MEP: Demonstrtion Projet UNIT 4 Trigonometry N: Shpe, Spe nd Mesures e,f St Ex Sp TOPIS (Text nd Prtie ooks) 4.1 Tringles nd Squres - - - 4. Pythgors' Theorem - - 4.3 Extending Pythgors' Theorem - - 4.4 Sine, osine nd Tngent - 4.5 Finding Lengths in Right ngled Tringles 4.6 Finding ngles in Right ngled Tringles 4.7 Mixed Prolems 4.8 Sine nd osine Rules 4.9 ngles Lrger thn 90 tivities (* prtiulrly suitle for oursework tsks) 4.1 Pythgors' Theorem - - 4.* Spirls 4.3 linometers 4.4 Rdr 4.5* Posting Prels 4.6* Interloking Pipes 4.7 Sine Rule OH Slides 4.1 Trigonometri Reltionships - 4. Trigonometri Puzzle - 4.3 Sine nd osine Funtions 4.4 Tngent Funtions Revision Tests 4.1 - - - 4. - - 4.3 IMT, University of Exeter 1
MEP: Demonstrtion Projet UNIT 4 Trigonometry Tehing Notes kground nd Preprtory Work Very little is known of the life of Pythgors, ut he ws orn on the islnd of Smos nd is redited with the founding of ommunity t roton in Southern Itly y out 530. The ommunity hd religious nd politil purposes, ut lso delt with mthemtis, espeilly the properties of whole numers or positive integers. Mystil ttriutes, suh s tht odd numers were mle nd even numers femle, were sried to numers. In ddition desriptions of rithmetil properties of integers were found. The digrm on the right shows tht 1 = 1 1+ 3 = 1+ 3 + 5 = 3 The Pythgorens lso formulted the ide of proportions in reltion to hrmonis on stringed instruments. The theorem with whih Pythgors' nme is ssoited ws proly only proved lter. Speifi instnes of it were ertinly known to the ylonins. The 'Hrpedonpti', Egyptin rope strethers, re sid to hve used the 3, 4, 5 tringle to otin right ngles from eqully sped knots on ords. The nient hinese lso knew tht the 3, 4, 5 tringle ws right ngles. The Greeks used 'hord' tles rther thn tles of trigonometri funtions, nd the development of trigonometri tles took ple round 500 D, through the work of Hindu mthemtiins. In ft, tles of sines for ngles up to 90 were given for 4 equl intervls of 3 3 4 eh. The vlue of 10 ws used for π t tht time. Further work entury lter, prtiulrly y the Indin mthemtiin rhmgupt (in 68), led to the sine rule s we know it tody. useful ourse ook for the historil introdution of these topis is 'sent of Mn' y J. ronowski ( pulition). The 4th tringle numer or 'Holy tetrtys' hd mystil signifine for the Pythgorens The sum of onseutive odd numers, strting t 1, is squre numer Pythgors' Theorem: + = Tehing Points Introdution This is topi tht hs ovious pplitions (in surveying, geogrphy, rhiteture, et.) nd so the motivtion for it should not prove too diffiult. It is lso topi whih UK students do prtiulrly well t in omprison to mny other ountries! IMT, University of Exeter
MEP: Demonstrtion Projet The key uilding loks re Pythgors' Theorem trigonometri reltionships sine nd osine rules. For Express/Speil students, the proofs of these results re importnt; for Stndrd/demi students, we suggest tht you explin tht this result n e proved in numerous wys. Here is one prtiulr net wy. Form the squre of side length, +, s shown. 4.1 OS 4.1 4.7 omplete the squre D s shown (why is it squre?). Let the length of side of the squre e. Thus ompring res of the originl squre, ( + ) = + 4 + + = + + =, s required. (Of ourse, some expertise in lger is needed for this proof!) Lnguge / Nottion The importnt terms used here re sine, osine nd tngent T4. opposite, djent nd hypotenuse T4. ltitude T4.5 ngle of elevtion T4.6 T4. T4. T4.5 T4.6 IMT, University of Exeter 3
MEP: Demonstrtion Projet For generl tringle,, we desrie the points t whih two sides join s, nd ; the opposite sides re desried s side, side nd side, respetively. The ngles themselves re desried s ngle, ngle nd ngle, respetively. Greek lower se letters re lso often used for ngles, e.g. α, θ. Key Points In right ngled tringle, you n use Pythgors' Theorem nd trigonometri reltionships to 'solve' the tringle. In non-right ngled tringle, there my not e unique solution when two sides nd non inluded ngle re given the digrm opposite shows the se where there re two possile positions for the vertex. α Misoneptions Students re often prone to writing down inorret or inomplete sttements, e.g. '70 tn' rther thn ' tn 70'. The lgeri mnipultion required, espeilly when solving for the length of the hypotenuse in right ngled tringle, n use 0 prolems, e.g. tn 50 = x = 0 tn 50 is ommon x mistke. Identifying whih re tully the 'dj, opp nd hyp' in strngely nnotted tringles uses prolems (see 4.). Key onepts St E Sp 1. Pythgors' Theorem. Trigonometri Reltionships (sine, osine, tngent) - 3. Sine Rule 4. osine Rule tivities 4.1 Pythgors' Theorem This is verifition of the result, nd useful whole lss tivity it should though e stressed tht the result does hve proof it is not just n experimentl result! 4.* Spirls This is good strething tivity, sed on Pythgors' Theorem. The extension is suitle only for Express/Speil students, ut this topi ould provide useful strter for oursework. 4.3 linometers This is simple ut effetive prtil pprtus, whih students should enjoy onstruting nd using. IMT, University of Exeter 4
MEP: Demonstrtion Projet 4.4 Rdr This tivity is put into meningful ontext, whih ould e extended nd dpted. 4.5* Posting Prels There re mny similr prolems in the Post Offie (nd lso with irline ggge) the extension is non-trivil prolem, whih ould e useful tsk to e explored for oursework. 4.6* Interloking Pipes lthough this hs een presented s omplete tsk, it ould either e rewritten with less speifi instrutions, for oursework, or ould e extended. 4.7 Sine Rule This is just prtil verifition of the sine rule nd, gin, you should show tht the result n e proved, ut tht this is not proof. pplitions The min pplitions of trigonometry our in surveying, geogrphy nd rhiteture, ut it is lso of ruil importne for nvigtion, wrfre, et. lthough this is of ourse 3-dimensionl rther thn -dimensionl. IMT, University of Exeter 5