Sinus va kosinuslar teoremalari
![Sinus va kosinuslar
teoremalari](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_1.png)
![FRANSUA VIY ET (1540 -1603)
Vi уе t t rigonomet riy ani
y arat ilishida k at t a hissa
qo'shgan. Ko'pgina
t rigonomet rik formulalar
birinchi mart a Viy et
t omonidan y ozilgan. 1593
y ilda u k osinus t eoremasini
og'zak i shak lda birinchi
bo'lib y arat di
Kosinus - bu lot in t ilidagi ex presssinus
ifodasining qisqarishi, y a'ni " k omplement ar
sinus" (y ok i boshqa y o'l bilan " t o'ldiruv chi
y oy ning sinusi" ; cosa = sin (90 ° - a)).](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_2.png)
![Sin x v a cos x belgilari
bilan zamonav iy sinus
v a k osinus y ozuv lari
birinchi mart a 1739 y ilda
Bernulli t omonidan
Sank t -Pet erburg mat emat igi L. Ey lerga y ozilgan
xat da k irit ilgan. Ushbu belgilar juda qulay
degan xulosaga k elib, ularni mat emat ik
ishlarida ishlat a boshladi. Bundan t ashqari,
Ey ler x burchak ning t rigonomet rik
funk t siy alari uchun quy idagi qisqart malarni
k irit adi: t ang x, cot x, sec x, cosecx.](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_3.png)
![Uchburchak y uzi formulasi
Uchburchak ning y uzi uning
ik k i t omoni v a ik k i t omoni
orasidagi burchak ning sinusi
y armiga t eng.
CBCACBBCABS sin
21
sin
21
BAC
CAB CACBAB
sinsin sinsin
A AC AB sin
2
1
AВС
sin
А В
С](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_4.png)
![Sinuslar t eoremasi
•
Uchburchak ning t omonlari qarama-
qarshi burchak larning sinuslariga
proporsionaldir
M FN
NMF
MNF
FMN
sinsinsin BAC
CAB
sinsin A
ВС
sin
А В
С
MN F uchburchak uchun
sinuslar t eoremasini y ozing](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_5.png)
![Uchburchaklarning sinusini toping
•
АВС
•
KLM
•
PQHB
АС
A
ВС
C
АВ
sin sin sin
K
LM
L
KM
M
KL
sin sin sin
P
QH
Q
PH
H
PQ
sin sin sin
](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_6.png)
![Izoh
Uchburchak t omonining qarama-
qarshi burchak sinusiga nisbat i
ay lana doirasining diamet riga
t engD R
B
АС
A
ВС
C
АВ
2
sin sin sin](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_7.png)
![R
AВC
2
sin A
B
C
1
A
O
a](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_8.png)
![Kosinuslar t eoremasi
•
Uchburchak istalgan tomoniningkvadrati
qolgan ikki tomoni kvadratlari yig’indisidan
shu ikki tomon bilan ular orasidagi burchak
kosinusi ko’paytmasining ikkilangani
ayirmasiga teng
M FNF FN MF FN MF MN cos 2
222](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_9.png)
![у
х(0;0) ( с;0 )( bcos A ; bsin A )
А С
Вb
c a
AbcAbaBC 222
22
sincos
22222
cos2sincos cAbcAbAb
2 2 2 2 cos2sincos cAbcAAb
A bc c b cos 2
2 2 Д ано:
ΔА ВС
А В=с
А С= b
BC=a
Д ок аз ать
:
A bc c b а cos 2
2 2 2
A bc c b а cos 2
2 2 2
Isboti :](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_10.png)
![Uchburchak lar uchun k osinus t eoremasini
y ozing:
•
АВС
•
KLPA AC AB AC AB BC
B BC AB BC AB AC
C BC AC BC AC АВ
cos 2
cos 2
cos 2 222 222 222
LLKMLLKMLMK KKMLKKMLKLM MMKLMMKLMLK
cos2 cos2 cos2
222 222 222](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_11.png)
![Kosinus t eoremasidan burchak k osinusini
ifodalangC BC AC BC AC АВ cos 2
222
222
cos 2 AB BC AC C ВС АС
BCAC ABBCAC
C
2cos 222](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_12.png)
![A AC AB AC AB BC
B BC AB BC AB AC
cos 2
cos 2222 222
I f oda l a n g
A B cos , cos](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_13.png)
![Umumlasht irilganPifagor t eorema
si.
Kosinust eoremasiba'zanumumlasht irilga
nPifagort eoremasi deb at aladi. Ushbu
nom
k osinust eoremasidamaxsusholat sifat ida
Pifagort eoremasiborligibilanizohlanadi.
Darhaqiqat , agar A BSdagiA
burchak t o'g'ribo'lsa, u holda cosA = cos 90
° = 0 v ak osinust eoremasibo'y icha
- 2bc cosa
niolamiz: = +, gipot enuzaning
k v adrat ik at et k v adrat lariy ig'indisigat eng
.](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_14.png)
![CMa sa l a
5,4,75,30 bCA Berilgan :
Topish kerak :
baB ,, А Ba
b
c](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_15.png)
![Dars zo’r bo’ldi! Darsga tushundim](/data/documents/86e5718b-f576-4d9a-8efb-928860351d08/page_16.png)
Sinus va kosinuslar teoremalari
FRANSUA VIY ET (1540 -1603) Vi уе t t rigonomet riy ani y arat ilishida k at t a hissa qo'shgan. Ko'pgina t rigonomet rik formulalar birinchi mart a Viy et t omonidan y ozilgan. 1593 y ilda u k osinus t eoremasini og'zak i shak lda birinchi bo'lib y arat di Kosinus - bu lot in t ilidagi ex presssinus ifodasining qisqarishi, y a'ni " k omplement ar sinus" (y ok i boshqa y o'l bilan " t o'ldiruv chi y oy ning sinusi" ; cosa = sin (90 ° - a)).
Sin x v a cos x belgilari bilan zamonav iy sinus v a k osinus y ozuv lari birinchi mart a 1739 y ilda Bernulli t omonidan Sank t -Pet erburg mat emat igi L. Ey lerga y ozilgan xat da k irit ilgan. Ushbu belgilar juda qulay degan xulosaga k elib, ularni mat emat ik ishlarida ishlat a boshladi. Bundan t ashqari, Ey ler x burchak ning t rigonomet rik funk t siy alari uchun quy idagi qisqart malarni k irit adi: t ang x, cot x, sec x, cosecx.
Uchburchak y uzi formulasi Uchburchak ning y uzi uning ik k i t omoni v a ik k i t omoni orasidagi burchak ning sinusi y armiga t eng. CBCACBBCABS sin 21 sin 21 BAC CAB CACBAB sinsin sinsin A AC AB sin 2 1 AВС sin А В С
Sinuslar t eoremasi • Uchburchak ning t omonlari qarama- qarshi burchak larning sinuslariga proporsionaldir M FN NMF MNF FMN sinsinsin BAC CAB sinsin A ВС sin А В С MN F uchburchak uchun sinuslar t eoremasini y ozing