1
00:00:00,800 --> 00:00:03,017
Le te bejme me shume shembuj

2
00:00:03,017 --> 00:00:07,036
vetem per t'u siguruar qe po ecim mire

3
00:00:07,036 --> 00:00:11,447
Te ndertojme ca trekendesha kenddrejte

4
00:00:11,716 --> 00:00:12,550
Te ndertojme ca trekendesha kenddrejte

5
00:00:13,668 --> 00:00:15,186
dhe dua te jem shume i qarte.

6
00:00:16,049 --> 00:00:18,732
Ajo qe kam thene vlen vetem per kenddrejte

7
00:00:20,279 --> 00:00:23,338
Ne raste funksionesh trig. te kendeve jo ne trekendesha kenddrejte

8
00:00:24,601 --> 00:00:27,646
duhet te ndertosh trekendesha kenddrejte

9
00:00:28,080 --> 00:00:28,953
Por te perqendrohemi tek kenddrejtet tani

10
00:00:29,439 --> 00:00:32,113
Pra, kemi nje trekendesh,

11
00:00:32,286 --> 00:00:35,128
le te themi gjatesia ketu eshte shtate

12
00:00:36,308 --> 00:00:38,749
dhe gjatesia ne kete ane ketu lart

13
00:00:38,749 --> 00:00:39,808
eshte, te themi, kater.

14
00:00:39,908 --> 00:00:41,729
Le te gjejme sa do te jete hipotenuza.

15
00:00:43,579 --> 00:00:45,641
Le ta quajme hipotenuzen "h,"

16
00:00:45,641 --> 00:00:51,765
h ne katror = 7 ne katror dhe 4 ne katror,

17
00:00:51,765 --> 00:00:55,610
e dime nga teorema e Pitagores,

18
00:00:55,610 --> 00:00:58,672
qe hipotenuza ne katror

19
00:00:58,672 --> 00:01:01,513
eshte sa shuma e katroreve te dy brinjeve.

20
00:01:01,513 --> 00:01:03,666
h ne katror =7 ne katror plus 4 ne katror.

21
00:01:03,666 --> 00:01:08,507
Pra, kjo eshte 49 plus 16,

22
00:01:08,507 --> 00:01:11,640
49 plus 16,

23
00:01:11,640 --> 00:01:17,017
49 plus 10 eshte 59, plus 6 eshte 65.

24
00:01:17,017 --> 00:01:23,165
Eshte 65. Pra, h ne katror,

25
00:01:23,165 --> 00:01:26,486
ta shkruaj: h ne katror, tjeter e verdhe,

26
00:01:26,486 --> 00:01:29,259
pra, kemi h ne katror baraz me 65.

27
00:01:29,259 --> 00:01:33,106
E bera mire? 49 plus 10 = 59, plus 6 = 65,

28
00:01:33,106 --> 00:01:37,487
ose h eshte =, me rrenjet e te 2 aneve,

29
00:01:37,487 --> 00:01:39,856
rrenja katrore

30
00:01:39,856 --> 00:01:42,759
rrenja e 65. Dhe s'mund ta thjeshtojme.

31
00:01:42,759 --> 00:01:44,820
Kjo eshte 13.

32
00:01:44,820 --> 00:01:47,728
Kjo eshte njesoj si 13 here 5,

33
00:01:47,728 --> 00:01:50,665
te dyja jane katrore perfekte dhe

34
00:01:50,665 --> 00:01:51,733
dhe jane numra prime, s'i thjeshtojme dot.

35
00:01:51,733 --> 00:01:55,580
Kjo eshte baraz me rrenjen e 65es.

36
00:01:55,580 --> 00:02:00,721
Tani gjejme funk. trig. per kete kend.

37
00:02:00,721 --> 00:02:05,124
Ta quajme kendim ketu lart theta.

38
00:02:05,124 --> 00:02:09,289
Kurdo qe ta beni

39
00:02:09,289 --> 00:02:12,142
gjithmone ta shkruani, une keshtu e bej,

40
00:02:12,142 --> 00:02:15,034
"sph knh tpn"

41
00:02:15,034 --> 00:02:16,523
sph...

42
00:02:16,523 --> 00:02:19,049
...sph knh tpn. Kam disa kujtime

43
00:02:19,049 --> 00:02:21,641
nga mesuesja e trigonometrise.

44
00:02:21,641 --> 00:02:23,665
Do ta kem lexuar diku. S'e di, ndoshta...

45
00:02:23,665 --> 00:02:25,666
1 princeshe indiane "Soh Kah Toa," s'e di

46
00:02:25,666 --> 00:02:27,798
por ia vlen ta mbani mend,

47
00:02:27,798 --> 00:02:30,549
mund te zbatojme "sph knh tpn"

48
00:02:30,549 --> 00:02:33,233
Ta zeme se duam te gjeme kosinusin.

49
00:02:33,233 --> 00:02:35,926
Duam te gjejme kosinusin e kendit tone.

50
00:02:35,926 --> 00:02:38,019
Te gjejme cos e kendit me "sph knh tpn!"

51
00:02:38,019 --> 00:02:40,452
P.sh., "knh." "Knh" per kosinusin,

52
00:02:42,000 --> 00:02:42,715
"Knh" na thote

53
00:02:45,079 --> 00:02:48,239
kosinusi = brinja Ngjitur mbi Hipotenuze

54
00:02:48,813 --> 00:02:51,740
Cos eshte Ngjitur permbi Hipotenuze (knh)

55
00:02:52,740 --> 00:02:55,487
Shohim theten; cila eshte brinja ngjitur?

56
00:02:56,333 --> 00:02:58,455
E dime qe hipotenuza,

57
00:02:59,117 --> 00:03:02,000
E dime qe hipotenuza eshte kjo brinja ketu

58
00:03:02,479 --> 00:03:05,438
S'eshte ana tjeter. E vetmja ngjitur qe s'eshte

59
00:03:06,270 --> 00:03:08,504
hipotenuza, eshte kjo katra.

60
00:03:09,006 --> 00:03:11,590
Brinja ngjitur ketu, kjo brinje eshte,

61
00:03:12,133 --> 00:03:14,800
eshte tamam ngjitur me kendin,

62
00:03:15,507 --> 00:03:18,121
eshte nja nga brinjet qe krijon kendin

63
00:03:18,449 --> 00:03:20,561
eshte 4 pjestuar me hipotenuze

64
00:03:21,016 --> 00:03:23,424
Hipotenuza eshte rrenja e 65es.

65
00:03:24,439 --> 00:03:27,030
Pra, eshte 4 permbi rrenjen e 65es.

66
00:03:28,091 --> 00:03:32,122
Hiq rrenjen nga emeruesi, dmth

67
00:03:32,786 --> 00:03:35,454
s'na pelqen te kemi rrenje ne emerues,

68
00:03:35,454 --> 00:03:37,062
si rrenja e 65es,

69
00:03:37,537 --> 00:03:40,067
nese do ta shkruash pa rrenje ne emerues,

70
00:03:40,648 --> 00:03:43,420
shumezo numeruesin me emeruesin,

71
00:03:43,420 --> 00:03:45,930
pra, me rrenjen e 65es.

72
00:03:45,930 --> 00:03:48,158
Kjo s'e ndryshon numrin,

73
00:03:48,158 --> 00:03:50,268
Se shumezon e pjeston me te njejtin numer,

74
00:03:50,268 --> 00:03:51,239
pra, po shumezojme me 1.

75
00:03:51,249 --> 00:03:54,158
Numri s'ndryshon, por emeruesi s'ka rrenje

76
00:03:54,158 --> 00:03:56,770
Numeruesi eshte

77
00:03:57,048 --> 00:03:58,973
4 here rrenja e 65es.

78
00:03:59,879 --> 00:04:02,862
dhe numeruesi, rrenja e 65es here vetveten eshte 65.

79
00:04:02,862 --> 00:04:06,819
S' shpetuam nga rrenja, eshte ende aty, por ne numerues.

80
00:04:09,158 --> 00:04:11,684
Te bejme funksione te tjera trigonometrike

81
00:04:11,684 --> 00:04:14,523
ose te pakten nje tjeter funksion baze.

82
00:04:14,523 --> 00:04:16,694
Do te mesojme me vone qe ka shume prej tyre

83
00:04:16,694 --> 00:04:17,374
por te gjitha vijne nga keto.

84
00:04:17,374 --> 00:04:20,187
C'shenje ka theta. "Sph knh tpn."

85
00:04:25,832 --> 00:04:30,592
"Spn" per sin. Sin = brinja perballe/hipotenuze

86
00:04:30,592 --> 00:04:33,122
Sinusi = brinja perballe mbi hipotenuze.

87
00:04:33,122 --> 00:04:35,139
Sinusi = brinja perballe mbi hipotenuze.

88
00:04:35,139 --> 00:04:37,648
Per kete kend cila eshte brinja perballe?

89
00:04:37,648 --> 00:04:40,407
Shohim perballe, eshte perballe shtates

90
00:04:41,147 --> 00:04:43,407
Shtata eshte brinja perballe.

91
00:04:44,068 --> 00:04:46,739
Ja, kjo, kjo eshte brinja perballe

92
00:04:47,147 --> 00:04:49,799
dhe pastaj hipotenuza, perballe mbi h.

93
00:04:49,799 --> 00:04:51,908
Hipotenuza eshte rrenja e 65es.

94
00:04:51,908 --> 00:04:53,607
Rrenja e 65es

95
00:04:53,607 --> 00:04:56,147
dhe nese do te heqim rrenjen nga emeruesi

96
00:04:56,538 --> 00:04:58,491
shumezojme me rrenjen e 65 /rrenjen e 65

97
00:04:59,560 --> 00:05:02,800
dhe tek numeruesi del 7 here rrenja e 65es

98
00:05:03,555 --> 00:05:06,439
dhe emeruesi del 65 perseri.

99
00:05:07,181 --> 00:05:09,866
Tani te gjejme tangentin!

100
00:05:10,719 --> 00:05:12,875
Te gjejme tangentin.

101
00:05:14,091 --> 00:05:15,609
Nese ju pyes per tangentin

102
00:05:16,499 --> 00:05:17,777
e kendit theta

103
00:05:18,267 --> 00:05:20,271
kthehemi tek "sph knh tpn."

104
00:05:21,103 --> 00:05:23,331
"Tpn" na tregon per tangentin

105
00:05:24,066 --> 00:05:25,365
na tregon...

106
00:05:25,408 --> 00:05:28,135
qe tangenti

107
00:05:28,135 --> 00:05:31,587
eshte brinja perballe mbi ate ngjitur

108
00:05:31,587 --> 00:05:33,786
eshte brinja perballe permbi

109
00:05:33,786 --> 00:05:36,244
brinja perballe permbi ate ngjitur

110
00:05:36,718 --> 00:05:39,062
Per kete kend, cila eshte brinja perballe?

111
00:05:40,030 --> 00:05:41,998
Eshte 7. Eshte 7.

112
00:05:43,240 --> 00:05:46,485
Kendi eshte perballe 7es.

113
00:05:47,532 --> 00:05:49,695
Pra, 7 permbi brinjen ngjitur.

114
00:05:49,808 --> 00:05:51,812
Kjo katra eshte ajo ngjitur.

115
00:05:51,892 --> 00:05:53,772
Katra eshte ngjitur. Brinja ngjitur 4.

116
00:05:53,968 --> 00:05:56,030
Pra, 7 permbi 4

117
00:05:56,030 --> 00:05:56,944
dhe mbaruam.

118
00:05:56,944 --> 00:05:59,635
I gjetem funk. trig. te thetes. Nje tjeter

119
00:06:02,770 --> 00:06:04,623
Bejme nje tjeter.

120
00:06:04,648 --> 00:06:05,452
Ta bej me konkrete, se tani vetem thame:

121
00:06:06,314 --> 00:06:08,270
"Cili eshte tg i x apo thetes?" Me konkret

122
00:06:08,270 --> 00:06:09,650
Te themi...

123
00:06:10,683 --> 00:06:13,362
te vizatojme nje trekendesh kenddrejte,

124
00:06:14,207 --> 00:06:15,999
nje tjeter kenddrejte.

125
00:06:15,999 --> 00:06:17,371
Do te kemi te bejme vetem me kenddrejte.

126
00:06:17,885 --> 00:06:20,216
Ta zeme hipotenuza eshte 4

127
00:06:20,811 --> 00:06:25,529
dhe kjo brinja eshte 2

128
00:06:25,529 --> 00:06:29,720
dhe kjo brinja eshte 2 here rrenja e 3.

129
00:06:29,720 --> 00:06:32,561
Ta verifikojme.

130
00:06:33,265 --> 00:06:36,306
Nqs kjo brinja ngrihet ne katror, kemi...

131
00:06:36,306 --> 00:06:40,152
2 here rrenja e 3shit e gjitha ne katror

132
00:06:40,601 --> 00:06:43,838
plus 2 ne katror baraz me cfare?

133
00:06:44,584 --> 00:06:48,239
Kjo eshte 2. 4 here 3.

134
00:06:48,854 --> 00:06:51,015
kater here tre plus kater

135
00:06:51,015 --> 00:06:53,970
dhe kjo do te jete 12 plus 4 baraz me 16

136
00:06:54,573 --> 00:06:57,092
e 16 = 4 ne katror. Kjo eshte 4 ne katror

137
00:06:57,758 --> 00:07:01,177
eshte 4 ne katror. Sipas Pitagores

138
00:07:01,588 --> 00:07:06,582
dhe i kujtoni trekendeshat 30, 60, 90

139
00:07:07,039 --> 00:07:09,210
qe mund te keni mesuar ne gjeometri,

140
00:07:09,770 --> 00:07:12,738
shihni qe ky eshte trekendesh 30, 60, 90.

141
00:07:12,738 --> 00:07:15,907
Ky trekendeshi ketu,

142
00:07:15,907 --> 00:07:17,831
duhet ta kisha vizatuar si kenddrejte

143
00:07:17,831 --> 00:07:21,408
ky eshte kendi yne 30 grade

144
00:07:22,454 --> 00:07:25,800
dhe ky ketu, ky kendi ketu eshte

145
00:07:25,800 --> 00:07:27,708
kend 60 grade

146
00:07:28,410 --> 00:07:30,546
dhe eshte 30, 60, 90, sepse

147
00:07:30,546 --> 00:07:33,714
brinja perballe 30 eshte 1/2 e hipotenuzes

148
00:07:33,714 --> 00:07:35,864
brinja perballe 60 eshte rrenja e 3 here tjetren

149
00:07:37,178 --> 00:07:39,121
Kjo s'eshte hipotenuza.

150
00:07:39,121 --> 00:07:40,770
Ne nuk do te...

151
00:07:40,770 --> 00:07:43,739
Qellimi s'eshte 30, 60, 90 dhe pse e permenda.

152
00:07:43,739 --> 00:07:46,530
Gjejme funksionet trig. per kende te tjera

153
00:07:46,530 --> 00:07:48,517
Nese dikush ju pyet...

154
00:07:49,149 --> 00:07:52,647
cili eshte sinusi i 30 gradeshit?

155
00:07:53,407 --> 00:07:57,365
Kujtoni 30 eshte kend i ketij trekendeshi

156
00:07:57,407 --> 00:08:00,500
Por kurdo keni kend 30 dhe rast kenddrejte

157
00:08:00,998 --> 00:08:03,453
Te flasim per perkufizime, por sin. i 30

158
00:08:04,335 --> 00:08:08,716
ky eshte 30, qe te perdor kenddrejtin

159
00:08:09,474 --> 00:08:12,757
dhe te mbajme mend "sph knh tpn"

160
00:08:13,147 --> 00:08:16,210
Ta rishkruajme "sph, knh, tpn."

161
00:08:16,800 --> 00:08:21,789
"sinusi na thote" (jo) sph = perballe/hipo

162
00:08:22,148 --> 00:08:25,331
sinusi i 30 eshte kendi perballe,

163
00:08:25,331 --> 00:08:29,754
kendi perballe, pra, 2 permbi hipotenuze

164
00:08:29,754 --> 00:08:32,122
Hipotenuza eshte 4 ketu.

165
00:08:32,122 --> 00:08:35,438
Eshte 2/4, pra 1/2.

166
00:08:35,438 --> 00:08:38,592
Sinusi i 30 gradeve do e shihni = 1/2.

167
00:08:38,592 --> 00:08:43,363
Po kosinusi?

168
00:08:43,363 --> 00:08:46,838
Cili eshte kosinusi i 30 gradeve?

169
00:08:46,838 --> 00:08:50,963
Kthehemi tek "sph knh tpn."

170
00:08:52,124 --> 00:08:54,182
"Knh" na ndihmon per kosinusin.

171
00:08:54,648 --> 00:08:57,302
Kosinusi = brinja ngjitur/hipotenuze.

172
00:08:57,648 --> 00:09:00,438
Per kendin 30 gradet kemi brinjen ngjitur.

173
00:09:00,731 --> 00:09:03,392
Kjo eshte brinja ngjitur. Afer kendit.

174
00:09:03,960 --> 00:09:06,924
S'eshte hipotenuza, por prane hipotenuzes.

175
00:09:06,924 --> 00:09:09,121
Eshte 2 here rrenja e treshit

176
00:09:09,121 --> 00:09:12,739
brinja ngjitur permbi hipotenuzen, permbi

177
00:09:12,739 --> 00:09:16,647
Pjestojme numeruesin dhe emeruesin me 2

178
00:09:16,647 --> 00:09:19,842
rrenja e 2shit permbi 2.

179
00:09:19,842 --> 00:09:22,518
Tani, bejme tangentin.

180
00:09:23,259 --> 00:09:26,608
Tangenti i 30 gradeve,

181
00:09:26,608 --> 00:09:29,907
kthehemi tek "sph knh tpn."

182
00:09:29,907 --> 00:09:31,618
"sph knh tpn"

183
00:09:31,618 --> 00:09:35,532
"tpn" per tg. Brinja perballe asaj ngjitur

184
00:09:36,446 --> 00:09:38,811
tg. i 30 gradeve, se ajo na duhet, tg i 30

185
00:09:38,811 --> 00:09:41,530
tg i 30. Brinja perballe eshte 2

186
00:09:41,530 --> 00:09:45,863
perballe ka 2 dhe ngjitur 2 rrenja e 3

187
00:09:45,863 --> 00:09:48,331
Ngjitur. Eshte ngjitur me te.

188
00:09:48,331 --> 00:09:50,424
Ngjitur, pra, brinja afer.

189
00:09:50,424 --> 00:09:52,501
2 here rrenja e 3shit

190
00:09:52,501 --> 00:09:54,438
kjo eshte...dyshat ikin

191
00:09:54,438 --> 00:09:56,709
1 mbi rrenjen katrore te 3shit

192
00:09:57,494 --> 00:10:00,302
shumezim numerues dhe emerues me rrenj. 3

193
00:10:01,240 --> 00:10:04,333
Kemi rrenjen e 3shit mbi rrenjen e 3shit

194
00:10:04,333 --> 00:10:07,685
kjo eshte sa numeruesi rrenja e treshit

195
00:10:07,908 --> 00:10:11,869
emeruesi del 3

196
00:10:11,907 --> 00:10:15,648
Pra, e beme racional me rrenj. 3 / rrenj.3

197
00:10:15,648 --> 00:10:15,898
Mire.

198
00:10:17,606 --> 00:10:20,647
Perdorim kete trekendesh dhe per 60 grade,

199
00:10:20,647 --> 00:10:22,999
meqe e vizatuam.

200
00:10:22,999 --> 00:10:26,469
sa...sa eshte kosinusi i 60 gradeve?

201
00:10:26,469 --> 00:10:30,147
Dhe shpresoj ta dini tani.

202
00:10:30,147 --> 00:10:33,031
Sinusi eshte perballe / hipotenuze. "Sph"

203
00:10:33,708 --> 00:10:37,091
Per 60 grade, cila eshte brinja perballe?

204
00:10:37,091 --> 00:10:40,498
Cili kend ka perballe 2 rrenja e 3shit?

205
00:10:40,498 --> 00:10:43,739
Pra, brinja perballe eshte 2 rrenja e 3

206
00:10:43,739 --> 00:10:45,680
dhe nga kendi 60 grade mbi ate ngjitur..jo

207
00:10:46,487 --> 00:10:49,300
mbi hipotenuzen, s'dua t'ju ngaterroj.

208
00:10:49,300 --> 00:10:51,517
Pra, mbi hipotenuze

209
00:10:51,517 --> 00:10:54,147
2 rrenja e 3 mbi 4. 4 per hipotenuzen.

210
00:10:54,546 --> 00:11:01,487
Pra, behet rrenja e 3 permbi 2.

211
00:11:01,487 --> 00:11:04,710
Sa eshte cos i 60 gradeve? Cos i 60

212
00:11:04,710 --> 00:11:09,239
"Sph tnh tpn." Cos = ngjit./hipotenuze

213
00:11:09,239 --> 00:11:13,130
ngjitur jane 2 brinjet, ngjitur me 60 .

214
00:11:13,147 --> 00:11:16,907
Pra, eshte 2 mbi hipotenuzen, qe eshte 4.

215
00:11:16,907 --> 00:11:19,895
Kjo eshte 1/2

216
00:11:19,895 --> 00:11:22,831
dhe ne fund, sa eshte tangenti?

217
00:11:22,831 --> 00:11:26,924
Tg i 60 gradeve?

218
00:11:26,924 --> 00:11:31,332
"Sph knh tpn." Tg eshte perballe/ngjitur

219
00:11:31,332 --> 00:11:35,424
perballe 60 gradeve

220
00:11:35,424 --> 00:11:37,739
eshte 2 rrenja e 3shit

221
00:11:37,739 --> 00:11:39,908
2 rrenja e 3shit

222
00:11:39,908 --> 00:11:41,862
dhe ngjitur me te

223
00:11:41,862 --> 00:11:43,468
ngjitur me te eshte 2shi.

224
00:11:43,887 --> 00:11:46,547
Ngjitur 60 gradeve eshte 2shi.

225
00:11:46,547 --> 00:11:49,709
Perballe/ngjitur, 2 rrenja e 2shit mbi 2

226
00:11:49,939 --> 00:11:52,301
qe eshte rrenja e 3shit.

227
00:11:52,770 --> 00:11:54,956
Doja te ... si lidhen keto?

228
00:11:54,956 --> 00:11:58,061
Sinusi i 30 eshte sa kosinusi i 60 gradeve

229
00:11:58,061 --> 00:12:00,769
Cos i 30 eshte sa sinusi i 60

230
00:12:00,769 --> 00:12:03,801
pra, jane te anasjellet e njeri-tjetrit

231
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dhe nese mendoni pak per kete trekendesh

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do ta kuptoni pse.

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Do te vazhdojme mbi kete teme dhe

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do te bejme me praktike ne videot e tjera.