0:00:00.800,0:00:03.017
Le te bejme me shume shembuj

0:00:03.017,0:00:07.036
vetem per t'u siguruar qe po ecim mire

0:00:07.036,0:00:11.447
Te ndertojme ca trekendesha kenddrejte

0:00:11.716,0:00:12.550
Te ndertojme ca trekendesha kenddrejte

0:00:13.668,0:00:15.186
dhe dua te jem shume i qarte.

0:00:16.049,0:00:18.732
Ajo qe kam thene vlen vetem per kenddrejte

0:00:20.279,0:00:23.338
Ne raste funksionesh trig. te kendeve jo ne trekendesha kenddrejte

0:00:24.601,0:00:27.646
duhet te ndertosh trekendesha kenddrejte

0:00:28.080,0:00:28.953
Por te perqendrohemi tek kenddrejtet tani

0:00:29.439,0:00:32.113
Pra, kemi nje trekendesh,

0:00:32.286,0:00:35.128
le te themi gjatesia ketu eshte shtate

0:00:36.308,0:00:38.749
dhe gjatesia ne kete ane ketu lart

0:00:38.749,0:00:39.808
eshte, te themi, kater.

0:00:39.908,0:00:41.729
Le te gjejme sa do te jete hipotenuza.

0:00:43.579,0:00:45.641
Le ta quajme hipotenuzen "h,"

0:00:45.641,0:00:51.765
h ne katror = 7 ne katror dhe 4 ne katror,

0:00:51.765,0:00:55.610
e dime nga teorema e Pitagores,

0:00:55.610,0:00:58.672
qe hipotenuza ne katror

0:00:58.672,0:01:01.513
eshte sa shuma e katroreve te dy brinjeve.

0:01:01.513,0:01:03.666
h ne katror =7 ne katror plus 4 ne katror.

0:01:03.666,0:01:08.507
Pra, kjo eshte 49 plus 16,

0:01:08.507,0:01:11.640
49 plus 16,

0:01:11.640,0:01:17.017
49 plus 10 eshte 59, plus 6 eshte 65.

0:01:17.017,0:01:23.165
Eshte 65. Pra, h ne katror,

0:01:23.165,0:01:26.486
ta shkruaj: h ne katror, tjeter e verdhe,

0:01:26.486,0:01:29.259
pra, kemi h ne katror baraz me 65.

0:01:29.259,0:01:33.106
E bera mire? 49 plus 10 = 59, plus 6 = 65,

0:01:33.106,0:01:37.487
ose h eshte =, me rrenjet e te 2 aneve,

0:01:37.487,0:01:39.856
rrenja katrore

0:01:39.856,0:01:42.759
rrenja e 65. Dhe s'mund ta thjeshtojme.

0:01:42.759,0:01:44.820
Kjo eshte 13.

0:01:44.820,0:01:47.728
Kjo eshte njesoj si 13 here 5,

0:01:47.728,0:01:50.665
te dyja jane katrore perfekte dhe

0:01:50.665,0:01:51.733
dhe jane numra prime, s'i thjeshtojme dot.

0:01:51.733,0:01:55.580
Kjo eshte baraz me rrenjen e 65es.

0:01:55.580,0:02:00.721
Tani gjejme funk. trig. per kete kend.

0:02:00.721,0:02:05.124
Ta quajme kendim ketu lart theta.

0:02:05.124,0:02:09.289
Kurdo qe ta beni

0:02:09.289,0:02:12.142
gjithmone ta shkruani, une keshtu e bej,

0:02:12.142,0:02:15.034
"sph knh tpn"

0:02:15.034,0:02:16.523
sph...

0:02:16.523,0:02:19.049
...sph knh tpn. Kam disa kujtime

0:02:19.049,0:02:21.641
nga mesuesja e trigonometrise.

0:02:21.641,0:02:23.665
Do ta kem lexuar diku. S'e di, ndoshta...

0:02:23.665,0:02:25.666
1 princeshe indiane "Soh Kah Toa," s'e di

0:02:25.666,0:02:27.798
por ia vlen ta mbani mend,

0:02:27.798,0:02:30.549
mund te zbatojme "sph knh tpn"

0:02:30.549,0:02:33.233
Ta zeme se duam te gjeme kosinusin.

0:02:33.233,0:02:35.926
Duam te gjejme kosinusin e kendit tone.

0:02:35.926,0:02:38.019
Te gjejme cos e kendit me "sph knh tpn!"

0:02:38.019,0:02:40.452
P.sh., "knh." "Knh" per kosinusin,

0:02:42.000,0:02:42.715
"Knh" na thote

0:02:45.079,0:02:48.239
kosinusi = brinja Ngjitur mbi Hipotenuze

0:02:48.813,0:02:51.740
Cos eshte Ngjitur permbi Hipotenuze (knh)

0:02:52.740,0:02:55.487
Shohim theten; cila eshte brinja ngjitur?

0:02:56.333,0:02:58.455
E dime qe hipotenuza,

0:02:59.117,0:03:02.000
E dime qe hipotenuza eshte kjo brinja ketu

0:03:02.479,0:03:05.438
S'eshte ana tjeter. E vetmja ngjitur qe s'eshte

0:03:06.270,0:03:08.504
hipotenuza, eshte kjo katra.

0:03:09.006,0:03:11.590
Brinja ngjitur ketu, kjo brinje eshte,

0:03:12.133,0:03:14.800
eshte tamam ngjitur me kendin,

0:03:15.507,0:03:18.121
eshte nja nga brinjet qe krijon kendin

0:03:18.449,0:03:20.561
eshte 4 pjestuar me hipotenuze

0:03:21.016,0:03:23.424
Hipotenuza eshte rrenja e 65es.

0:03:24.439,0:03:27.030
Pra, eshte 4 permbi rrenjen e 65es.

0:03:28.091,0:03:32.122
Hiq rrenjen nga emeruesi, dmth

0:03:32.786,0:03:35.454
s'na pelqen te kemi rrenje ne emerues,

0:03:35.454,0:03:37.062
si rrenja e 65es,

0:03:37.537,0:03:40.067
nese do ta shkruash pa rrenje ne emerues,

0:03:40.648,0:03:43.420
shumezo numeruesin me emeruesin,

0:03:43.420,0:03:45.930
pra, me rrenjen e 65es.

0:03:45.930,0:03:48.158
Kjo s'e ndryshon numrin,

0:03:48.158,0:03:50.268
Se shumezon e pjeston me te njejtin numer,

0:03:50.268,0:03:51.239
pra, po shumezojme me 1.

0:03:51.249,0:03:54.158
Numri s'ndryshon, por emeruesi s'ka rrenje

0:03:54.158,0:03:56.770
Numeruesi eshte

0:03:57.048,0:03:58.973
4 here rrenja e 65es.

0:03:59.879,0:04:02.862
dhe numeruesi, rrenja e 65es here vetveten eshte 65.

0:04:02.862,0:04:06.819
S' shpetuam nga rrenja, eshte ende aty, por ne numerues.

0:04:09.158,0:04:11.684
Te bejme funksione te tjera trigonometrike

0:04:11.684,0:04:14.523
ose te pakten nje tjeter funksion baze.

0:04:14.523,0:04:16.694
Do te mesojme me vone qe ka shume prej tyre

0:04:16.694,0:04:17.374
por te gjitha vijne nga keto.

0:04:17.374,0:04:20.187
C'shenje ka theta. "Sph knh tpn."

0:04:25.832,0:04:30.592
"Spn" per sin. Sin = brinja perballe/hipotenuze

0:04:30.592,0:04:33.122
Sinusi = brinja perballe mbi hipotenuze.

0:04:33.122,0:04:35.139
Sinusi = brinja perballe mbi hipotenuze.

0:04:35.139,0:04:37.648
Per kete kend cila eshte brinja perballe?

0:04:37.648,0:04:40.407
Shohim perballe, eshte perballe shtates

0:04:41.147,0:04:43.407
Shtata eshte brinja perballe.

0:04:44.068,0:04:46.739
Ja, kjo, kjo eshte brinja perballe

0:04:47.147,0:04:49.799
dhe pastaj hipotenuza, perballe mbi h.

0:04:49.799,0:04:51.908
Hipotenuza eshte rrenja e 65es.

0:04:51.908,0:04:53.607
Rrenja e 65es

0:04:53.607,0:04:56.147
dhe nese do te heqim rrenjen nga emeruesi

0:04:56.538,0:04:58.491
shumezojme me rrenjen e 65 /rrenjen e 65

0:04:59.560,0:05:02.800
dhe tek numeruesi del 7 here rrenja e 65es

0:05:03.555,0:05:06.439
dhe emeruesi del 65 perseri.

0:05:07.181,0:05:09.866
Tani te gjejme tangentin!

0:05:10.719,0:05:12.875
Te gjejme tangentin.

0:05:14.091,0:05:15.609
Nese ju pyes per tangentin

0:05:16.499,0:05:17.777
e kendit theta

0:05:18.267,0:05:20.271
kthehemi tek "sph knh tpn."

0:05:21.103,0:05:23.331
"Tpn" na tregon per tangentin

0:05:24.066,0:05:25.365
na tregon...

0:05:25.408,0:05:28.135
qe tangenti

0:05:28.135,0:05:31.587
eshte brinja perballe mbi ate ngjitur

0:05:31.587,0:05:33.786
eshte brinja perballe permbi

0:05:33.786,0:05:36.244
brinja perballe permbi ate ngjitur

0:05:36.718,0:05:39.062
Per kete kend, cila eshte brinja perballe?

0:05:40.030,0:05:41.998
Eshte 7. Eshte 7.

0:05:43.240,0:05:46.485
Kendi eshte perballe 7es.

0:05:47.532,0:05:49.695
Pra, 7 permbi brinjen ngjitur.

0:05:49.808,0:05:51.812
Kjo katra eshte ajo ngjitur.

0:05:51.892,0:05:53.772
Katra eshte ngjitur. Brinja ngjitur 4.

0:05:53.968,0:05:56.030
Pra, 7 permbi 4

0:05:56.030,0:05:56.944
dhe mbaruam.

0:05:56.944,0:05:59.635
I gjetem funk. trig. te thetes. Nje tjeter

0:06:02.770,0:06:04.623
Bejme nje tjeter.

0:06:04.648,0:06:05.452
Ta bej me konkrete, se tani vetem thame:

0:06:06.314,0:06:08.270
"Cili eshte tg i x apo thetes?" Me konkret

0:06:08.270,0:06:09.650
Te themi...

0:06:10.683,0:06:13.362
te vizatojme nje trekendesh kenddrejte,

0:06:14.207,0:06:15.999
nje tjeter kenddrejte.

0:06:15.999,0:06:17.371
Do te kemi te bejme vetem me kenddrejte.

0:06:17.885,0:06:20.216
Ta zeme hipotenuza eshte 4

0:06:20.811,0:06:25.529
dhe kjo brinja eshte 2

0:06:25.529,0:06:29.720
dhe kjo brinja eshte 2 here rrenja e 3.

0:06:29.720,0:06:32.561
Ta verifikojme.

0:06:33.265,0:06:36.306
Nqs kjo brinja ngrihet ne katror, kemi...

0:06:36.306,0:06:40.152
2 here rrenja e 3shit e gjitha ne katror

0:06:40.601,0:06:43.838
plus 2 ne katror baraz me cfare?

0:06:44.584,0:06:48.239
Kjo eshte 2. 4 here 3.

0:06:48.854,0:06:51.015
kater here tre plus kater

0:06:51.015,0:06:53.970
dhe kjo do te jete 12 plus 4 baraz me 16

0:06:54.573,0:06:57.092
e 16 = 4 ne katror. Kjo eshte 4 ne katror

0:06:57.758,0:07:01.177
eshte 4 ne katror. Sipas Pitagores

0:07:01.588,0:07:06.582
dhe i kujtoni trekendeshat 30, 60, 90

0:07:07.039,0:07:09.210
qe mund te keni mesuar ne gjeometri,

0:07:09.770,0:07:12.738
shihni qe ky eshte trekendesh 30, 60, 90.

0:07:12.738,0:07:15.907
Ky trekendeshi ketu,

0:07:15.907,0:07:17.831
duhet ta kisha vizatuar si kenddrejte

0:07:17.831,0:07:21.408
ky eshte kendi yne 30 grade

0:07:22.454,0:07:25.800
dhe ky ketu, ky kendi ketu eshte

0:07:25.800,0:07:27.708
kend 60 grade

0:07:28.410,0:07:30.546
dhe eshte 30, 60, 90, sepse

0:07:30.546,0:07:33.714
brinja perballe 30 eshte 1/2 e hipotenuzes

0:07:33.714,0:07:35.864
brinja perballe 60 eshte rrenja e 3 here tjetren

0:07:37.178,0:07:39.121
Kjo s'eshte hipotenuza.

0:07:39.121,0:07:40.770
Ne nuk do te...

0:07:40.770,0:07:43.739
Qellimi s'eshte 30, 60, 90 dhe pse e permenda.

0:07:43.739,0:07:46.530
Gjejme funksionet trig. per kende te tjera

0:07:46.530,0:07:48.517
Nese dikush ju pyet...

0:07:49.149,0:07:52.647
cili eshte sinusi i 30 gradeshit?

0:07:53.407,0:07:57.365
Kujtoni 30 eshte kend i ketij trekendeshi

0:07:57.407,0:08:00.500
Por kurdo keni kend 30 dhe rast kenddrejte

0:08:00.998,0:08:03.453
Te flasim per perkufizime, por sin. i 30

0:08:04.335,0:08:08.716
ky eshte 30, qe te perdor kenddrejtin

0:08:09.474,0:08:12.757
dhe te mbajme mend "sph knh tpn"

0:08:13.147,0:08:16.210
Ta rishkruajme "sph, knh, tpn."

0:08:16.800,0:08:21.789
"sinusi na thote" (jo) sph = perballe/hipo

0:08:22.148,0:08:25.331
sinusi i 30 eshte kendi perballe,

0:08:25.331,0:08:29.754
kendi perballe, pra, 2 permbi hipotenuze

0:08:29.754,0:08:32.122
Hipotenuza eshte 4 ketu.

0:08:32.122,0:08:35.438
Eshte 2/4, pra 1/2.

0:08:35.438,0:08:38.592
Sinusi i 30 gradeve do e shihni = 1/2.

0:08:38.592,0:08:43.363
Po kosinusi?

0:08:43.363,0:08:46.838
Cili eshte kosinusi i 30 gradeve?

0:08:46.838,0:08:50.963
Kthehemi tek "sph knh tpn."

0:08:52.124,0:08:54.182
"Knh" na ndihmon per kosinusin.

0:08:54.648,0:08:57.302
Kosinusi = brinja ngjitur/hipotenuze.

0:08:57.648,0:09:00.438
Per kendin 30 gradet kemi brinjen ngjitur.

0:09:00.731,0:09:03.392
Kjo eshte brinja ngjitur. Afer kendit.

0:09:03.960,0:09:06.924
S'eshte hipotenuza, por prane hipotenuzes.

0:09:06.924,0:09:09.121
Eshte 2 here rrenja e treshit

0:09:09.121,0:09:12.739
brinja ngjitur permbi hipotenuzen, permbi

0:09:12.739,0:09:16.647
Pjestojme numeruesin dhe emeruesin me 2

0:09:16.647,0:09:19.842
rrenja e 2shit permbi 2.

0:09:19.842,0:09:22.518
Tani, bejme tangentin.

0:09:23.259,0:09:26.608
Tangenti i 30 gradeve,

0:09:26.608,0:09:29.907
kthehemi tek "sph knh tpn."

0:09:29.907,0:09:31.618
"sph knh tpn"

0:09:31.618,0:09:35.532
"tpn" per tg. Brinja perballe asaj ngjitur

0:09:36.446,0:09:38.811
tg. i 30 gradeve, se ajo na duhet, tg i 30

0:09:38.811,0:09:41.530
tg i 30. Brinja perballe eshte 2

0:09:41.530,0:09:45.863
perballe ka 2 dhe ngjitur 2 rrenja e 3

0:09:45.863,0:09:48.331
Ngjitur. Eshte ngjitur me te.

0:09:48.331,0:09:50.424
Ngjitur, pra, brinja afer.

0:09:50.424,0:09:52.501
2 here rrenja e 3shit

0:09:52.501,0:09:54.438
kjo eshte...dyshat ikin

0:09:54.438,0:09:56.709
1 mbi rrenjen katrore te 3shit

0:09:57.494,0:10:00.302
shumezim numerues dhe emerues me rrenj. 3

0:10:01.240,0:10:04.333
Kemi rrenjen e 3shit mbi rrenjen e 3shit

0:10:04.333,0:10:07.685
kjo eshte sa numeruesi rrenja e treshit

0:10:07.908,0:10:11.869
emeruesi del 3

0:10:11.907,0:10:15.648
Pra, e beme racional me rrenj. 3 / rrenj.3

0:10:15.648,0:10:15.898
Mire.

0:10:17.606,0:10:20.647
Perdorim kete trekendesh dhe per 60 grade,

0:10:20.647,0:10:22.999
meqe e vizatuam.

0:10:22.999,0:10:26.469
sa...sa eshte kosinusi i 60 gradeve?

0:10:26.469,0:10:30.147
Dhe shpresoj ta dini tani.

0:10:30.147,0:10:33.031
Sinusi eshte perballe / hipotenuze. "Sph"

0:10:33.708,0:10:37.091
Per 60 grade, cila eshte brinja perballe?

0:10:37.091,0:10:40.498
Cili kend ka perballe 2 rrenja e 3shit?

0:10:40.498,0:10:43.739
Pra, brinja perballe eshte 2 rrenja e 3

0:10:43.739,0:10:45.680
dhe nga kendi 60 grade mbi ate ngjitur..jo

0:10:46.487,0:10:49.300
mbi hipotenuzen, s'dua t'ju ngaterroj.

0:10:49.300,0:10:51.517
Pra, mbi hipotenuze

0:10:51.517,0:10:54.147
2 rrenja e 3 mbi 4. 4 per hipotenuzen.

0:10:54.546,0:11:01.487
Pra, behet rrenja e 3 permbi 2.

0:11:01.487,0:11:04.710
Sa eshte cos i 60 gradeve? Cos i 60

0:11:04.710,0:11:09.239
"Sph tnh tpn." Cos = ngjit./hipotenuze

0:11:09.239,0:11:13.130
ngjitur jane 2 brinjet, ngjitur me 60 .

0:11:13.147,0:11:16.907
Pra, eshte 2 mbi hipotenuzen, qe eshte 4.

0:11:16.907,0:11:19.895
Kjo eshte 1/2

0:11:19.895,0:11:22.831
dhe ne fund, sa eshte tangenti?

0:11:22.831,0:11:26.924
Tg i 60 gradeve?

0:11:26.924,0:11:31.332
"Sph knh tpn." Tg eshte perballe/ngjitur

0:11:31.332,0:11:35.424
perballe 60 gradeve

0:11:35.424,0:11:37.739
eshte 2 rrenja e 3shit

0:11:37.739,0:11:39.908
2 rrenja e 3shit

0:11:39.908,0:11:41.862
dhe ngjitur me te

0:11:41.862,0:11:43.468
ngjitur me te eshte 2shi.

0:11:43.887,0:11:46.547
Ngjitur 60 gradeve eshte 2shi.

0:11:46.547,0:11:49.709
Perballe/ngjitur, 2 rrenja e 2shit mbi 2

0:11:49.939,0:11:52.301
qe eshte rrenja e 3shit.

0:11:52.770,0:11:54.956
Doja te ... si lidhen keto?

0:11:54.956,0:11:58.061
Sinusi i 30 eshte sa kosinusi i 60 gradeve

0:11:58.061,0:12:00.769
Cos i 30 eshte sa sinusi i 60

0:12:00.769,0:12:03.801
pra, jane te anasjellet e njeri-tjetrit

0:12:04.147,0:12:06.711
dhe nese mendoni pak per kete trekendesh

0:12:06.711,0:12:08.723
do ta kuptoni pse.

0:12:08.723,0:12:10.344
Do te vazhdojme mbi kete teme dhe

0:12:10.344,0:12:11.284
do te bejme me praktike ne videot e tjera.