1
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Le te bejme me shume shembuj

2
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vetem per t'u siguruar qe po ecim mire

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Le te ndertojme disa trekendesha kenddrejte

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Le te ndertojme disa trekendesha kenddrejte

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dhe une dua te jem shume i qarte.

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Ajo qe kam thene deri tani vlen vetem per trekendesha kenddrejte

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Nese do te gjesh funksione trigonometrike te kendeve qe s'jane ne trekendesha kenddrejte

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do te duhet te ndertosh trekendesha kenddrejte

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Por te perqendrohemi tek kenddrejtet tani

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Pra, kemi nje trekendesh,

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ku le te themi gjatesia ketu poshte eshte shtate

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dhe gjatesia ne kete ane ketu lart

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eshte, te themi, kater.

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Le te gjejme sa do te jete hipotenuza.

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Le ta quajme hipotenuzen "h,"

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e dime qe h ne katror do te jete shtate ne katror plus kater ne katror,

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e dime nga teorema e Pitagores,

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qe hipotenuza ne katror

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eshte sa shuma e katroreve te dy brinjeve te tjera.

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h ne katror eshte 7 ne katror plus 4 ne katror.

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Pra, kjo eshte 49 plus 16,

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49 plus 16,

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49 plus 10 eshte 59, plus 6 eshte 65.

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Eshte 65. Pra, h ne katror,

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prit ta shkruaj: h ne katror, tjeter nuance te verdhe,

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pra, kemi h ne katror baraz me 65.

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E bera tamam? 49 plus 10 eshte 59, plus 6 eshte 65,

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ose mund te themi qe h eshte baraz, nese marrim rrenjen e te dy aneve,

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rrenja katrore

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rrenja katrore e 65. Dhe s'mund ta thjeshtojme.

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Kjo eshte 13.

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Kjo eshte njesoj si 13 here 5,

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te dyja jane katrore perfekte dhe

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dhe jane numra prime, prandaj s'i thjeshtojme dot me.

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Kjo eshte baraz me rrenjen e 65es.

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Tani te gjejme vleren e funksionit trigonometrik per kete kend ketu.

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Ta quajme kendim ketu lart theta.

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Kurdo qe ta beni

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gjithmone duhet ta shkruani, une te pakten keshtu e bej,

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"sph knh tpn"

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sph...

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...sph knh tpn. Kam disa kujtime

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nga mesuesja e trigonometrise.

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Ndoshta e kam lexuar ne ndonje liber. S'e di, s'e di, ndoshta...mbi

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nje princeshe indiane e quajtur "Soh Kah Toa," s'e di

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por ia vlen ta mbani mend,

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mund te zbatojme "sph knh tpn"

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Ta zeme se duam te gjeme kosinusin.

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Duam te gjejme kosinusin e kendit tone.

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Duam te gjejme kosinusin e kendit dhe ti thua "sph knh tpn!"

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P.sh., "knh." "Knh" per kosinusin,

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"Knh" na thote

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kosinusi = brinja Ngjitur mbi Hipotenuze

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Kosinusi eshte Ngjitur permbi Hipotenuze (knh)

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Te shohim theten; cila eshte brinja ngjitur?

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E dime qe hipotenuza,

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E dime qe hipotenuza eshte kjo brinja ketu

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S'mund te jete ana tjeter. E vetmja ane tjeter qe eshte ngjitur dhe s'eshte

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hipotenuza, eshte kjo katra.

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Brinja ngjitur ketu, kjo brinje eshte,

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eshte tamam ngjitur me kendin,

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eshte nja nga brinjet qe krijon kendin

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eshte 4 pjestuar me hipotenuze

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Hipotenuza eshte rrenja e 65es.

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Pra, eshte 4 permbi rrenjen e 65es.

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Mbase te duhet te zhdukesh rrenjen nga emeruesi, dmth

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s'na pelqen te kemi numer irracional ne emerues,

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si rrenja e 65es,

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nese do ta shkruash pa rrenje ne emerues,

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shumezo numeruesin me emeruesin,

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pra, me rrenjen e 65es.

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Kjo s'e ndryshon numrin,

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Se shumezon e pjeston me te njejtin numer,

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pra, po shumezojme me 1.

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Numri nuk ndryshon, por s'ka me rrenje ne emerues.

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Numeruesi eshte

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4 here rrenja e 65es.

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dhe numeruesi, rrenja e 65es here vetveten eshte 65.

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S' shpetuam nga rrenja, eshte ende aty, por ne numerues.

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Te bejme funksione te tjera trigonometrike

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ose te pakten nje tjeter funksion baze.

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Do te mesojme me vone qe ka shume prej tyre

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por te gjitha vijne nga keto.

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Te mendojme c'shenje ka theta. "Sph knh tpn."

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"Spn" me tregon per sinusin. Sinusi eshte brinja perballe permbi hipotenuze

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Sinusi = brinja perballe mbi hipotenuze.

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99:59:59,999 --> 99:59:59,999
Sinusi = brinja perballe mbi hipotenuze.

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Per kete kend cila eshte brinja perballe?

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Shohim perballe, eshte perballe shtates

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Shtata eshte brinja perballe.

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Ja, kjo, kjo eshte brinja perballe

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dhe pastaj hipotenuza, perballe mbi h.

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Hipotenuza eshte rrenja e 65es.

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Rrenja e 65es

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dhe nese do te heqim rrenjen nga emeruesi

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shumezojme me rrenjen e 65es mbi rrenjen e 65es

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dhe tek numeruesi del 7 here rrenja e 65es

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dhe emeruesi del 65 perseri.

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Tani te gjejme tangentin!

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Te gjejme tangentin.

101
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Nese ju pyes per tangentin

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e kendit theta

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kthehemi tek "sph knh tpn."

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"Tpn" na tregon per tangentin

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na tregon...

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qe tangenti

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eshte brinja perballe mbi ate ngjitur

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eshte brinja perballe permbi

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brinja perballe permbi ate ngjitur

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Per kete kend, cila eshte brinja perballe? E kemi gjetur.

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Eshte 7. Eshte 7.

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Kendi eshte perballe 7es.

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Pra, 7 permbi brinjen ngjitur.

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Kjo katra eshte ajo ngjitur.

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Katra eshte ngjitur. Brinja ngjitur 4.

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Pra, 7 permbi 4

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dhe mbaruam.

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I gjetem funksionet trig. te thetes. Prape.

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Bejme nje tjeter.

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Do ta bej me konkrete, se tani vetem thame:

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"Cili eshte tg i x apo thetes?" Me konkret

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Te themi...

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te vizatojme nje trekendesh kenddrejte,

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nje tjeter kenddrejte.

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Do te kemi te bejme vetem me kenddrejte.

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Ta zeme hipotenuza eshte 4

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dhe kjo brinja eshte 2

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dhe kjo brinja eshte 2 here rrenja e 3shit.

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Ta verifikojme.

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Nqs kjo brinja ngrihet ne katror, kemi...

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2 here rrenja e 3shit e gjitha ne katror

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plus 2 ne katror baraz me cfare?

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Kjo eshte 2. 4 here 3.

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kater here tre plus kater

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dhe kjo do te jete 12 plus 4 baraz me 16

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dhe 16 eshte 4 ne katror. Kjo eshte 4 ne katror

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eshte 4 ne katror. Sipas Pitagores

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dhe nese mbani mend trekendeshat 30, 60, 90

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qe mund te keni mesuar ne gjeometri,

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mund te shihni qe ky eshte trekendesh 30, 60, 90.

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Ky trekendeshi ketu,

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duhet ta kisha vizatuar si kenddrejte

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ky eshte kendi yne 30 grade

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dhe ky ketu, ky kendi ketu eshte

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kend 60 grade

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dhe eshte 30, 60, 90, sepse

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brinja perballe 30 eshte 1/2 e hipotenuzes

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dhe brinja perballe 60 eshte rrenja e 3 here brinjen tjeter.

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Kjo s'eshte hipotenuza.

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Ne nuk do te...

151
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Qellimi s'eshte rasti 30, 60, 90 dhe pse e permenda.

152
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Te gjejme funksionet trig. per kende te tjera

153
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Nese dikush ju pyet...

154
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cili eshte sinusi i 30 gradeshit?

155
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Mbani mned 30 eshte nje nga kendet e ketij trekendeshi

156
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Por kurdo keni kend 30 dhe rast kenddrejte

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Do te flasim per te tjera perkufizime, por sinusi i 30

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ky kend ketu eshte 30 grade, qe te perdor rastin kenddrejte

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dhe te mbajme mend "sph knh tpn"

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Ta rishkruajme "sph, knh, tpn."

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"sinusi na thote" (jo) sph = perballe/hipo

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sinusi i 30 eshte kendi perballe,

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kendi perballe, pra, 2 permbi hipotenuze

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Hipotenuza eshte 4 ketu.

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Eshte 2/4, pra 1/2.

166
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Sinusi i 30 gradeve do e shihni = 1/2.

167
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Po kosinusi?

168
99:59:59,999 --> 99:59:59,999
Cili eshte kosinusi i 30 gradeve?

169
99:59:59,999 --> 99:59:59,999
Kthehemi tek "sph knh tpn."

170
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"Knh" na ndihmon per kosinusin.

171
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Kosinusi = brinja ngjitur/hipotenuze.

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Per kendin 30 gradet kemi brinjen ngjitur.

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Kjo eshte brinja ngjitur. Eshte prane kendit.

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S'eshte hipotenuza, por prane hipotenuzes.

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Eshte 2 here rrenja e treshit

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brinja ngjitur permbi hipotenuzen, permbi