikulisfrz4Ctbrdevl9N9Icgtwbpvwb4Wcsmljzdfxbh Saw4J2Vabqhaf3T 4N5R7G2Paoaea2Xsdpil4Wzeo1Tffazmrg19U Ynv00Hiptsiy 8J4L9Zicmm2Howxstr9Lkkr4C9Dllxqmwlpxllqaqn6Oacve0Qjtp21V Wov7Lav27Zliy8= 2
Gadtlxsel7Ntx4Mcm5R3Dorydxfv9Bv Sloakdnu7N5Ffzc7Pnuiq 9Lhasix F1Wtwb1Lw3Jjnl84Wfotny4Wikulisfrz4Ctbrdevl9N9Icgtwbpvwb4Wcsmljzdfxbh SawXi6J7Lbqyavozx5Ysxbzkvp1S9Veu9Fg4Jzn7Vzh7Xg 3Rigtxwtsh70Dghw4=2
Xi6J7Lbqyavozx5Ysxbzkvp1S9Veu9Fg4Jzn7Vzh7Xg 3Rigtxwtsh70Dghw4=La6Xlrobsf0Fzsuy2Xzakuinop2E305Efzu2Idv9Nmnr3 X 7N9Bbj1Q4Qnz34N5Lvmgskt02Fmdh7Pscnsh
Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk+ x =Jcbza2T53G2E8Wy1Qr6Joyag04C Nbipmvwwtsywpwxfuvv4Qabvrn1Hg Xzit6Pnul275Zjvxombx5O4Tg8Z2Zelsws3Z0Jwacbf Stpk08Lphbpze2Cf Cbn2W9Cwjvhvvlm=68.2°
Using the general solution of cosine
Qkkq2Jeqhekxn P3Ejardvis2Vuhm5Ij52Wtgtjcbf Mqljawr6Hwnsrlrqdmu1Trlc9P Ss2C7I Twie8Zw2Wqmzglldfme06Pp2Htw0Nhl7Nkas6Zphxypmruvultcaecukb8
Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk+ x =360°n Y7I19Cdqqikhrkd1Khb 2Yge8Mzoiupc3Usiay1Vexsymsy68.2°
X = Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk
Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk= 68.2°
X= – 21.8
n =2Mf3Ennuscq0D6Ka4Emkcttwirqsk Fwx5Pe9Jkj Cvbsbjeq8Oeoczouzcdksiu0Woybpdrogeywciqnb
OTHER KIND OF QUESTIONS USING THE TRANSFORMING INTO A SINGLE FUNCTION CONCEPT
Example:1 Express
i) 4cosx – 5sinx in the form of Rcos(x +
ii) 2sinx + 5cosx in the form of Rsin(x +
Solution(i)
4cos x-5sinx =Rcos(x +
Bjipelpo5W7Ifhitmfk4T9 Zig0Qfoof
4cosx = RcosXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpkcosx
RcosXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk = 4 ……… (i)
5sinx = RsinXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpksinx
RsinXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk =5 …………..(ii)
Dividing (ii)by (i)
Nkowe5Ilqfn4Lhpbo58B=6Mx8Dt0Ienc2B 8Uae7Yp92Zpbo0Ttqiefveajxmrjoh9Ohiwjxdufkeu3Ggakij8Moxfrh7Etxtyfajq5Xn = 6Lxwvibmsr5Jy4Jvicch610A Qoqswklhxuopbgaa5Ilgerk0Sfkw5Kgqqh1Ashtf3P3Bsz0Hcj5Apol1Qfesjdmcojgehnbfyknomo1N1Upphyvsz5Ititmz 9Lbo7Dsc1O4Gq= tan Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk
7Yzqjc6Kf3Ehd8Ztullx7Rqyunsrbswjfe6 Ylz0Tihqeoekdidtuqrunbycb0Ieemvmrvlefo=Ufpxdaczkx5Zywt1Xhc8Bn4C9Cns6C3Lkplgsj5Iaad Yipthl6Kdrjju3Blpuohwkj98Xv2Gtca
Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk= tan⁻¹
Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk=
Squaring equations (i) and (ii) then add
U47Szry12Ea7Ujvz4Dplx1Evqmewu Xxnolgxnkro00Fojppcma95Ykfuy Ngnyjcknx 4N7Bjniaiylm1Jcyu1Q Cquwx Qm9Ue 7Rxougl65Eqjfdm5Eplgmohcg8Cm6Sed Q+ X9Te6T0Ieu6U8Oeyiarre2Shvd4Odunhrs8Qv7Awwwxb0Tftsnxjgg9Gmdepe4Ri 0Ah8Jkuvh 1Trxsplgheyqlj3Nwqrsvwegyuqkzuukj X4L1Ekmyq3Gjkq2Lfogltz Nfu= 4² + 5²
R²cosXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk + R²Iry1Op1Eewtgrfwv31Jdjpvoqai1 Qrya Xg5Q0Nlw82W0P8A 0Ionhg31Edyhevdlwmeu4Aifd39Vk6 Hgqcmvsqmbpszeproayiot8Nlbqmepggjm 480Adnl I Pjxvhha = 16 + 25
RC Cz1Zmtsa8Iz0Mrmoz32Pv2I4 Rm0Nqyvop5Qukms8Fwklfodqufym1Buk2Udxf2B8Uc6W04Dcppozh1 Kcrayro5Aneufzh 2Hizcugj0Mkkmlexr87Vqd Usbdpvrls7Tvba = 41
R=41, R=Jqkpmwwbkj Pnch1Zjuimzsbiktqpi7Yakl4Vnmd996Mjtl3I086Njvir Fduurrrcg3Tum2E7D8Opiwflhxttdi44Oyirbbzys4T3Cqmjh8Fv7Pyhpg3Eyivs8D7Qjorqiujh0
Dzs Sz9Ew7Jsndgbp4R12Cflsrdhjv7Fhohsozxmivgxwl85Idvok9Jtihoenb9Pu8Zj3Jil24F9Enbatqnqnrnkqdm4Mihj318Wcu9R6S8Apshhtjsgvbwgrshrgkhgt G6 Ag4cos x -5 sin x = Jqkpmwwbkj Pnch1Zjuimzsbiktqpi7Yakl4Vnmd996Mjtl3I086Njvir Fduurrrcg3Tum2E7D8Opiwflhxttdi44Oyirbbzys4T3Cqmjh8Fv7Pyhpg3Eyivs8D7Qjorqiujh0cos(x+ )
RcosXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpksinx = 2sinx
RcosXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk =2 …………(i) and
RcosxsinXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk = 5cosx
RsinXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk = 5 ………….(ii)
Dividing (ii) by (i)
Kzasdaqimjgowcfdqlpltc6Df8Ogsnvfa0Zwne8L Rbhwe5Wbd7E6Gzpqh7Dkxgjucywe Ev5Gpbpzuus4Sqlao9Bllcytxmnhzgibf9Mli4Yhe8O Xbqfliiidntawy= See4 Xxsu6Yszkzouf4Wecrsiyacogijilbsbpudjfby36Hq Otnxpdbvrtv4Stosio70Rjurv6Ha8Fmomkriimwpyup4Yu8T= Ay8Qjvvb Cplf7Hmz7Wionjawtnq91Vrpxvehchvvuok6Zze Eivx53Pquhfad8Sfct4X9Vpo6Os5Cs Uhhmyu39Ezt
Tan Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk= , 3Obdxvltsgy1O5Uakigkxkldxchrbx6Vxi4Z6Nllm1U0Zuk2Pyqm 4Bbr8Hrfzmwotjqql V6 4Sdtbm2 Ofexvijclyfqyzsyilv0 Pjvaanrpkij Fzhjwbgm0Htwnlzcft9W= Nlwmpidbl3Tyxjbjqwzc Ubl85Kggk7Otmcpjwhcmqh3W23C1Cgrktvw273Mqbutp75Rmwg0Owuaju1Swtbmzhrwws0Ch7H Ncmgg9 5Hwphvkvetzq8Kfdjxm6Ihtir Jg08Mg
Squaring equations (i) and (ii) then add
U47Szry12Ea7Ujvz4Dplx1Evqmewu Xxnolgxnkro00Fojppcma95Ykfuy Ngnyjcknx 4N7Bjniaiylm1Jcyu1Q Cquwx Qm9Ue 7Rxougl65Eqjfdm5Eplgmohcg8Cm6Sed Q+ X9Te6T0Ieu6U8Oeyiarre2Shvd4Odunhrs8Qv7Awwwxb0Tftsnxjgg9Gmdepe4Ri 0Ah8Jkuvh 1Trxsplgheyqlj3Nwqrsvwegyuqkzuukj X4L1Ekmyq3Gjkq2Lfogltz Nfu= 2² + 5²
R²cos²Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk + R² sin²Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk = 4 + 25
RTlhgcib6Tw1Oegxykx9Tramofwq9Mqldj0Aw0Qudben32Dypbbl6Kktiakskdkvyiuvtw Ydcx Qsgf28Vzrqqzfjvk7Tdtfzgqektsx06Mfnqyhesjpdsonmfd Jr4 Qydr81O =29
But cos²Hrpcnyrbrrxshrj3Kqx0F Zx
R²(1)=29
Wze2Fteoffap0Ib1Si7Cp65Rezxvtjsqfbkmecofrygmr0Ug2Vnxci Vnwdeo9Qhus1Oisozuo
Example. Find the maximum value of 24sinx -7cosx and the smallest positive value of x that gives this maximum value.
Solution. 24sin x -7cosx = Rsin(x –
Oq3Wilaxwf9Rz8Ycofmqdg M Gb Ziv8Rwanjtfd Khvid8E3Q Zwt7Iqmkiz7Dcsydwtalba0Dqvxth0Xnneink0Ol9Chfxpok9Sgd03Giogqtuakwjklm4Pujeox0Ym Ahju
24sinx = RcosXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpksinx
RcosXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk =24, 7cosx = RsinXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpkcosx
RsinXmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk =7 ………(ii)
Js1Aydjmzp7Lpiopp8Dgx4Bemzmn Bp8Rrff 4P2R6Zlfuyvbzmxwm8U8Hdxeirhvfdasebw4K Hs2Jmdou Lhbfrnxri2Fdiwxgczrrjywe Hk44Egre34Iaihqo7Kkk258Qqu= Sockk1Grxgfyfm Wtt4Fw4Bzqootnp2Tqn
Xjj4Q9 Hexe Gh94Pcpzdcivyeglrxxcfjpxauomnqb5Gcojmol2Ap92Gbyngodv9Gqrx8Feyk= 7Yzqjc6Kf3Ehd8Ztullx7Rqyunsrbswjfe6 Ylz0Tihqeoekdidtuqrunbycb0Ieemvmrvlefo=Sockk1Grxgfyfm Wtt4Fw4Bzqootnp2Tqn
Xmxrunbomlsnviroqrgnn Cjmtvbhyzycc3Btog4K5Vfo Wo5Xcnmqehcg V3Prklq5Pd5Hokclxmwdtsgt4Wxwbsvpdsjnnupith3Yvrl 349Xx6Yeynrvoluuikni Gh2Gpk=Gouzhw0Rhozszqw4Ybwkebyfgyoe4F8Lp4Fssaygranzpt2248B4H6Qblf65Jwx7X Le9 Vzfyzuibav2Puaihx1Ju7Ldnb Uavudl9Pjzf6Xealzwd0W Gy5Zp8Pw Pmpuz8U= 16.26°
Squaring equation (i) and (ii) then add
+ X9Te6T0Ieu6U8Oeyiarre2Shvd4Odunhrs8Qv7Awwwxb0Tftsnxjgg9Gmdepe4Ri 0Ah8Jkuvh 1Trxsplgheyqlj3Nwqrsvwegyuqkzuukj X4L1Ekmyq3Gjkq2Lfogltz Nfu=Nru5F24Gi7Ojqtalbxttqlrmimm A9Ezcxqp 6Qb1Jj Aqjag1Ghkrkxaupnw4Hnd Ghsasacpohl6Nhansir7Oz4Iycl9Mhksej 1Prxmlwjvvoqw5Pnedptuo Azy5R Djpua+Dbkxscd Bjvu2Oqvc5Quo9Pqqhotmt777Lylnc5Mceh Fox5Pxgydmnf91Bcuiltyc174Tzshpmnahfcz5Ua2Lsagc6Vv Zv
R79A6Vivwxevjrjqigmzizwdks2 =625
RGjrptyodylccw6Gcdkijhhoxg64 Nqjll3Nok4Rtudpwxtimlnaid1Xv9Luda Br1Dwkgq1Xke76G I1Osr64Alpjipbfmra Dd 6 Halfn2Zy4S1I Dgzivyci3X6Cl81E S0K =625
R²=625, R=Vdu4Whdctietfrrywylpvwbwhkz5Zryirwip9Cfn Lm235Zy8T9Hh9Dhnuhqshetzmj7Ky Lxnujssgteesj6Qh7Fb3Kwe4Qwd0Sue5Blxnh6Kbx3Mwcoltab8Qjzd Hvfa9Gq8
R =25
24Husd1Qdsunx7T733Fm Yeembhh6Ts6Hkwewpp2Xfwdjqklqo8Adnhz Fppv Byqtwy07Qgihb0Jatyqxh8Xwxjhxkr Cn6Jzjjao1Agjbu8Umd O1M9Gi11 B1 – 7cosx = Rsin
=Hrumn8Ytllmj8 Jaxjcqhffjybi4Gjuyzls9Nhaxz535Solpvm3D0 Oljubymw
=25sin 0Qfyutz2Rxildf33Mgdboes2Pwxeng6Ubuatdngtm Tqau4Bnw6E6Pugahnjfucigs5Lylkafkgfzwbmn9Bhridwywx2I3M7Aj0Gxmrihstxjl48Stfdhki250Pgvvrjo82Xy G
24sinx – 7cosx = 25sin0Qfyutz2Rxildf33Mgdboes2Pwxeng6Ubuatdngtm Tqau4Bnw6E6Pugahnjfucigs5Lylkafkgfzwbmn9Bhridwywx2I3M7Aj0Gxmrihstxjl48Stfdhki250Pgvvrjo82Xy G
f(x)= 25sin(x – 16.26°)
Max value of sine function is when
Sin
Zo Quxw3V4Tthqobwzlllopmrkbohhjrjaz 5Gsvv4Zdfdjlxyambeqpmxa7 Jqwwo9Nshfgl9Z3Vfc6Cuyox5Tuqueuy0Y Tjqqdyvwt1Sfgplun3Xwkbtmgt Sw5Qynimokpw
X – 16.26°=90°
X = 90° + 16.26°
X= 106.26°
Hence max value fHzuhljcf9Sslbgla2Zxk50Uuyigv5Vj4P3Gqobdirqftznpfs87Vx5Ojy 0Aymtg3Vttcx8Y5Gucbcgd8Fqgahaztwqu Jifo3Njoygxsmoh92X Zoesmin=y=25 sin 90°
=25
Cs6Rzjkjdqik9O0Q5Aupoqotcl22V25C8C9Betncat9Hkodyeuhjymepk Vqzyrb5Qffjt4Zbayyknahggsayutzewdvrlesk Kftfvdqoqwa01K Ttdhs7MozzoegsetcretcqThe maximum value is 25 obtained when x = 106.26°
Note. The maximum values of
Ryaibd1V Ulwqcibdfgfiuvdr2R9U5Gqk6Krvdskku
Problems to work on
Using t formula and R –formula solve the following.
Ntexeepeey6Yoawt Aybpgra76Ev3Bfaheabrkhyjkvj0Mpsfzjc2Urb7Hz 6Wkzwb6U1Jbpmpu301Uehm Qmmvg6Zvhnfooew6Buf81Pukshco Ecvgbtp2Rqbzq30Zd Ydha4
3. 6sinx + 8cosx =6
4. Express 7cosθ+ 24 sinθ in the form of Rcos(10 –
5. Solve for θ
3cosθ + 4sinθ =2
6. 5cos2θ– sin 2θ=2
Note: If the question has no limits/boundaries write the answer using the general solution
FACTOR FORMULAE (SUM AND DIFFERENCE FORMULAE)
The concept here is to express the sum or difference of sine and cosine functions as product and vice versa
Refer
Sin(A +B) = sin AcosB + cosAsin B ……….(i)
Sin(A –B) = sinAcosB –cosAsinB ………….(ii)
Cos(A + B) =cosAcosB – sinA sinB …………(iii)
Cos(A+ B) =cosAcosB + sinAsinB ……………(iv)
Add (i) and (ii)
Sin(A + B) + sin(A +B) =2sin AcosB
Let f = A + B ………(i)
Q =A-B …….(ii)
(a) +(b) 2A = P+Q, A= Okvc8Syoowuh0Pfoju5Dhpetrkgzbrlmr Clnkfhkmzp83Izx3Oj2Psnikckios2Sas0Q2Wd Ddzegu9H2Xclv3Fg6Z8Ykb8S0Jxnmwrhrxgjvqg Y3Ybyxa3Kl8Xz Exym 9Tu
(a) –(b) 2B =P-Q, B=Vf Hdrovlgfzp78Li69Ijf1G54Imuocwbc7Ebniqtgoo659M Ntwjory0Cuc Lbv Y F8Uwyyojrfa 7Fuhkr4Z Rwnosaev6Lsykbzuehtsonla9Pmiv0Pgsgzizkilmk1Nmta
Therefore sin(A+B)+sin(A-B)=2sinAcosBbecome
Pilu8Vmaquwjz2Q5Qlhfp48Z 4Lepotynlobyjmydgfm7Fyz Mhjr4Oeukshnesoed H20Usfeqb3Ugwkasfoudgslhe5Vh1Jaajjdvvoo6Ntwxqvuwfpxt2Nmomhv3Bkgxsmgm
SinP + sinQ= 2sinT4Nmd1 E To Eobtft4cos …(1)
Substract(i) –(ii)
Sin(A+B) –sin(A-B) = 2cosA sinB
But P=A+B, Q=A-B
9Xj6Rfvrel Wfnqh9Ore Cyxhfkcsvjujxvlogug
Add (iii) and (iv)
Cos(A+B)+cos(A-B) = 2cosAcosB
CosP + cosQ = 2cosT4Nmd1 E To Eobtft4cos
2Htupm3Qwemsdnwoll8Gku2Zwlggigmowmbilm1Usp2Lwk5Yxwyzcrfbawdkh9Gaiyjnfkgwzgt1Cklsvdpnnc7Gfmdnouxcvj Ldzrfsxaje60Cmbf Adyb0Omdseszrsip8Og
Substract (iii) – (iv)
Cos(A + B) –cos(A-B) = -2sinAsin B
Expressions (1) (2) (3) and ( 4) are called factor formulae
APPLICATIONS OF THE FACTOR FORMULAE
a) Proving problems
Examples
i) Seaef2Xe7Yfqaezdcrayf4E Monhfsldsw6Rzfassajcixk8Mufb Qhwnu1Mheuv4Rnsrnw0V2Easudu7Kne2D0 K Dyan19Iaszkk0Mn Duyaqjf 8Ppg0Mxfmsfrsrnflo2Vi= cot 2x
ii) W4Ylvl890Oae0Xrlhx1J8Y7Zlig3Hzrowxhfftsc 6Qtleac 2Mwh9A4Q5Rmitiydwu Od Gzzbzyomb8Ad1Ma15Nf8Lakrygy264L9Qc7Coz= cot Kjd1Dswov4Tvhjjiy7Hhbqubfckv5Gd0Rbhh0P5Wr3Uvwu9 Rlz1Htz31Z0Itgdvyn Zafj8 Xacuxyv0Sxtflmnfn
iii) 3Pgtldxsmvtdlweillw0Watl5Ydlsurxnbvkusqflz9Itf9F9Iza3Uuaxdsmkmrjc0Opeond87Vh 3Uxb8Lwhsx Mucl4L3Hhwz1Ismahh10Vgg8Tzjrew0H8Dfehrjabl5Lnk8= tanHz2Ydichwbmeai2R6Zvffxqdvmcxvbqqa8Rsv Mpomz
v) If A, B and C are angles of a triangle prove that
cosA +cosB + cosC -1 = 4sin sinsin N635Aamdqbdvyusujb9Ldshxfotqy 0Az3B0Golub0Q9Ochtnun0D6Sy Ymqriau5Jdokeoivxno9Edv89Oab8Lqnx1Xkh31R05Kxplae0 Pgpehiahxj0U8Gl8Nybj Mhadlai
vi) If A, B and C are angles of a triangle prove that
cos2A + cos2B + cos2C + 1 = 4cosAcosBcosC
vii) Jktcci8Nyiem=tan A
viii) 6Ss5Noawbzqaj4Foz6Nz6Y Wo1A7Obdmqov Xq3A3Esykhoohmkb692Rbzmqj2Uljyuy3Abakxwysu5Txyj L58Gb6Vfssgfvv9Uqi9Qqouwya Yi8Rwygtouv Vr7Bbhhwqzi4=
Solution (i)
2U24Nsjyscaszjl2Q6Jkinidj1C39Fcyjn95Uoyso7Fka9Uit Wepc0Nahczrcegsrxxj 4Gpp3Oanj9Bthemngefyf1G7Xws8Rl Bgnrgk3Mnumou Gasz6Gkyiokwcuvu824O(L.H.S)
= 599Vbrwy Mzbli0Qfdslvvitt3Edomzftpczcz4Bp3275 Qm1Flby7L Thokjpfzicisgd8 5Mouajnlw6Zu86Yvwyoil Qnlibdnjr39 Dngpfqrafdrgoa8Ipwbxmivjiijis
= Zo5Kpz6Ingmgq Iu1Ow2Jxgnf5Ahkdbmiez93M2Tymnhiniw5Oin92Brz At1Nmsrpjvii Y1Kfc61Qdly4B 2Kgsbq5Xvlwii X9A75Y6Oawhucfum4Nzi3Hzvuerz9Jlh04Lu
But Pzhc2Xzym Dbkzkka0L3Gndkqtpx9Hktdp6Wv7Vpdacw61Ul3Iotg2Teid6 Wbarnqzpj9Euezbsovlndfhkvusowsuyfhavm6 Hxar4J5F9Lxcdcwn Nrftg4X9Yiqno6Lc6TyHusd1Qdsunx7T733Fm Yeembhh6Ts6Hkwewpp2Xfwdjqklqo8Adnhz Fppv Byqtwy07Qgihb0Jatyqxh8Xwxjhxkr Cn6Jzjjao1Agjbu8Umd O1M9Gi11 B1
=
= S1H9Gjf4W9553Tutsqvd4Ggalxorsxzmi3Ek20Rzybxocb1Ln2Ustm 1U9Dmrogmwm6Ufowq= Su88I4Sbx
=
=
Solution(ii)
Tmmpyc S2D2Pjnnyro60Dozaqym1U0Cvkwbvzowid S Zpfglw7Tbm3Atisqretom Rdnkhqf5Clon6Gnqooodjscvpecm6Cvpign9P8Xyj6Cmipja Yck,
= Veea46Zco0Yklgri0N6Wlssbuznbwy4Jxopid5Gr Jwi7Ambggvlmg8063Aw Cw2Bvw1Npx5Wv11Qt26Txuugec0Quueywldixi Bhigoo9Eaqldlidoap1P3Szq1Lfdgm 83B4
= Lkb Eonr Ktavd1Xp9Yxfrm0Kciur9Xrkigfmvvh60Lws86X3Zgctthpe8Lg0Gcg0W

Bmmtnbawblo Ckoixceppbmtmuljotqcohv0Ez0J6Pb0Uuksppzwzsvlq Ly4Oiyhjadr6Xvhgd57Wab3Xyumx3H7Zwdrlkf Qzbppzdybacjb


Solution (iii)
= Fdalurft39F1Bfrp3Dso2Ccs8Oiurryscxyc Jaj0Tinuhsxklinwowtoiqpdltdgzbxslurqcggadwvve7Tcrymf7Er5Sewc Up7R9E40Jizrgf9Omdyuzenqcn9Mnooms2Eum
= M Axzyoajsicarul8Gt9Ott Ggf Pdtrxh1Utx27 Jezcu9G9Poxza9R6C1T42De9Rebappp70Qfmw U5Thprvoe1Os7L1S4Oqqnapjvumocawf Nov2Ptmrfted2QvwombsfnsR.H.S
=Hwemycrw Jdnppv9P7Z07Gv85Uewq5Ubrrmgljxd Mpdu Pce1Zwuuxmqe Yhcz V6Jcrzbjga1Sqyolgv Jl0Tfy8Ws4Nybxqurmfc6Hz1Dpenworgxhwbrcmu7Uxv0X Gh31S
Solution(iv)
= 4Pyt33Ip5Fpyonaf3Z4Jc 80Jksq2Ceilx0C 712Klfatglejng342Kh9Akloe Bufwaeqyhhekgekjf7Tmw8Ncwpzp2Xjlrwn9Lknvpnbhwrqwzrj8Flyjryvcizyardjzzqp U
Ylq5Glwr0Ljrecky29N6Ejl6G5Wh75Gsigxxvs7Jh9Mrnpepxphcqdj50J49Dlnebhztzko6E93Vwpyfvqbygmtdtbxa7Yj9Jylh O Tpbdkgoilxjmzvrposl58Bzsp Ch81Mc+3A = 2Khgthik0D Zaw62Felmjgiyreevxvqd2Ifampothpcefy0Jbwyneugxmjwujlilvzrqtvr7Wv4Vg3Kjohs64Cfwoerj1Abvkn5Gejaasvx 5Syzjurkrexpsoxz 3J6Od9S0Mq0
=2Qg Qiujdovpq9Kt0Bgok7
=2cos2Acos
=2Rmf4Uzwaozhg4Ljsjivi3W8X72N49V3Ncwxi8Fsnhwno5Wxiuyag4Vfazkzeifywhmjzgsvscekysfbz74Qc7Uvxt63Klh7Psfmkqcrrp3D2Zuu45Iasnvp5Yryf
Appdsnzwzdozdlgmkuoylert1N6Ix8Pvch Gdz8Nrldywbpi5Vcd9Epu91E0L396Rw7Nlhdld8Txfyzpy2Bbmv Iaofwij2 8Jr7Ku2Q4Zxvy0Wzymv4Fmydce Aqv7Csnrotxm+5Shscsbkhuat3Udsvjjod7Hpnqqub2Xf V6Sjizjwplx6Wpriapk4Wnihl8Emgjjl74Njehkbpekv7 N=2Jq7Btbyvt Crd7Hm21A06Eadgku659Rmtufbqxs3Lrmcc
=2Foy6Gncpqvq3B2X35Lgbwrz1V9Lxvw
=2
=V2Losv0Sk8Xicf0Bghuqcrqipe6V1Tkgt0Ukfaxqykys Mrgeb2E3Gc1Rnxnumzbbgj8Ge Md5Sgxmlhso9Yeke7L1 Becy 4Nytvwfluxqxr Tzwtsoqgmynqjmrmvlrj096Ve=Ylq5Glwr0Ljrecky29N6Ejl6G5Wh75Gsigxxvs7Jh9Mrnpepxphcqdj50J49Dlnebhztzko6E93Vwpyfvqbygmtdtbxa7Yj9Jylh O Tpbdkgoilxjmzvrposl58Bzsp Ch81Mc
=2Wc16Zv51Nwmluwqep3Ct Q3Uwskt0Jgat
Then
=2Rmf4Uzwaozhg4Ljsjivi3W8X72N49V3Ncwxi8Fsnhwno5Wxiuyag4Vfazkzeifywhmjzgsvscekysfbz74Qc7Uvxt63Klh7Psfmkqcrrp3D2Zuu45Iasnvp5Yryf + 2Wc16Zv51Nwmluwqep3Ct Q3Uwskt0Jgat
=24Tqtcjkms8Djazoayrrg5X7Ybeevfsi 8Ivrb Huimn Nrbfgem4Nilteud Iarzqgasv Hti
=2Zevfyhryxbhefucjhgqditbmcwnhad7Eggctehokfjjwqfvrfwafm Xovp 5O738Dak0Jau2Vlzp2Pya5Xuwxhf4Hl6Audshscc 7Kfo0Lcpotsipcew 5Hijaujnq9Hudlhhyc
=2Cqnngr6Kwpc
=22Afsbohm8Pcvvcdzeui1Fh Juel6Y4Dsf1W
=2
=4Pyt33Ip5Fpyonaf3Z4Jc 80Jksq2Ceilx0C 712Klfatglejng342Kh9Akloe Bufwaeqyhhekgekjf7Tmw8Ncwpzp2Xjlrwn9Lknvpnbhwrqwzrj8Flyjryvcizyardjzzqp U R.H.S
Solution(V).
A, B, C are angles of a Ou59A5Sypk9Az6Uyx2J1Mt1 Oa2Fusx6Oxk
Ccfo Oplikeygb28Xrleakvmrlfh9Oobgl9Zuobollz2Pwwkuq Pxizxhm4Diksng17Vvmbil441Zisgjj0U8Qhfcmfg2Tjp222Vfaizxe H0Pcfgi6U2Fkoulvuvucvlpz1I8+ 8X8Jmkzbz D6Y4Vlkemzjmcdtuhz4Xx
L.H.S
CosA + cosB + cosC – 1
Pmu Wvtmyh3Frvdj2 B9Xev3Guuoeemyia3Lq9Ncbxfwjoelcxooiz Bfvqq9Bn Bbaiwn9Jsmd4Fihyc D9G2Qvh79Wuzbbey3Intbilw4Lareil13Mjr1By9Nrsqkyyypi4 W
2
=2Xni30 M1Zokybzw5Bbro0 Gb0Lpykavxqiezknyza2Ojxd8Bjbxdx Q1Gbalnccfdeeszbig7Zjc9Pktxzfg7Drqmozsdzzquyoh3Urkvd6Czo02P Cvsyv Ooyvcmgs2O 6Aes-219Xgupxirnhc8Yolmfpcg67110Lr4Kcyqqwatc9Htumj9Ulfpdgx2Z8Kim Emdis0Mhvp64Xbhefegmdt6Aha1Slr Ufevp6Sriq 0Pzinktmtpx 1Hzoxms35Jvcl3 Hv Hoiw ………….(i)

But A + B + C= 180°
(Degree angle in Xzx6Md1Sdthxi Opc71Atayceaircaes 59Ijfii Urpx8Cg4I6Y1Xwsvsjsbzscmhtzir1Ocsxwfs8Snffrbss8Q 37Phu59A6Bkyjqtxjoitiyala0Iftzb Bykic7Mlmpd6K
A + B = 180°-C
Pw9Uyvvgvy6Euzzc Vc5Fn2Xytam15Bapp0Wctmjv74Htc7Q4Rfwozdo9Xqgesvaowjktucrkfiaanrkuu7Gt5Veqxg=Ucx8Aclbd0J6Khkwpzhixdquli6Yoor3Shdgkfya2Peys3Wgvq Hiizi2Mvbotwrsbvxi13Tgt0Nomucyrtcczsef0Vvszy55T1Xtv1Chgxvd Ytlnaknz9Faxjwa 7Z7Zjoi3S
90 –T7Rvb8Jjtakk9Rc6Daeizyigslrplmdjdl Tdeur Xsc2Ibniqj7Uo 49Ibvffnls = Pw9Uyvvgvy6Euzzc Vc5Fn2Xytam15Bapp0Wctmjv74Htc7Q4Rfwozdo9Xqgesvaowjktucrkfiaanrkuu7Gt5Veqxg
Apply cos
cos0Fydj6Neh Adw7C Tnvo Ogsr3Wtmacr2Ywq4Ni3Qgcy Zjfgta4Hkcadp3Yzw2Dy4Hagzv27G5Ld65Htycy Rvwe1Lgxh13Xdcoxc Gxj2Lml8A38T0Nhficsudaiqln M3Moc= cos5Axszc9Cv3T88Vfb7Dnz5Gmc94Vtqufq33Jphjqoky5U8De G5Qcgcl33Xzis5A5Wojnljw4Vbmqxgz45Wwe0Tcvkojqyls Et1Df1G0Omvqt Bejdoybsr72Svnl4Hvsiz85Co
Cos5Axszc9Cv3T88Vfb7Dnz5Gmc94Vtqufq33Jphjqoky5U8De G5Qcgcl33Xzis5A5Wojnljw4Vbmqxgz45Wwe0Tcvkojqyls Et1Df1G0Omvqt Bejdoybsr72Svnl4Hvsiz85Co= 6H45Yy
2Vn3 Jm4Fumeglw7Zaiwtpkvb61A Te
But D8Crlyktufb0R80Rbx4Q7Hcaf1Dx20Fiauihy6Yoigw3F9Culkrspccwbcdr7
=1 –
= 1 – 2
Substitute (ii) into (i)
=2390Ydtldagttqlyxxgxez6Yci7Pcailyog4Yfqqz Xk2Txadn2V Wbdwubfykup P7Lu459Ie4Sdty P8V5Rmpzhyl6Idsgmzkuc6Girb7Odyk Nybzbt Gmasdv75 Py1Bws60cos-2sinTozesskov3M11Rag2Oc4L2Mymvx1Hjykkxxuh4Oukb Zrmdhkoitxvqcucobdfjsk3Xdqubk Vr4E4Z9Alzruetpabmwtab Lntrauj9J9Hccbqr9F80Zepmybj5Hiqfjmt Fhg
= 2Zviycjlvu1Jaqfg 0Omjrptg D Vrtfnlbaq4Vqgkptjyrn1Vabkqeaqgx Nhzenhmgxynxyvqawftjckweivi Rp8Chr0Zauqts3Hwiwdat Zyled3Egxrzaabgu Qvuwpzqam-219Xgupxirnhc8Yolmfpcg67110Lr4Kcyqqwatc9Htumj9Ulfpdgx2Z8Kim Emdis0Mhvp64Xbhefegmdt6Aha1Slr Ufevp6Sriq 0Pzinktmtpx 1Hzoxms35Jvcl3 Hv Hoiw
=2
=22Yl4C7Ach343H
But Ahtglc Roaiwgymsjvhmg1Abvug7Nlna1R Wq0Qquhjuo6K Kvhlxvwws8Ppgkks1N0Yi Rrv5Xzp3Uikch2N Dsxwjio6T=390Ydtldagttqlyxxgxez6Yci7Pcailyog4Yfqqz Xk2Txadn2V Wbdwubfykup P7Lu459Ie4Sdty P8V5Rmpzhyl6Idsgmzkuc6Girb7Odyk Nybzbt Gmasdv75 Py1Bws60
Using factor formula
2Y9Ddtutth Ekfkvotav1Jv9Bhjkgp4Prdcn9Jz9Dyeqrlftflrnypn2Wudommoswsu1Lvgf0Ie30Vj7Dsonwi Lhzfeo4Cz7Kjhwp2Mjkjohwumh6Fexnvw7Edt42Gyrs1 Oimk
2Lz6Eik Yqjstsmd8Etrz6 Ucxcrud1Exxcw97Vtozwf52Hwmclcxw79 Kolkdd Ogwea8Kc10Yg1Ttx30Poltk7Aqx33Lsl2Pzi4Yaaf4Xvlabxwb3Zip1Bwn 1N Es9Axuiltg
2
2Oknuvijnw Nxnvf3Qgzxgz 5E0Tlabsp Ldg28Qi1Dqaxjgjkne0Xsysdt2Yhn8V9Rk Wnedk0B9Hnf8Yqjiyjamg4M2Jh015Jvspedh37Qj254Jxaljk9Lnuhkeoqw I4Vjm4
But 9P 5Vx M9Gvoek69Gg5Qmv1I1
2Tdzci2Yoyyl08Ht5Uefrtgbr3Q I U
2R6Rdjhzqo7 Lndlt3Cp7Ycn8Bxclsegei4Lvhk0Z8Vcb V2Xhs37Hmt1T1Dkhehhvyunxtjqsmw2Qn5Hokibdjp58Nmeebmxwu7Lxa
=3Upmepr3O Omfy1Sjubn6Gfgfbhcpjl2Ltrermr78Mkj5Zbqx7Jprcp6Bsiriyrb39Vrjb5Ywu3Hr97Llhve
=4Fjzzrk7L7Yiekdelscziwzg3I
solution(VI).
Iwh1Aepokq5Ej Quritcpfar3Du 7 Fwpdkigfye
= 4Nf 19Zwr590Ghgf0Hl8Gqizpadr8C6Qhkd6Jtacthw5Jdnsacie6S3Hb7W3Ns0Gw J0Srqett9V6Tfzflnx8Ld1Orap60Qlten6Zdnp4Fjr91D5Ju6Aqo7Bus W54Yta03Xzry
From factor fomulae
=Rkzopaqhfh6T50Ucp06Kbcyzwyvvn 2A0Peaxj5Uhb54Dpktxc5Ibyet Iy Lfrrmt3Adamlz
=2P25Fcphgc8C Zeux
=2
But A + B +C = 180° (H5H2H5Lamutrluzj8Xf2 )
A +B = 180° -C
CosBrstbg1Y8Hk9Rsfhrgvdhjlb585Vmuocqhyqxplzhv6J66Egly1K2Xe30G
=Pcl96Yhvwiz3M58Xygna0S3Zmjst8Wymgrsnjheogdv2Wbldjvauwaqmsvxo1H7Hotzxsth5 Ixgjvglhmprkjaio4V63Flfexzee Je08Lwtzjeib6Owjqsgijeh5Kfuui6Mau +
= –Nvcwxk0Ggw7Venckhei5G5Baxp8Ck3Owxvks3Mve0Wkh32Gzbciuixsb3Tb6315Iwn5Zwjuk1B Libwonptowm7Flcany K K6Ndataokdzxnab5Au9If7Qtvp3Ql Gsr1C32F0+ 0
5Ockur141Tt 8Achttd13X1D14Yfllpchxzrrosvsb0Nuvqfj2D4Rge7Khdmosfqnsai7Ptb3Pi8Pkioiwh1Wprimved2A9Dw4Vujlpovcei31Px L Zzjmo6Vmlpmwknhsfxg0= –Nvcwxk0Ggw7Venckhei5G5Baxp8Ck3Owxvks3Mve0Wkh32Gzbciuixsb3Tb6315Iwn5Zwjuk1B Libwonptowm7Flcany K K6Ndataokdzxnab5Au9If7Qtvp3Ql Gsr1C32F0
Substitute into (i)
=-2G70Mzrijczohoa3Wv1Y2H24Du3C336T45Gmx9Mb0Garg72Tztwigzvdbuq Iz 8Y3Nxraxooia Kjc71Qsxadwuxj Terblgyofy Yzdi1Sfkfwbui2Qtf4N6Gfn0Xxhmojiv8C+Fjh6F5Shqzp4Jjhrul1Ucycbnwxx5S8Yhws40Ofser7Otyhyi2 Zxjgu3Gjhbptloui5Yoffgt6Vr9Ix81Sq2Fl8Adawbsdvw2G Ducxh1Y6Uu5 Ix3Rmv Woy4N1Oexrpfguwk + 1
Fjh6F5Shqzp4Jjhrul1Ucycbnwxx5S8Yhws40Ofser7Otyhyi2 Zxjgu3Gjhbptloui5Yoffgt6Vr9Ix81Sq2Fl8Adawbsdvw2G Ducxh1Y6Uu5 Ix3Rmv Woy4N1Oexrpfguwk=Yhsd 7G
Fjh6F5Shqzp4Jjhrul1Ucycbnwxx5S8Yhws40Ofser7Otyhyi2 Zxjgu3Gjhbptloui5Yoffgt6Vr9Ix81Sq2Fl8Adawbsdvw2G Ducxh1Y6Uu5 Ix3Rmv Woy4N1Oexrpfguwk=Ie5Wspmqq3U9Fdksffpmvbopvp8Z5Zkdz7Lekiybkrrfxgrh Bdopckjsoxcsa6Ei8Msfsmnpw2Bmwcnha7Vt1Vsy0Munxpdzhjqmfbpcgrbhebkxegxqmoje00Chmpxoxwxf0
=8K8J6Hnucukz39Eaeesk72Wxyroz4Mb0Ajjfpdulvbycyzzmb3Hl2Na9Pevxzupwvljmazow2C059 Lqraagq8Nzjddnxe 8Vgn0Pvculusngcfklzu8Yimhj4Inpi3Mocqyoc
=2Iknwfd5K9Jc Evjf8Vhqpq3Emxuybpqaicwgiedcv8Miw7Mghamnchvtv1Mqur1Mkuxmflppcflo Hoh0Oqh2Rgwh3 O4Bb Guaey0 Kviuymwrn1P8Hmrlfdqte I3Masac4
= -2H1D8L4Kqcr Utz8Nzlg1Hupqe7Twkkqdwgswl6Y3Ezwly 1H6Vsq8J2L32M6Ql Sqjphuu 6Fmx8Jdvhxgtinhpwif8Tx9Twjh6So Xlns
= -297Vqmynznbdkmltpu9Zib Igj1X4Njo3 Zzjzbt7Ur5Jyt6Kqxmgov Wmb8Ndgller7Wjwkfznqqce7Nr8Fmkuizcyta J7K9Ujcerwkihpo9Jjurjo4Ajwrgz Hoyhmm6Ja Pc
= -2Xqxdb Reoyiizrzn0Kp Yzdmxvoqpawy9Nppdht Lfp2Ygktsnrsq 8C3L0Mtslgwqwpuqkjxlhnziti7 Rz9Mp9Lyv87Az G0Tenf Lfu 901Pkqncjswxprnx4Gao6Rxlnjqg+2Xibv06Kc7Vpoq Jr9Axlc10Sbc9Bmwgvjz Ipltvewth8Irwnkvzs6T Lfllugdpmoaicohszpcik1Uupy Kzqllgemogn52Cpkabdgstw2U2Mlwsb1Ozrs2Ohlr1S Uc6Ee36E
=2Pdsi1Rjosst
ButA9Gdwl Ltforrdnleukhwbcyhe Rohmtiwakx3Wgujzgmq0De= –5Ockur141Tt 8Achttd13X1D14Yfllpchxzrrosvsb0Nuvqfj2D4Rge7Khdmosfqnsai7Ptb3Pi8Pkioiwh1Wprimved2A9Dw4Vujlpovcei31Px L Zzjmo6Vmlpmwknhsfxg0
2Doxdbdwr8 Rp4G6Oequexlbrz5Rc Hj0Qozl9Iajpttll49Cddvnatrgkkcjkofljrjtjrttftchsma1Qhied3Etadwzshie4Golirnhmdqn9Un8L Shm0Y Jxi9Fzthn4Bt8C
= -2K Fpzsfvngdiicj Acii7Mvqzs3Sdqxsmuasnwor Uzzfbkvycs9Uff7Q Pyvdh83Bulf Ibnekvbqe3M9Cizoidnwf4 1 Wqdcw7Amwei Ntsavffxoyvwvy6Je5Ybe9Libv0
= -2C U0Qdjjnpdapsad0N6J5Rt Ltlqpqns8F Dg9Bi9Lf Vd Ebvdusq1U1H0Jyrzhsaav5Smnrcht U Cxuxsle3Qjd4Wltg3Phf1Kxm Nrxtwuk85Rxl Mjhmpvcl Lu Gbs10
= -2Zntkjuyiurd9Zmcgnzpzmzbfgd4Bqqjvfiepgqux Utit Nwi0C0Qeyjje9W5Nzejejehohb46Dyawtau0Jexlk Mxjiwulyfejzsswbajchijwe97K2Qvlpy
= -2Hi2Ivm23Ohc0Xlygn0Dw T 7Vfv7J275Qskqtcghmyxxt4Xpbabxlfoozcglrsykx3Mrioj P811Lkvz0Zaqt Xl0Ax0Rjymkfgfwa
=
= -4Gyiphmvyq7Egutv0Q6Vtq3Ujlwzy1Vutwd21Whf6S55L74Jorf6C8Gxs Zrotnzhulux3J Ggrqld2Ff8 9Nf6Rxpucfbekycfnpb 6Hevaymbfpb0Viyhu0F7Y9Gsoiadn Bho
=Kmg2A0Fpvoc1O7I Fl8Agi85Pp56Wbmg Eljop1Cxro4Frdx9 Hirbcsojczmkqkvbtyijajj1Mubplxmxl7Ssuvowvsiubwsrd8F0Babiss9Ssqm1B583Gm Wlmorhrptrwpiq
=Idhnmszad5O5Psw Phqklblf16Wyg3Tfmjrovvxsisqzvofbtz7E 9Yee O4Sx Th Znerhvlmouyogloillzpfsx F Z15Ukkzyb
=
=Qzuevm7Ybuomu3375Flripcmke4Hvxq6Lrzmjhicgu Quo2Sgjepp0Ppsv7Reyj61 Yjm2Kik5Umrduggu9Dn3Xboid408B0Wzqsazbl3Omxbsaeelcuavwjrqg Bhaz8Fjox I
Liijhlyieec Mzzjkaiqvk Yve4Dion0Yamx8Shhbhte3Ea Cviafmibq8Zqyalrrrd4Hzo7S9Fz1Xb3Acwt1V5Rb9Vlmjeset9Ev Dda17Bx5Wgqfjo Nmblarrwhmwaql7Fyk
Solution (vi)
Xelniaki0O
L.H.S changing the products into sin or difference
Numerator: Y8L1Ykxqmqwymeafhthqyj8Anadr3Nnr4Gd0Kduthaghnujhlbjlvmvglodn4Hzejrsl Zea1Jwd07Bmi9L9Seskwyjl5Ptonmyutdlm1S0Etjbcizj7J842Mocbxl86Vj8Gsrq
From sinP +sinQ=2Ex055S3Ouj5Akx6H6Zwfqyh9Kdqk3Ikfocxtamebduevzxcqp Yrncmmbbiomivlanwbts787Qv Rbvs9V9Micp2Cg6Ycmts4Dkyorksbfxu Gx6O Bvg2Nfgfbddcqbsrbrwjw
Hbpglgngjtqudfrv5Xjcfkfwiwyg Ir
= Y8L1Ykxqmqwymeafhthqyj8Anadr3Nnr4Gd0Kduthaghnujhlbjlvmvglodn4Hzejrsl Zea1Jwd07Bmi9L9Seskwyjl5Ptonmyutdlm1S0Etjbcizj7J842Mocbxl86Vj8Gsrq
K1Admz4Vnwio Q2Mdtdjsxs3U8Zywftjyzjsuwsy2B6Pw9Yzn3Wt8K0N8C8L50Zbnxbf J3Kewptzwfnpdqufnoh46Ivgkunlughqwocul4Bh
Wmf63Cg Lyxvqk7E2Khfufzzmrsmxcqpqi15C8Rgjfkmylxioeihlm1R0Ekjaedufsxepc43Btcnc7Nvpcywxbejgnenkovirwv7Qsgup Dnttzcz9U9Cz5Wpsvydksrd4MuupmY6Rfgjvgbu5Mm= Ylbylypmjixqoeqdjtigeq5Ihyqyord8Qr3Vkl2Qbqih3Kmius6Lofvzkuumc5Uj4Nbleocg02Cwekpkre Elv6Rgidicluuydnomivuodrbchwpfl Jyiw7Ig1Lcrbdhdwfjcq
Similarly Fghuayd1Ytyvlj8Etzs52D5Sxtuqoelsiduwn8Jxlgodcgdct Tl5Ychv=Gqt3Iixzfo I5Xk6Evezu1Bnarhxwdwjmq54Hfong7Pfd9Zvw7Qowolwu9Do8Ijv24Xz1Ztjqbnjhd Wxrg2Apqicp6C49Nmk3Amdjpbibgfr4 Ybr2Dynjvxpxa Le4Y2W3Q40
Denominator
Fwgokxjnlphn98Lzpnv9Mpcyqsuaft Ou75C Uj2Rmo6Gsabpeibnhpyhs4C5F8Gscw3Hmada9Sm6Hkloupzyek2GltY6Rfgjvgbu5Mm=Rhcgyqwvkrgqiqrbuhputxrjpq6Elrbzf5X2T97Kc0Pa79Qlyughbu1Aghoqeltbyqjlnimdaikl7Tgpxm8Yyeeaal8W
74Hwfa2H9Ekdxs1Vteo 8Vnjtlwgo Wge2Uau Akodqzktw Klgmp3 Wwkaqdehpuklhmpfgqafr
Vbjs Wzfbr 0 Bp6Fcoszyjrcrmcmtnd5Wv8Yjzajvjnqwzsvyr66 Bbnusdgvq5F6Nc1Fgftmrmykyybe5Sqgwvtx Bm6Jvpmqxkmu5Ztzmow
Dfyjppb3Nhdchae8C4S6Cdqtmsh42Nz Mnghh26Q5A2Sbiojlsy Quz4Blyc91Nc1Kbwnod8N22Hornpigabsdrdyjn7Fupz2Onxw Yywxubhkooyzo1I Sya9R6T2H4Zar1Cmw
=Udzxmxitwxqziie6Uxy3Stzgaqn5Ehorkw4E8Bjium8
=Wsglncjoz Cuul3Poxb4 A1Dektd15Zmhzxqrtugsugsi1Nkr Xcfr0Pz3J Q Tmrw6Ggdyp6Pgposagefkcig9Epdrjvtr1Pc Shniyb4Jgsmenxrmerk Ubui9Gjppwcjroci
=W5Qj9K5 Ukamdb2Zrr0Vqxdxfkdnbgtyikmgync8 Fr8Cmxkftqm9Zo4Q
Wcwzrs Lqpf Jkrq42Ky2D9Ezrphtpkc72A4Lgprswjtuclwctctfunewfsibd9Opusx Abmo8Ig7Gj2S9Vpdlf8Ck1Mxym7Wvdqohitnyeia1Ee8Hi6But 0Fsjlf P1Uvrdo=Fwgokxjnlphn98Lzpnv9Mpcyqsuaft Ou75C Uj2Rmo6Gsabpeibnhpyhs4C5F8Gscw3Hmada9Sm6Hkloupzyek2Glt
=8Pipyoqj13Snsjebsxj6N5P2Clkwjv8Wtolaek3Llh 2Bdhztlmyzzrn5I Ju0V8Hbc 7Fbpmdytwgwsqsnugsesaefa0Zohpeogqrtdpsnwqqd1Gqy15Llbkoastum 8I9Fef4
=Sktlsycsmfhmtpwg9C Upxfg4Be A8Ombs7J11Qvebg7Tkjvl4Dojkzmum J73Bs 92Mif8Sbmwltzvts0Dxfcbh8Unkkqv468Lc6Okkghq 8184Izeetazvfgvkl8 Anu9J2K8 RHS
Examples (i) solve for x if
Tyuu8H6P2H86Vjyfgiz97Alwkgcgfdu90L9Vmwcmnt6Itndpfny8Ybw94Uyb7Xjxwh2Dsqczl6Kuc Onoxh8Xil3Lux3Gy4Duhrjf3Yf5Oehrtkcgtmjncmfpih9X8 Dkpqv0K0+Fwnlwk6Roe1Av4Bzmfhnejsroou91Hfj3J9Zb9Bv9Ns5Nacdqv77Plbe5Rknevvj8Tn 5Uc Bdf9Envaz4Unte75Ejzmn4Mrlp Wlys3Tiwkg26Tpk6Rsdq8Pl=Bniexsowksunixx6C5Xdx9Iiy Znn7Aa L 8Lxiq53Evvbr S2Bggo5S2Fhflae23K0J6Iyvhrtstese4Ri1Hj0Cqgbjrs Ksc9Zr Tbsty3Pgkoqptdw0Lu Jenq Ynxni2Hvg for 0°Kqvyzwvaeem4Gdt5Bx4Xcxjizruvjw8Ursmniqklhdw4Azck 2Tq2L4Jfnt 8Zv522Auh6Pzo1Axswwuuwuell33By6Obxo77Jgbzarvvz2Fvl Tzdh9Cd
ii)
For 3A6Fw1Y40Etpicyfcoxnykq0Wmiys771Ciabngbyezw0Lyvkz0Cpizs8V1N85Ie0Wywd3Vh
iii) 5Fzizq5Wpc1Pn Ih5Ftx9Gswdhaq Xzpb1X
For Rsq3
Solution (i)
02Ipzli4Ollrwesnqk6P3Iw8Xzw8Z6Z32 Hmrhxzdljlutwbyg3Mmmuwj9Utsdc6Fivz1 Itzamdspn3Q Snqaxow2Bestgnqrz+ Fwnlwk6Roe1Av4Bzmfhnejsroou91Hfj3J9Zb9Bv9Ns5Nacdqv77Plbe5Rknevvj8Tn 5Uc Bdf9Envaz4Unte75Ejzmn4Mrlp Wlys3Tiwkg26Tpk6Rsdq8Pl=Bniexsowksunixx6C5Xdx9Iiy Znn7Aa L 8Lxiq53Evvbr S2Bggo5S2Fhflae23K0J6Iyvhrtstese4Ri1Hj0Cqgbjrs Ksc9Zr Tbsty3Pgkoqptdw0Lu Jenq Ynxni2Hvg
Writing using factor formulae
X7Yxi25Ii Jlmdriff6Pvz1Cneivc Sjymv4Eu2Vhudblexok07U4Pappsrakcz4Dtf576U5I0Nmgygsbqawbyeugtfpz2Ttxju8Vsxj0 W7Uwaubuv81Bn Igdfqyio36Q7Uso=2Qwymuwi6Smyhbzvlph7Bisdtt8Ke0C0Vlf10Fybrcywkj5F7Voi0Kkxksgjww B079 Tnxef7A95Osm9Nlo70Viaujoqfozmltqs8V7Xaqtxaawr4Kfzypao9I6I3Mmidznbok
=2
=2
=2Kdphdjg5Z1Wk7Oplafbbmemoau Sj1E W3Hv9Eoh7N6Wavh K5Bctimwc0Rv5Szsmn1Vkmhwuqhw2Q9Qe2Hsgw0Qu Zg1Xvvjqnoms5Wmpzoqu3R8Wmj Gi5Vw6Oxuiddh Hlw4
28Bdymmw5Nv564Wicfn Iysim Kgl0M4 2W8Mqosdjkv6Ep0Lfrzyalshbpbhdrc
2Hyysqkouda4 H9Soyfjxoet Smwzslnvgfb Zbu6 Kqfcfiungty5Cbeow0Dy5Jxvjssuzauudhiwqmgpct Cf9Rdyrfkafl2Wmchll5Ghzljk3Dvipt55Un5Sn09Dzn 88Ew K
Qglf8Ytl A99Vtovz1Gakqabl88C3Owz24Tkb3Wofabe Pisx1Fdgnokuu0Qfyeaiwmhge9Q3Ndvzzv8Jnaebxibcmtxg1Apswhlgeb7Duxykanj G Ij Jnmtq5Lvuaw1BtcpmUdfwsghafjoczd9Zyd0Eqavv2Wl Ma0W6Lvz73Yxbukvip6Ujv9Za0Divebrhf S772Deynt24O1Mfa Vwpea8Gojtre9Ju1Rf24Vygbvxsfnicvrkuxkypvaa6Ccbmjztx6J1Q=0
Jili7 2Uagjh6Narfvaulhhedrus30Kpl3Dxiogrjda Jczesufgxckosmw6Ktf4Nenmfd2S1Qcspcwcgvpwagq D G1G1Dv1B43Kj4 Rutjpgbpz5G0 7Ze9Cftzbcmjlaus2O=0, 2Bwe3Px6Rlzumb6Zhcujgzhrefvvcte7Tgooh1Oezmxrfdntblbs3R7F5Xb
202Uurgip6Xfcireoua Kvg18Sjwqyi Vs1Wolybh4Opuxqg8Cgxzok Hqq0Gfoom7Fvwnyioei5D6Bbljgiubgbp Qbou3L 2Ruvedriokpvm 3B8Owl Ukl6Ht1Hmw Ys3F9Qq=1
Bniexsowksunixx6C5Xdx9Iiy Znn7Aa L 8Lxiq53Evvbr S2Bggo5S2Fhflae23K0J6Iyvhrtstese4Ri1Hj0Cqgbjrs Ksc9Zr Tbsty3Pgkoqptdw0Lu Jenq Ynxni2Hvg=0 02Uurgip6Xfcireoua Kvg18Sjwqyi Vs1Wolybh4Opuxqg8Cgxzok Hqq0Gfoom7Fvwnyioei5D6Bbljgiubgbp Qbou3L 2Ruvedriokpvm 3B8Owl Ukl6Ht1Hmw Ys3F9Qq= Xbrcuyluzc5Oc7Psq8P35M3Ohoskkktecambu7Hjd7Qwfkq2Clh22Iewrori0Sawcvrrmuvzp7Dvvpnrju29Tavoqtxw6Q5F3
3x = =0°, 180, 360°
X= Nbukynpz0Jb8Tqabqm1Vpxgjmbk Sgcdlzr Yyucd540°
=0°,60°,120°, 180°
02Uurgip6Xfcireoua Kvg18Sjwqyi Vs1Wolybh4Opuxqg8Cgxzok Hqq0Gfoom7Fvwnyioei5D6Bbljgiubgbp Qbou3L 2Ruvedriokpvm 3B8Owl Ukl6Ht1Hmw Ys3F9Qq== 60°,300°
X= Tfmjjxcmc8Et8S5Inqw1Bn7Rwyhimkwhwlr7Ofo7H22Wzwnfcutja189Urw1Yunbmu5Opi3
X=30°, 150°
Px2Akas2Datavj6En5Mk Cv4Aie0Kk Xxf7 Ymumoaoodpay Oi6 Khuvkskx=Cha6Ff845N8Seyyzgdnnvnversshv S6S1B Os6Wb3Mojyomoj2H41Urezghoc Dr
iv) =Qmzn99Ak4Xsshsmx5Uq Xac00Jcxlqs2Yha0Tr2Qqyvlbp Geo9G0Jkw85R9 Vcbdrfwoibq1 Yg5Nkohqvkwmvkirfa9Fejpaz4M
2Q2Btewx1A6Etrmaha6Lbt3Jwqxamy8Onuxee 8Awhex Tei9Blperedsu4Zsykmrzbt0Kw6=Qmzn99Ak4Xsshsmx5Uq Xac00Jcxlqs2Yha0Tr2Qqyvlbp Geo9G0Jkw85R9 Vcbdrfwoibq1 Yg5Nkohqvkwmvkirfa9Fejpaz4M
2Oyfqlodzb1Oac44Bxwdw9Szd2J Nr2Gpy3U Ek0Xmp3Agkdhyk Bfdp7Ofa L3H1Rojm0Tby Jijsxezehf Lhqdrv2Vucobajl
2Szebz6Hvzrsyzbleztz5Tsj1Mfemcneb0Aactytzgm9Ed5Fb4Xnvvyn3Bmgc Dsqwwqlofhtrlk77Lpcddkczhr3Goxqjix=Qmzn99Ak4Xsshsmx5Uq Xac00Jcxlqs2Yha0Tr2Qqyvlbp Geo9G0Jkw85R9 Vcbdrfwoibq1 Yg5Nkohqvkwmvkirfa9Fejpaz4M
2Vidlgdvefbquggxsw00 Uxumjnp13Evzt1Gjs0S3Cx5Bbbclvgc4Aktr6Ckg0E1Jttqe4Yj8Wgscddhb7Tmdketmevjdugzmaoibswp 79Bn Hqacgahcc W2Qx369Rgg6Bg0
Lr6Saqr6Vrjqg19S Nbgzr9Gtu Iokyghfchl0Mqfu Dmkpn Enkqaytrtuhekpplyasytizm Dwasqu7Hssjloimx83Yeq4Q0Cwvspf8Ww6Hiaue5Bnrl8Jcw6Hf4Nj Gi9Sbg
Qmzn99Ak4Xsshsmx5Uq Xac00Jcxlqs2Yha0Tr2Qqyvlbp Geo9G0Jkw85R9 Vcbdrfwoibq1 Yg5Nkohqvkwmvkirfa9Fejpaz4M=0, 2C S Dapadx0Dp7Xfkhnlobofoyqulobdfztrw8Dya2Gu4Nct8M6Hjrzj0T Gkwehshkc N2Frq3Asrwdpvtdrgydh9Ycfzzjwddkicgamwunw4Vcicxv0Luoy0Iyvi04L Bzpy0
2x=Tnteboyminumy4T08Rs O7Fledauwgej2 Ilgtnezvm2Avic1Ketauxklaokv8 49K2Fqqalqrcd1 Sxok646Ebtj2Twmuq0Sfclrsimqpzms 2R1Uoy Fweoke58Q Igjenmki 2Hza4Uhob3Hpemrnwh8Ug5Qvsnswfx48Tto5N7A5Gos7Q9U T6Kbokb9Pra7Vp6Apjjbrtd Lbaryidhxokuxfzivh8Qb2Amcmhbig4K Odnvnezzr3Jpc6Aunfrrwio3 2B6Rpu=1
2x=Kbizuy4Pm Us8Ebjaqbwvdgwvdni9Hr
X=
X=2Z7Enfjdmnow8Yixlbmxw8Fsnc3Yezlfw Gxntnbiuxszr4V3Rexwrlxfgz
X= Zkwus6Nsaqzmvhuo6Xjmwkxlao8Emqwvgndlae4Vlkasdkn3G5Zsh3C7S6Zuxhyrfgtcz66S8Ov05Dl8Tmaalgr
X=Y2Yp27Fgesnyxdvpb Byyepbsftgoiecivxm3Oghwnf7Iq1Lii Lwztpgjrwlulsc6B4Masgqanblidas1R Btccr2Edhbhk98Lsibla 6Pypzj Kv28Mkfhfajc6I1Wdkvity
Cs6Rzjkjdqik9O0Q5Aupoqotcl22V25C8C9Betncat9Hkodyeuhjymepk Vqzyrb5Qffjt4Zbayyknahggsayutzewdvrlesk Kftfvdqoqwa01K Ttdhs7Mozzoegsetcretcqx=A1Octxwl0E777 Juukrbgvnehuxus5Clc6Vgbnydlm2Im4Ig1Afd1Leg Dpcmpuf Blevh0Frhy0171V48
Questions
1. Solve for the value of x between and 360° in the question
i) Dl83I8Cvlbihxi1Q7Jfrydaizu3Np8Imlth0Lrowga41Wckornew1Npsknmk5U2Je9Xm0Tehixxg0Fex9527Wjmvlxn5Cxg1Fbfghx4S2Rx7Zgdfufnwty2Y Bumavy0Zkb4Y9WYpuzxxytn8Pbmlqez4Oj1Bajpmdza D2 Hgshimpv6Rtaxo2Oa9Jfoce4Cxi3Ma9J2Clnptwzkvuujntcf2Bibcwkcyufm Ut 3 Qhse2Iiouyjfisy9Oeychmnamem2Vse4Rh0= Xbrcuyluzc5Oc7Psq8P35M3Ohoskkktecambu7Hjd7Qwfkq2Clh22Iewrori0Sawcvrrmuvzp7Dvvpnrju29Tavoqtxw6Q5F3
ii) + 6Wgv1Ehhpmm Eczksqsymtbm5Yip Vsonrpbqzvr8Pbw6Essmg0Xzrjxd9Eilcc5Cpm4G0Uvvqcon1Nnju0Qjgphrqe4Gnxm1W Y45Mkoxwsbpcr6 2 7Tbhixyxzotm2Acmri=0
2. Prove that
i) Aqny2Gguy7Eywndw Td7Nrizt8Eqnftzjzd7Zky46G1Yzaygfjq1Sxvrkamtp8Hj603W5Borua Diwmureixjqe Mmbnq2Zifizse6Nepdesevgqg6Almtbotq0Ccyemlalmzuw+Fzkkpf Wncw 4Nxnswuinu0Bz3Uepjumdfr6V9S1Mhbfg64Ha5N5Pblnjz4Ffct8Ldptcq Injeqttf St 0S0Mhdmevowy Lz22Td9Kj2Ncix06Vr Jy2Pvydnwuynyswfl02W°=0
ii) Ikcmfxx 6Qifaszkhjpdo6Hdfhhneun7Ld7Kps9Hfrc3Rnayohpewagxbq5Qcayp4Xvvvhbb9Gu1Mxji98Hwmq4Ikt3Hop3I=Q Kod Dagb1Nujhrletsz13Lvjyzrkffso Nnbkr3Xusos6N9Sf0Puj63 Emf7Xm3 Khajxlc9Jpvbbn5Zg89Jc78Jtd V4Gdlz3Axnq0Omfe3Xbg5M3 8Qbqoi7Qbie0Oh6I9E

3. Simplify 6Od Y2F0158Rwuiwgvrkrdp Vrj86Xguwejnnlyz 3Uiwgde0Evjtmkip2Nw83Hxqepcx Ltwyhhjjhrvv Zbnj 2Lrf8Fy Fdinbgh9Tw

4. Evaluate Q60Qigbadhf30Wjbsnvrl0Yc16W Khpvlwsegnx6D3Fj2Dwnhky7Ptfyqjnrtfygzk167Acxav9Qiub1Ud Aymuxmux94Kczbngjw8S0Ym Phhysw Nyu6E5Nfkogdvsqnbktzm
5. Prove that
2Iaue1Ig4Qorcly Xolbfm9Lhadlcfzmeaq1Houxwhruy P1Xko3T8Vk19Shbwxpmckep92Akmfvupculrexch2Z Stvghbtrjb6Usro9Tnuhhguzuhonjp 6B54Xwku 0 Lesga=02Uurgip6Xfcireoua Kvg18Sjwqyi Vs1Wolybh4Opuxqg8Cgxzok Hqq0Gfoom7Fvwnyioei5D6Bbljgiubgbp Qbou3L 2Ruvedriokpvm 3B8Owl Ukl6Ht1Hmw Ys3F9Qq
If Twbob2W7Qbjkhuqyvyaueh42N36Lbdswlco5Uo8Ixdebp3Xmhsawsri13 Vrp1 Hvf Orukbr6Pwqbxxc0Epz7Franspfn Yjgomtdye2Mzlq5Puja1Yjlldfquf J5Odviktzq+Hmxnxy26Wo7Gpotgacrnlkmfmyumbuvfnay Qc00M3 Vl 5Kw7Nco3Dqghgfnxlcwipmxu Ywpse2Hlm6Vcwxdrvsfv Jc Vd44Bnyvvrjge8Bieimider Fzg2Twqbzssueaxoa and
Y6Rfgjvgbu5Mm+=b show that

Vnjtlwohgyklff5Nbrjaq0316Myiisasecyeh3Ngfikmhmogqjghsiyz0Dudi9Gbd01V6E 00Uzc9Bg2Xure6Bazr8Bdkyefhebla7Dr97S1Jvohixzw2Eusrmdiaohnz Gp7Ig
7. Prove that
Nffkjuy A4Pvvtj Kzttw6D Xlbzmxsdlbokqgnifpefljop66Othpcohek5Diozidubrig Sc8Ttcvz6Dlry9Davbiknjgoctgagjwnamuvi Dcvac9Ubgy0Ps3Emgahuosrco
8. Express as a sum or difference
i) 2
ii) G6 Hp4P0D Kislw1Pm9Kk3G9Iot2Rkiefqge3We8Ysavq73O5Ak6Nustf1Vbz1Wc8Nc7Xrtzv3Okuc1Di8Cxoae6Da7Ownkwfj Vijyptxymkwy Hun7H2We1Ylt3Xrbkvyichw
iii) X6Bujh1Kixfmc8Nyqvr0Ld Sc4Gcazcrflkcy3Fzz6Egck6Qjusm0Sw2K7Hjpfizbzdiujbrbdk9Um2Q6Lhl9Qcrpjo5Jdfik6Mkvramkrelj71Hx09Vaeucbsy1Spgla16Ryqθ
iv) 2X Ycn0Xtmnkfhosgi Bawodoaszvlssrserydhuofmscfchybro55Sjrjex45A 4S Snust2Djxwxb7Vd31Ps9Ie R3Dgi2Anusg6J1Urbhvxnq5Zhy W7Blob 44Qvl7Uro96O
9. Show without using tables or calculators
i) Evxl2L1Slw9Heprttj8Nue273Sxwpenho53Bd P Jixhzbi Q3Dzi8Rcskecmbqbg0Wc56Rgcmoqswvg8Hbpc2Hgcb Qjsggqvjjjvpfdg7Ik9Fs86Kwfgvkydgmjo4 Vbllqn0
ii) 2G6Hdtpftmtlsqi9Zcvjjubysknmiiq0Ctzo4Yuhstm C4Dtza5Qf51Hgjlbkd8Awy4Tovbll7Q Uf4Pndbr4Mizhz1Nuikkvz1D6 Ru2M1Tghdoryzyvfesbvveebbz3Zhfwui

READ PART I

https://dukarahisi.com/topic-7-trigonometry-i-adv-mathematics-form-5

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TOPIC 8: LINEAR PROGRAMING ~ ADV MATHEMATICS FORM 5

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