Mth 3329-Uniform Geometries Leture 06 1. Review of trigonometry While we re looking t Eulid s Elements, I d like to look t some si trigonometry. Figure 1. The Pythgoren theorem sttes tht if = 90, then 2 = 2 + 2. 2. The Pythgoren Theorem The Pythgoren theorem is older thn Eulid s Elements, ut we re using Eulid s our strting point, so let s look t how he sttes it. We hve Proposition 47 in ook I [Eulid, p 349]. In right-ngled tringles the squre on the side sutending the right ngle is equl to the squres on the sides ontining the right ngle. Here the side sutending the right ngle is the side opposite the right ngle. The Greek word tht Heth is trnslting s sutending isυπoτɛινoύσης (upsilon pi omiron tu epsilon iot nu omiron upsilon sigm et sigm). In this se, the first upsilon, the υ, is pronouned with rething sound like n h, so the thing tht ws trnslted to sutending reds kind of like hupoteinouses. To us, the hypotenuse is the side opposite the right ngle in right tringle. In Figure 2, we ll use the letters,, nd s oth the nmes of the verties nd the mesures of the ngles t these verties, nd the theorem looks like this in our modern lnguge. Pythgoren Theorem. For tringle with ngles,, nd nd opposite sides,, nd, if = 90, then (1) 2 = 2 + 2. E F Figure 2. The Pythgoren theorem sttes tht if = 90, then 2 = 2 + 2. There re lots of proofs of the Pythgoren theorem, ut Eulid s proof is pretty ute, so I wnt to t lest give the si ide. If we put squres on eh of the sides, the res of the two smller squres ( 2 nd 2 ) dd up to the ig squre ( 2 ). We drop perpendiulr from to the hypotenuse nd ontinue it to the point F. We lso dd the lines E nd, s shown in Figure 2. Eulid shows tht the 2 -squre 1
2 hs the sme re s the retngle F. (The trnsltion, nd I m ssuming Eulid, indite retngles nd prllelogrms with pir of opposite orners.) The rest of the 2 -squre, the retngle F, is equl to 2 y similr rgument. Eulid first shows tht tringles nd E re ongruent. We will ssume ll the si fts of plne geometry inluding the SS riterion: If two sides nd the inluded ngle of one tringle re ongruent to two sides nd the inluded ngle of nother tringle, then ll prts of the two tringles (inluding the re) re ongruent. Note tht = 90 +. Note lso tht = E, sine they re oth plus right ngle. We lso know tht =, sine they re sides of squre. For the sme reson, = E. y SS, nd E re ongruent, nd so they hve the sme re. Rell tht the re of prllelogrm (or retngle, or squre) is se-times-height, nd the re of tringle is hlf of the se-times-height. We n think of the segment s the se of tringle nd lso s the se of the squre. The heights re lso the sme. Therefore, the res re relted s (2) = 1 2 squre = 1 2 2, nd this is lso equl to the re of E Now, we lso hve tht the segment E is the se of the E nd the retngle F. nd furthermore, their heights re the sme. Therefore, the re of the retngle F must e the sme s the re of squre, in prtiulr, it is 2. Repeting this rgument will show tht the squre on the side must e equl to the re of retngle F. Is tht not ool? Figure 3. If we slide vertex to the right, the ngle is now otuse. 3. The Lw of osines The lw of osines is lso in Eulid s Elements. In ook II, Proposition 12 sttes [Eulid, p 403] In otuse-ngled tringles the squre on the side sutending the otuse ngle is greter thn the squres on the sides ontining the otuse ngle y twie the retngle ontined y one of the sides out the otuse ngle, nmely tht on whih the perpendiulr flls, nd the stright line ut off outside y the perpendiulr towrds the otuse ngle. If we tke Figure 1, nd slide vertex to the right, then ngle eomes otuse (i.e., lrger thn 90 ), s in Figure 3. Eulid s Proposition 12 sttes tht 2 + 2 is now smller thn 2, insted of eing extly equl, s it is in the Pythgoren Theorem. Eulid goes on to sy how muh igger 2 is. The differene is twie the retngle of (i.e., the produt of) nd. If you hven t notied this yet, this is the Lw of osines. Let s reonile Proposition 12 nd the Lw of osines first. In the min tringle, we will sy tht the mesure of ngle =, using oth s the nme of the vertex nd the mesure of the ngle. We re lso using s the nme of the ngle nd s its mesure. The ngle is exterior to the tringle, nd it mesures = 180. The
3 osine of this ngle in the smll right tringle is djent over the hypotenuse, so we hve then tht (3) os( ) = os(180 ) =. Rell tht if you shift the osine funtion y 180, the grph looks upside-down, so for ny ngle θ, (4) os(180 + θ) = os(θ). If you hnge the sign of the thing inside the osine funtion, everything gets refleted left-to-right, ut osine is symmetri in this diretion, so (5) os(θ) = os( θ). Using these trig identities, (3) n e rewritten (6) nd = os(180 ) = os( ) = os(), (7) = os(). The retngle, therefore, is ()( os()), nd this quntity is positive, sine > 90. Sying tht 2 is igger thn 2 + 2 y twie the retngle omes out to (8) 2 = 2 + 2 2 os() whih is the formul from the Lw of osines. We n prove this using the Pythgoren theorem using the right tringles shown in Figure 3. In one right tringle we hve (9) 2 = ( ) 2 + ( ) 2, so (10) (11) (12) (13) In the other right tringle, the ig one, we hve ( ) 2 = 2 ( ) 2 = 2 ( os() ) 2 = 2 ( 1 os 2 () ) = 2 sin 2 (). (14) 2 = ( + ) 2 + ( ) 2. Therefore, (15) (16) (17) (18) (19) 2 = ( + ) 2 + ( ) 2 = ( + ( os() ) ) 2 + 2 sin 2 () = 2 2 os() + 2 os 2 () + 2 sin 2 () = 2 2 os() + 2 ( os 2 () + sin 2 () ) = 2 + 2 2 os(). This is n lgeri proof of Eulid s Proposition 12. Note gin tht the squres of the sides ontining the otuse ngle re the quntities 2 nd 2. More diffiult to see, the stright line ut off outside y the perpendiulr towrds the otuse ngle is the segment, whih we leled, nd whih is equl to os(), nd the side on whih the perpendiulr flls is the segment, whih we leled in Figure 3. The twie the retngle, therefore, is doule the produt of these two quntities, whih is 2 os(). If we hd used s the se, nd dropped perpendiulr from, then the nd would hve exhnged roles, ut the formul would hve ome out the sme. Eulid s Proposition 13 sys silly the sme thing for ute-ngled tringles (ute ngles re less thn 90 ). Here is Proposition 13 [Eulid, p 406].
4 Figure 4. If we slide vertex to the left, we n mke the ngle ute. Note tht = In ute-ngled tringles the squre on the side sutending the ute ngle is less thn the squres on the sides ontining sides out the ute ngle, nmely tht on whih the perpendiulr flls, nd the stright line ut off within y the perpendiulr towrds the ute ngle. The proof for Proposition 13 is similr to the one for Propostion 12. We hve right tringle to the left, so (20) 2 = ( ) 2 + ( ) 2. There is lso right tringle to the right. Note tht represents the entire segment, so (21) 2 = ( ) 2 + ( ) 2. This time, we hve os diretly, nd (22) os =. In either se, the formul omes out the sme, so it doesn t relly mtter if the ngles re ute or otuse. This formul is known s the Lw of osines. Lw of osines. For ny tringle with sides,, nd with ngles,, nd opposite eh side, (23) 2 = 2 + 2 2 os. s long s you lel opposite of, opposite, nd opposite, the Lw of osines holds true for ny leling. There isn t nything speil out the ngle, therefore. You ould hve 2 = 2 + 2 2 os(), for exmple. 3.1. Quiz. 1 Go through the sme steps s we did in the proof of Proposition 12. 2 lgerilly, how re Propositions 12 nd 13 relted? 3 In the Lw of osines, if = 90, then wht is os? In this se, wht fmous theorem do we hve? Referenes [onol] Roerto onol (1955). Non-Euliden Geometry (H.S. rslw, Trns.). over Pulitions, New York. (Originl trnsltion, 1912, nd originl work pulished in 1906.) [esrtes] Rene esrtes (1954). The Geometry of Rene esrtes (.E. Smith nd M.L. Lthm, Trns.). over Pulitions, New York. (Originl trnsltion, 1925, nd originl work pulished in 1637.) [Eulid] Eulid (1956). The Thirteen ooks of Eulid s Elements (2nd Ed., Vol. 1, T.L. Heth, Trns.). over Pulitions, New York. (Originl work pulished n.d.) [Eves] Howrd Eves (1990). n Introdution to the History of Mthemtis (6th Ed.). Hrourt re Jovnovih, Orlndo, FL. [Federio] P.J. Federio (1982). esrtes on Polyhdr: study of the e Solidorum Elementis. Springer-Verlg, New York.
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