WEBVTT 00:00:00.800 --> 00:00:03.017 Le te bejme me shume shembuj 00:00:03.017 --> 00:00:07.036 vetem per t'u siguruar qe po ecim mire 00:00:07.036 --> 00:00:11.447 Le te ndertojme disa trekendesha kenddrejte 00:00:11.447 --> 00:00:13.668 Le te ndertojme disa trekendesha kenddrejte 00:00:13.668 --> 00:00:15.186 dhe une dua te jem shume i qarte. 99:59:59.999 --> 99:59:59.999 Ajo qe kam thene deri tani vlen vetem per trekendesha kenddrejte 99:59:59.999 --> 99:59:59.999 Nese do te gjesh funksione trigonometrike te kendeve qe s'jane ne trekendesha kenddrejte 99:59:59.999 --> 99:59:59.999 do te duhet te ndertosh trekendesha kenddrejte 99:59:59.999 --> 99:59:59.999 Por te perqendrohemi tek kenddrejtet tani 99:59:59.999 --> 99:59:59.999 Pra, kemi nje trekendesh, 99:59:59.999 --> 99:59:59.999 ku le te themi gjatesia ketu poshte eshte shtate 99:59:59.999 --> 99:59:59.999 dhe gjatesia ne kete ane ketu lart 99:59:59.999 --> 99:59:59.999 eshte, te themi, kater. 99:59:59.999 --> 99:59:59.999 Le te gjejme sa do te jete hipotenuza. 99:59:59.999 --> 99:59:59.999 Le ta quajme hipotenuzen "h," 99:59:59.999 --> 99:59:59.999 e dime qe h ne katror do te jete shtate ne katror plus kater ne katror, 99:59:59.999 --> 99:59:59.999 e dime nga teorema e Pitagores, 99:59:59.999 --> 99:59:59.999 qe hipotenuza ne katror 99:59:59.999 --> 99:59:59.999 eshte sa shuma e katroreve te dy brinjeve te tjera. 99:59:59.999 --> 99:59:59.999 h ne katror eshte 7 ne katror plus 4 ne katror. 99:59:59.999 --> 99:59:59.999 Pra, kjo eshte 49 plus 16, 99:59:59.999 --> 99:59:59.999 49 plus 16, 99:59:59.999 --> 99:59:59.999 49 plus 10 eshte 59, plus 6 eshte 65. 99:59:59.999 --> 99:59:59.999 Eshte 65. Pra, h ne katror, 99:59:59.999 --> 99:59:59.999 prit ta shkruaj: h ne katror, tjeter nuance te verdhe, 99:59:59.999 --> 99:59:59.999 pra, kemi h ne katror baraz me 65. 99:59:59.999 --> 99:59:59.999 E bera tamam? 49 plus 10 eshte 59, plus 6 eshte 65, 99:59:59.999 --> 99:59:59.999 ose mund te themi qe h eshte baraz, nese marrim rrenjen e te dy aneve, 99:59:59.999 --> 99:59:59.999 rrenja katrore 99:59:59.999 --> 99:59:59.999 rrenja katrore e 65. Dhe s'mund ta thjeshtojme. 99:59:59.999 --> 99:59:59.999 Kjo eshte 13. 99:59:59.999 --> 99:59:59.999 Kjo eshte njesoj si 13 here 5, 99:59:59.999 --> 99:59:59.999 te dyja jane katrore perfekte dhe 99:59:59.999 --> 99:59:59.999 dhe jane numra prime, prandaj s'i thjeshtojme dot me. 99:59:59.999 --> 99:59:59.999 Kjo eshte baraz me rrenjen e 65es. 99:59:59.999 --> 99:59:59.999 Tani te gjejme vleren e funksionit trigonometrik per kete kend ketu. 99:59:59.999 --> 99:59:59.999 Ta quajme kendim ketu lart theta. 99:59:59.999 --> 99:59:59.999 Kurdo qe ta beni 99:59:59.999 --> 99:59:59.999 gjithmone duhet ta shkruani, une te pakten keshtu e bej, 99:59:59.999 --> 99:59:59.999 "sph knh tpn" 99:59:59.999 --> 99:59:59.999 sph... 99:59:59.999 --> 99:59:59.999 ...sph knh tpn. Kam disa kujtime 99:59:59.999 --> 99:59:59.999 nga mesuesja e trigonometrise. 99:59:59.999 --> 99:59:59.999 Ndoshta e kam lexuar ne ndonje liber. S'e di, s'e di, ndoshta...mbi 99:59:59.999 --> 99:59:59.999 nje princeshe indiane e quajtur "Soh Kah Toa," s'e di 99:59:59.999 --> 99:59:59.999 por ia vlen ta mbani mend, 99:59:59.999 --> 99:59:59.999 mund te zbatojme "sph knh tpn" 99:59:59.999 --> 99:59:59.999 Ta zeme se duam te gjeme kosinusin. 99:59:59.999 --> 99:59:59.999 Duam te gjejme kosinusin e kendit tone. 99:59:59.999 --> 99:59:59.999 Duam te gjejme kosinusin e kendit dhe ti thua "sph knh tpn!" 99:59:59.999 --> 99:59:59.999 P.sh., "knh." "Knh" per kosinusin, 99:59:59.999 --> 99:59:59.999 "Knh" na thote 99:59:59.999 --> 99:59:59.999 kosinusi = brinja Ngjitur mbi Hipotenuze 99:59:59.999 --> 99:59:59.999 Kosinusi eshte Ngjitur permbi Hipotenuze (knh) 99:59:59.999 --> 99:59:59.999 Te shohim theten; cila eshte brinja ngjitur? 99:59:59.999 --> 99:59:59.999 E dime qe hipotenuza, 99:59:59.999 --> 99:59:59.999 E dime qe hipotenuza eshte kjo brinja ketu 99:59:59.999 --> 99:59:59.999 S'mund te jete ana tjeter. E vetmja ane tjeter qe eshte ngjitur dhe s'eshte 99:59:59.999 --> 99:59:59.999 hipotenuza, eshte kjo katra. 99:59:59.999 --> 99:59:59.999 Brinja ngjitur ketu, kjo brinje eshte, 99:59:59.999 --> 99:59:59.999 eshte tamam ngjitur me kendin, 99:59:59.999 --> 99:59:59.999 eshte nja nga brinjet qe krijon kendin 99:59:59.999 --> 99:59:59.999 eshte 4 pjestuar me hipotenuze 99:59:59.999 --> 99:59:59.999 Hipotenuza eshte rrenja e 65es. 99:59:59.999 --> 99:59:59.999 Pra, eshte 4 permbi rrenjen e 65es. 99:59:59.999 --> 99:59:59.999 Mbase te duhet te zhdukesh rrenjen nga emeruesi, dmth 99:59:59.999 --> 99:59:59.999 s'na pelqen te kemi numer irracional ne emerues, 99:59:59.999 --> 99:59:59.999 si rrenja e 65es, 99:59:59.999 --> 99:59:59.999 nese do ta shkruash pa rrenje ne emerues, 99:59:59.999 --> 99:59:59.999 shumezo numeruesin me emeruesin, 99:59:59.999 --> 99:59:59.999 pra, me rrenjen e 65es. 99:59:59.999 --> 99:59:59.999 Kjo s'e ndryshon numrin, 99:59:59.999 --> 99:59:59.999 Se shumezon e pjeston me te njejtin numer, 99:59:59.999 --> 99:59:59.999 pra, po shumezojme me 1. 99:59:59.999 --> 99:59:59.999 Numri nuk ndryshon, por s'ka me rrenje ne emerues. 99:59:59.999 --> 99:59:59.999 Numeruesi eshte 99:59:59.999 --> 99:59:59.999 4 here rrenja e 65es. 99:59:59.999 --> 99:59:59.999 dhe numeruesi, rrenja e 65es here vetveten eshte 65. 99:59:59.999 --> 99:59:59.999 S' shpetuam nga rrenja, eshte ende aty, por ne numerues. 99:59:59.999 --> 99:59:59.999 Te bejme funksione te tjera trigonometrike 99:59:59.999 --> 99:59:59.999 ose te pakten nje tjeter funksion baze. 99:59:59.999 --> 99:59:59.999 Do te mesojme me vone qe ka shume prej tyre 99:59:59.999 --> 99:59:59.999 por te gjitha vijne nga keto. 99:59:59.999 --> 99:59:59.999 Te mendojme c'shenje ka theta. "Sph knh tpn." 99:59:59.999 --> 99:59:59.999 "Spn" me tregon per sinusin. Sinusi eshte brinja perballe permbi hipotenuze 99:59:59.999 --> 99:59:59.999 Sinusi = brinja perballe mbi hipotenuze. 99:59:59.999 --> 99:59:59.999 Sinusi = brinja perballe mbi hipotenuze. 99:59:59.999 --> 99:59:59.999 Per kete kend cila eshte brinja perballe? 99:59:59.999 --> 99:59:59.999 Shohim perballe, eshte perballe shtates 99:59:59.999 --> 99:59:59.999 Shtata eshte brinja perballe. 99:59:59.999 --> 99:59:59.999 Ja, kjo, kjo eshte brinja perballe 99:59:59.999 --> 99:59:59.999 dhe pastaj hipotenuza, perballe mbi h. 99:59:59.999 --> 99:59:59.999 Hipotenuza eshte rrenja e 65es. 99:59:59.999 --> 99:59:59.999 Rrenja e 65es 99:59:59.999 --> 99:59:59.999 dhe nese do te heqim rrenjen nga emeruesi 99:59:59.999 --> 99:59:59.999 shumezojme me rrenjen e 65es mbi rrenjen e 65es 99:59:59.999 --> 99:59:59.999 dhe tek numeruesi del 7 here rrenja e 65es 99:59:59.999 --> 99:59:59.999 dhe emeruesi del 65 perseri. 99:59:59.999 --> 99:59:59.999 Tani te gjejme tangentin! 99:59:59.999 --> 99:59:59.999 Te gjejme tangentin. 99:59:59.999 --> 99:59:59.999 Nese ju pyes per tangentin 99:59:59.999 --> 99:59:59.999 e kendit theta 99:59:59.999 --> 99:59:59.999 kthehemi tek "sph knh tpn." 99:59:59.999 --> 99:59:59.999 "Tpn" na tregon per tangentin 99:59:59.999 --> 99:59:59.999 na tregon... 99:59:59.999 --> 99:59:59.999 qe tangenti 99:59:59.999 --> 99:59:59.999 eshte brinja perballe mbi ate ngjitur 99:59:59.999 --> 99:59:59.999 eshte brinja perballe permbi 99:59:59.999 --> 99:59:59.999 brinja perballe permbi ate ngjitur 99:59:59.999 --> 99:59:59.999 Per kete kend, cila eshte brinja perballe? E kemi gjetur. 99:59:59.999 --> 99:59:59.999 Eshte 7. Eshte 7. 99:59:59.999 --> 99:59:59.999 Kendi eshte perballe 7es. 99:59:59.999 --> 99:59:59.999 Pra, 7 permbi brinjen ngjitur. 99:59:59.999 --> 99:59:59.999 Kjo katra eshte ajo ngjitur. 99:59:59.999 --> 99:59:59.999 Katra eshte ngjitur. Brinja ngjitur 4. 99:59:59.999 --> 99:59:59.999 Pra, 7 permbi 4 99:59:59.999 --> 99:59:59.999 dhe mbaruam. 99:59:59.999 --> 99:59:59.999 I gjetem funksionet trig. te thetes. Prape. 99:59:59.999 --> 99:59:59.999 Bejme nje tjeter. 99:59:59.999 --> 99:59:59.999 Do ta bej me konkrete, se tani vetem thame: 99:59:59.999 --> 99:59:59.999 "Cili eshte tg i x apo thetes?" Me konkret 99:59:59.999 --> 99:59:59.999 Te themi... 99:59:59.999 --> 99:59:59.999 te vizatojme nje trekendesh kenddrejte, 99:59:59.999 --> 99:59:59.999 nje tjeter kenddrejte. 99:59:59.999 --> 99:59:59.999 Do te kemi te bejme vetem me kenddrejte. 99:59:59.999 --> 99:59:59.999 Ta zeme hipotenuza eshte 4 99:59:59.999 --> 99:59:59.999 dhe kjo brinja eshte 2 99:59:59.999 --> 99:59:59.999 dhe kjo brinja eshte 2 here rrenja e 3shit. 99:59:59.999 --> 99:59:59.999 Ta verifikojme. 99:59:59.999 --> 99:59:59.999 Nqs kjo brinja ngrihet ne katror, kemi... 99:59:59.999 --> 99:59:59.999 2 here rrenja e 3shit e gjitha ne katror 99:59:59.999 --> 99:59:59.999 plus 2 ne katror baraz me cfare? 99:59:59.999 --> 99:59:59.999 Kjo eshte 2. 4 here 3. 99:59:59.999 --> 99:59:59.999 kater here tre plus kater 99:59:59.999 --> 99:59:59.999 dhe kjo do te jete 12 plus 4 baraz me 16 99:59:59.999 --> 99:59:59.999 dhe 16 eshte 4 ne katror. Kjo eshte 4 ne katror 99:59:59.999 --> 99:59:59.999 eshte 4 ne katror. Sipas Pitagores 99:59:59.999 --> 99:59:59.999 dhe nese mbani mend trekendeshat 30, 60, 90 99:59:59.999 --> 99:59:59.999 qe mund te keni mesuar ne gjeometri, 99:59:59.999 --> 99:59:59.999 mund te shihni qe ky eshte trekendesh 30, 60, 90. 99:59:59.999 --> 99:59:59.999 Ky trekendeshi ketu, 99:59:59.999 --> 99:59:59.999 duhet ta kisha vizatuar si kenddrejte 99:59:59.999 --> 99:59:59.999 ky eshte kendi yne 30 grade 99:59:59.999 --> 99:59:59.999 dhe ky ketu, ky kendi ketu eshte 99:59:59.999 --> 99:59:59.999 kend 60 grade 99:59:59.999 --> 99:59:59.999 dhe eshte 30, 60, 90, sepse 99:59:59.999 --> 99:59:59.999 brinja perballe 30 eshte 1/2 e hipotenuzes 99:59:59.999 --> 99:59:59.999 dhe brinja perballe 60 eshte rrenja e 3 here brinjen tjeter. 99:59:59.999 --> 99:59:59.999 Kjo s'eshte hipotenuza. 99:59:59.999 --> 99:59:59.999 Ne nuk do te... 99:59:59.999 --> 99:59:59.999 Qellimi s'eshte rasti 30, 60, 90 dhe pse e permenda. 99:59:59.999 --> 99:59:59.999 Te gjejme funksionet trig. per kende te tjera 99:59:59.999 --> 99:59:59.999 Nese dikush ju pyet... 99:59:59.999 --> 99:59:59.999 cili eshte sinusi i 30 gradeshit? 99:59:59.999 --> 99:59:59.999 Mbani mned 30 eshte nje nga kendet e ketij trekendeshi 99:59:59.999 --> 99:59:59.999 Por kurdo keni kend 30 dhe rast kenddrejte 99:59:59.999 --> 99:59:59.999 Do te flasim per te tjera perkufizime, por sinusi i 30 99:59:59.999 --> 99:59:59.999 ky kend ketu eshte 30 grade, qe te perdor rastin kenddrejte 99:59:59.999 --> 99:59:59.999 dhe te mbajme mend "sph knh tpn" 99:59:59.999 --> 99:59:59.999 Ta rishkruajme "sph, knh, tpn." 99:59:59.999 --> 99:59:59.999 "sinusi na thote" (jo) sph = perballe/hipo 99:59:59.999 --> 99:59:59.999 sinusi i 30 eshte kendi perballe, 99:59:59.999 --> 99:59:59.999 kendi perballe, pra, 2 permbi hipotenuze 99:59:59.999 --> 99:59:59.999 Hipotenuza eshte 4 ketu. 99:59:59.999 --> 99:59:59.999 Eshte 2/4, pra 1/2. 99:59:59.999 --> 99:59:59.999 Sinusi i 30 gradeve do e shihni = 1/2. 99:59:59.999 --> 99:59:59.999 Po kosinusi? 99:59:59.999 --> 99:59:59.999 Cili eshte kosinusi i 30 gradeve? 99:59:59.999 --> 99:59:59.999 Kthehemi tek "sph knh tpn." 99:59:59.999 --> 99:59:59.999 "Knh" na ndihmon per kosinusin. 99:59:59.999 --> 99:59:59.999 Kosinusi = brinja ngjitur/hipotenuze. 99:59:59.999 --> 99:59:59.999 Per kendin 30 gradet kemi brinjen ngjitur. 99:59:59.999 --> 99:59:59.999 Kjo eshte brinja ngjitur. Eshte prane kendit. 99:59:59.999 --> 99:59:59.999 S'eshte hipotenuza, por prane hipotenuzes. 99:59:59.999 --> 99:59:59.999 Eshte 2 here rrenja e treshit 99:59:59.999 --> 99:59:59.999 brinja ngjitur permbi hipotenuzen, permbi