tom, Tu usasrulod klebadi geometriuli progresiis
jams aRvniSnavT S-iT, (1) tolobidan miiReba
b
S= 1 .
1− q
$50. lvUyvsj!tjejeffcjt!sbejbovmj!{pnb!
!
kuTxeebis gazomvis dros gamoiyeneba zomis sxvadasxva
erTeuli: 1o , 1′, 1′′ da a. S. SemoviRoT kuTxis zomis kidev
erTi erTeuli.
hbotb{Swsfcb/ im centraluri kuTxis sidides, romlis
Sesabamisi rkalis sigrZe radiusis tolia, radiani
ewodeba.
radgan naxevarwrewiris sigrZe πr –is tolia, xolo
misi Sesabamisi centraluri kuTxis sidide 180 o -ia, amitom
radiani π –jer naklebia 180 o -ze, e. i.
180 o
1 rad =
≈ 57 o17′45′′
π
da piriqiT
π
rad ≈ 0,017 rad.
1o=
180
SemdgomSi aRniSvnas “radiani” gamovtovebT da davwerT
π
π
π
π
o
360 = 2π , 90o = , 60o = , 45o = , 30o =
da a. S.
2
3
4
6
$51. usjhpopnfusjvmj!gvordjfcj!
!
rogorc cnobilia, sibrtyis iseT gadaadgilebas, roca
nebismier OA sxivsa da mis Sesabamis OA1 sxivs Soris
kuTxis sidide α aris mudmivi, ewodeba mobruneba O
wertilis garSemo α kuTxiT da R0α simboloTi aRiniSneba.
Tu mobruneba xdeba saaTis isris moZraobis sawinaaRmdego
α > 0,
xolo
mimarTulebiT,
maSin
miRebulia,
rom
Tu_saaTis isris moZraobis mimarTulebiT, maSin α < 0 ;
α=0
Seesabameba
sibrtyis
igivur
asaxvas.
aqedan
gamomdinare, −π ≤ α ≤ π , magram, Tu mobrunebas ganvixilavT,
rogorc brunvis Sedegs, maSin mobrunebis kuTxe SeiZleba
iyos nebismieri; amasTan cxadia, rom
R0α + 2 πk = R0α , k ∈ Z .
114
ganvixiloT wrewiri, romlis centri koordinatTa
saTaveSia, xolo radiusi 1-is tolia. aseT wrewirs
erTeulovani wrewiri ewodeba.
vTqvaT P0 aris wertili koordinatebiT (1;0). Pt –Ti
aRvniSnoT
wertili,
romelSic
y
aisaxeba P0 koordinatTa saTavis
1
Pt
garSemo t radianiT mobrunebisas
yt
(nax. 44). Pt wertilis yt ordinats
P0
t
t
ricxvis ( t
radiani kuTxis)
xt 1
x
sinusi ewodeba da sin t simboloTi
aRiniSneba, xolo xt abscisas t
radiani
kuTxis)
ricxvis
(t
nax. 44
kosinusi
ewodeba
da
cos t
simboloTi aRiniSneba, e. i.
yt = sin t , xt = cos t .
sin t
da cos t warmoadgenen t
argumentis funqciebs,
romlebic gansazRvrulia nebismieri t –saTvis, radgan
nebismieri t radianiT mobrunebas Seesabameba erTaderTi
savsebiT gansazRvruli Pt wertili.
sin t da cos t funqciebis mniSvnelobaTa simravlea [-1;1],
radgan erTeulovani wrewiris wertilebis TiToeuli
koordinati Rebulobs mxolod [-1;1] segmentis yvela
mniSvnelobas.
t ricxvis (radianis) sinusis Sefardebas kosinusTan t
ricxvis (radianis) tangensi ewodeba da tgt simboloTi
aRiniSneba. e. i.
sin t
.
tgt =
cos t
cxadia, rom tgt funqcia gansazRvrulia, roca cos t ≠ 0 ,
magram cos t = 0 mxolod maSin, rodesac Pt wertili Oy
π
RerZze mdebareobs, e. i. roca t = + πk , k ∈ Z .
2
amrigad, tgt funqcia gansazRvrulia yvelgan, garda
π
t = + πk , k ∈ Z ricxvebisa.
2
ganvixiloT x = 1 wrfe, romelsac tangensebis RerZi
π
t ≠ + πk ,
k∈Z
ricxvs SeiZleba
ewodeba. nebismier
2
115
SevusabamoT tangensebis RerZis erTaderTi At wertili,
romelic warmoadgens OPt wrfis tangensebis RerZTan
gadakveTis wertils (nax. 45). t ricxvis tangensi udris
tangensebis RerZze misi Sesabamisi At wertilis ordinats.
y
y
Pt
t
0
At
At
Pt
P0
0
x
t
P0
x
nax. 45
nax. 46
sin t da cos t funqciebisagan gansxvavebiT, tgt funqciis
mniSvnelobaTa simravlea R , radgan tangensebis RerZis
nebismieri At wertilis ordinati warmoadgens P0OAt
kuTxis tangenss (nax. 45).
t ricxvis (radianis) kosinusis Sefardebas sinusTan t
ricxvis (radianis) kotangensi ewodeba da ctgt simboloTi
aRiniSneba, e. i.
cos t
.
ctgt =
sin t
cxadia, rom ctgt funqcia gansazRvrulia, roca sin t ≠ 0 ,
magram sin t = 0 mxolod maSin, rodesac Pt wertili Ox
RerZze mdebareobs, e. i. roca t = πk , k ∈ Z .
amrigad, ctgt funqcia gansazRvrulia yvelgan, garda
t = πk , k ∈ Z , ricxvebisa.
ganvixiloT y = 1 wrfe, romelsac kotangensebis RerZi
ewodeba.
nebismier
t ≠ πk ,
k ∈Z ,
ricxvs
SeiZleba
SevusabamoT kotangensebis RerZis erTaderTi At wertili,
romelic warmoadgens OPt wrfis kotangensebis RerZTan
gadakveTis wertils (nax. 46). t ricxvis kotangensi udris
At
wertilis
kotangensebis RerZze misi Sesabamisi
abscisas.
ctgt funqciis mniSvnelobaTa simravlea R , radgan
kotangensebis RerZis nebismieri At wertilis abscisa
warmoadgens P0OAt kuTxis kotangenss (nax. 46).
116
y
+
y
_
+
_0 _
_0 +
x
sinusis niSnebi
x
kosinusis niSnebi
nax. 48
nax. 47
sin t , cos t , tgt da
funqciebi ewodeba.
+
ctgt
funqciebs trigonometriuli
trigonometriul
funqciaTa
gansazRvrebis safuZvelze SeiZleba davadginoT maTi niSnebi im
_
meoTxedebSi, romlebadac iyofa
+
sibrtye sakoordinato RerZebiT.
0 _
x
sin t funqcia I da II meoTxedSi
+
dadebiTia,
xolo
III
da
IV
meoTxedSi_uaryofiTi, radgan I
da II meoTxedSi moTavsebuli
tangensis da kotangensis
niSnebi
nebismieri wertilis ordinati
dadebiTia,
xolo
III
da
IV
nax. 49
meoTxedSi moTavsebuli nebismieri wertilis ordinati_uaryofiTi (nax. 47).
analogiurad SeiZleba vaCvenoT, rom cos t funqcia I da
IV meoTxedSi dadebiTia, xolo II da III meoTxedSi _
uaryofiTi (nax. 48).
radgan sin t da cos t funqciebs I da III meoTxedSi
erTnairi niSani aqvT, xolo II da IV meoTxedSi _
mopirdapire, amitom tgt da ctgt funqciebi I da III
meoTxedSi dadebiTia, xolo II da IV meoTxedSi _
uaryofiTi (nax. 49).
y
117
$52. usjhpopnfusjvmj!gvordjfcjt!nojTwofmpcbUb!
hbnpUwmb!bshvnfoujt!{phjfsUj!nojTwofmpcjtbUwjt!!
!
!
gamovTvaloT trigonometriul funqciaTa mniSvnelo3
π π π π
π argumentebisaTvis.
bebi, 0, , , , , π da
6 4 3 2
2
ganvixiloT erTeulovani wrewiris P0 (1;0) , Pπ (0;1) ,
2
Pπ (−1;0) da P3π (0;−1) wertilebi (nax. 50). trigonometriul
2
gansazRvrebidan gamomdinare gvaqvs: sin 0 = 0 ,
π
sin 0 0
cos 0 = 1 , tg 0 =
= = 0,
ctg 0
ar arsebobs; sin = 1 ,
cos 0 1
2
π
cos
π
π
π
2 = 0 = 0 ; sin π = 0 ,
cos = 0 , tg
ar arsebobs, ctg =
2 sin π 1
2
2
2
funqciaTa
y
y
Pπ
2
B
Pπ
P0
0
π
6
0
x
A
x
P3π
2
nax. 51
nax. 50
cos π = −1 , tgπ = 0 ,
ctgπ
ar arsebobs; sin
3π
= −1 ,
2
cos
3π
=0,
2
3π
3π
= 0.
ar arsebobs, ctg
2
2
trigonometriul funqciaTa gansazRvrebidan gvaqvs:
1
1
π
sin = AB = OB =
6
2
2
o
marTkuTxa samkuTxedSi 30 -iani kuTxis mopirdapire
kaTetis Tvisebis Tanaxmad (nax. 51);
π
cos = AO = OB 2 − AB 2 =
6
tg
118
1
3
=
4
2
(piTagoras Teoremis Tanaxmad);
= 1−
π
π
3
1
cos
sin
π
π
1
3
6
6
2
2
=
= 3.
; ctg =
tg =
=
=
=
1
6 sin π
6 cos π
3
3
3
2
6
6
2
π
π
sin = AB = OA = cos , radgan OAB samkuTxedi tolferdaa
4
4
(nax. 52). piTagoras Teoremis Tanaxmad
2 AO 2 = OB 2 = 1 ⇒ OA =
1
2
2
.
2
=
e. i.
sin
π
π
2
= cos =
;
4
4
2
y
y
B
B
π
4
0
π
3
P0
A
0
x
nax. 52
A
x
nax. 53
π
π
sin
cos
π
π
4 = 1 ; ctg =
4 = 1.
tg =
4 cos π
4 sin π
4
4
π
1
1
cos = AO = OB =
3
2
2
(marTkuTxa samkuTxedSi 30 o -iani kuTxis
kaTetis Tvisebis Tanaxmad (nax. 53));
1
3
π
= AB = OB 2 − OA 2 = 1 − =
3
4
2
(piTagoras Teoremis Tanaxmad);
sin
119
mopirdapire
π
π
3
1
sin
cos
π
π
3 = 2 = 3 ; ctg =
3 = 2 = 1 = 3 .
tg =
1
π
3 cos π
3
3 sin
3
3
2
3
3
2
trigonometriul funqciaTa gamoTvlili mniSvnelobebis safuZvelze SevadginoT Semdegi cxrili:
α
sin α
cos α
tgα
ctgα
0 = 0o
π
= 30o
6
π
= 45 o
4
π
= 60 o
3
π
= 90 o
2
π = 180 o
3π
= 270 o
2
0
1
2
2
2
3
2
1
0
−1
1
3
2
2
2
1
2
0
−1
0
0
3
3
1
3
_
0
_
_
3
1
3
3
0
_
0
$53. y = sin x gvordjjt!Uwjtfcfcj!eb!hsbgjlj!
!
1. y = sin x funqciis gansazRvris area D( y ) = R , xolo
mniSvnelobaTa simravle_ E ( y ) = [− 1;1] .
2. y = sin x funqcia kentia, radgan nebismieri x –isaTvis
y
sin( − x) = − sin x , vinaidan abscisTa RerZis mimarT simetriuli Px da P− x
Px
wertilebis ordinatebi gansxvavdebian mxolod niSniT (nax. 54).
0
x
3. y = sin x funqcia periodulia
da misi umciresi dadebiTi periodia
P− x
2π . marTlac, radgan, nebismieri x –
isaTvis R0x+ 2 π = R0x , amitom 2π warmoanax. 54
dgens erT-erT periods. vaCvenoT,
rom igi umciresi dadebiTi periodia. Tu davuSvebT, rom
aris periodi, maSin nebismieri
x –isaTvis
l ∈ ]0;2π [
sin( x + l ) = sin x . kerZod, roca x = 0 , miviRebT tolobas
sin l = 0 ,
120
romelic marTebulia ]0;2π [ Sualedis mxolod erTi l = π
mniSvnelobisaTvis, radgan es ricxvi aris erTaderTi
aRniSnuli Sualedidan, romlis Sesabamisi wertili
erTeulovan wrewirze, abscisTa RerZze Zevs. magram π ar
π
warmoadgens sin x –is periods, radgan sin = 1 , xolo
2
3π
⎛π
⎞
sin ⎜ + π ⎟ = sin
= −1 .
2
⎝2
⎠
⎡ π π⎤
4. y = sin x funqcia zrdadia ⎢− ; ⎥ SualedSi, radgan
⎣ 2 2⎦
π
π
rodesac x izrdeba − –dan
–mde, Px -is ordinati
2
2
izrdeba –1-dan 1-mde. analogiurad SeiZleba vaCvenoT, rom
⎡ π 3π ⎤
sin x funqcia ⎢ ; ⎥ SualedSi klebulobs 1-dan –1-mde.
⎣2 2 ⎦
radgan sin x periodulia periodiT 2π , amitom cxadia, rom
π
⎡ π
⎤
igi zrdadia nebismier ⎢− + 2πk ; + 2πk ⎥ , k ∈ Z , saxis
2
⎣ 2
⎦
3π
⎡π
⎤
+ 2πk ⎥ ,
SualedSi, xolo klebadia nebismier ⎢ + 2πk ;
2
⎣2
⎦
k ∈ Z , saxis SualedSi.
y = sin x
ganxiluli Tvisebebis safuZvelze avagoT
funqciis grafiki.
⎡ π⎤
y
vinaidan ⎢0; ⎥ SualedSi sin x
⎣ 2⎦
funqcia izrdeba 0-dan 1-mde, xolo
π
π
π
,
da
–ze
misi mniSvnelobebi
6
4
3
1
2
3
π
π π π
0
Sesabamisad aris
,
da
,
x
2
6 4 3
2
2
2
amitom, am SualedSi y = sin x funqcinax. 55
is grafiks aqvs saxe (nax. 55).
π
sin x funqciis grafiki simetriulia x =
wrfis
2
mimarT, radgan nebismieri x –isaTvis
⎛π
⎞
⎛π
⎞
sin ⎜ − x ⎟ = sin ⎜ + x ⎟ ;
⎝2
⎠
⎝2
⎠
121
y
marTlac, Pπ
π
2
2
Pπ
da Pπ
−x
2
wertilebi
+x
simetriulia Oy RerZis mimarT (nax.
56), amitom maTi ordinatebi tolia. aqedan
gamomdinare, y = sin x funqciis grafiks
0
x
[0; π ] SualedSi aqvs saxe (nax. 57).
sin x funqciis grafiki simetriulia
koordinatTa saTavis mimarT, radgan
nax. 56
igi kenti funqciaa. amitom [− π ; π ]
SualedSi am funqciis grafiks aqvs saxe (nax. 58).
sin x
funqcia periodulia periodiT
2π , amitom
sabolood mis grafiks eqneba saxe (nax. 59). y = sin x
funqciis grafiks sinusoida ewodeba.
Pπ
2
+x
2
−x
y
y
0
π
2
π
−
x
π
2
0
π
2
x
nax. 57
nax. 58
y
−5π
3π
2
−π
2
2
− 2π
−3π
2
−π
0
π
2
π
3π
2π
5π
2
x
nax. 59
rogorc sin x funqciis grafikis agebis sqemidan Cans,
sinusoidis asagebad sakmarisia vicodeT y = sin x funqciis
⎡ π⎤
grafiki ⎢0; ⎥ SualedSi.
⎣ 2⎦
122
$54. y = arcsin x gvordjjt!Uwjtfcfcj!eb!hsbgjlj!
!
⎡ π π⎤
⎢− 2 ; 2 ⎥ SualedSi, amitom
⎣
⎦
arsebobs misi Seqceuli funqcia am SualedSi, romelsac
arksinusi ewodeba da arcsin x simboloTi aRiniSneba.
⎡ π π⎤
radgan sin x funqcia ⎢− ; ⎥ SualedSi Rebulobs yvela
⎣ 2 2⎦
mniSvnelobas [-1;1] Sualedidan, amitom y = arcsin x funqciis
y = sin x
gansazRvris
funqcia zrdadia
are
D(arcsin x) = [− 1;1] ,
xolo
mniSvnelobaTa
⎡ π π⎤
simravle_ E (arcsin x) = ⎢− ; ⎥ .
⎣ 2 2⎦
cxadia, Tu −1 ≤ a ≤ 1 , maSin arcsin a warmoadgens im
⎡ π π⎤
kuTxis sidides ⎢− ; ⎥ Sualedidan, romlis sinusi
⎣ 2 2⎦
udris a –s, e. i.
sin(arcsin a ) = a .
SeiZleba vaCvenoT, rom
arcsin(− x) = − arcsin x ,
amitom y = arcsin x funqcia kentia.
y = arcsin x funqcia zrdadia, rogorc zrdadi funqciis
Seqceuli funqcia da radgan urTierTSeqceul funqciaTa
grafikebi simetriulia y = x wrfis mimarT, amitom am
funqciis grafiks aqvs saxe (nax. 60).
y
π
2
-1
1
x
−
π
2
nax. 60
123
$55. y = cos x gvordjjt!Uwjtfcfcj!eb!hsbgjlj!
1.
y = cos x
!
funqciis
gansazRvris
area D( y ) = R , xolo mniSvnelobaTa
simravle_ E ( y ) = [−1;1] .
2. y = cos x funqcia luwia, radgan
y
Px
0
x –isaTvis
cos(− x) = cos x ,
nebismieri
x
vinaidan
abscisTa
RerZis
mimarT
P− x
simetriuli Px da P− x wertilebis
abscisebi tolia (nax. 61).
nax. 61
3. y = cos x funqcia periodulia da
misi umciresi dadebiTi periodia 2π . marTlac, radgan
nebismieri x –isaTvis R0x+ 2 π = R0x , amitom 2π warmoadgens
erT-erT periods. vaCvenoT, rom igi umciresi dadebiTi
periodia. Tu davuSvebT, rom l ∈ ]0;2π [ aris periodi, maSin
nebismieri x –isaTvis cos( x + l ) = cos x . kerZod, roca x = 0 ,
miviRebT tolobas
cos l = 1 ,
l –is
arcerTi
romelic
ar
aris
marTebuli
mniSvnelobisaTvis ]0;2π [ Sualedidan.
4. y = cos x funqcia zrdadia [− π ;0] SualedSi, radgan
rodesac x izrdeba −π –dan 0-mde, Px -is abscisa izrdeba
-1-dan 1-mde. analogiurad SeiZleba vaCvenoT, rom cos x
funqcia [0; π ] SualedSi klebulobs 1-dan –1-mde. radgan
cos x periodulia periodiT 2π , amitom cxadia, rom igi
zrdadia nebismier [− π + 2πk ;2πk ] , k ∈ Z , saxis SualedSi,
xolo klebadia nebismier
[2πk ; π + 2πk ] , k ∈ Z , saxis
SualedSi.
ganxiluli Tvisebebis safuZvelze avagoT y = cos x
funqciis grafiki.
y
⎡ π⎤
vinaidan ⎢0; ⎥ SualedSi y = cos x
⎣ 2⎦
funqcia klebulobs 1-dan 0-mde,
π π
,
da
xolo misi mniSvnelobebi
6
4
π
π π π
0
x
π
3
2
2
6 4 3
–ze Sesabamisad aris
,
da
3
2
2
nax. 62
124
1
, amitom am SualedSi
2
saxe (nax. 62).
y = cos x funqciis grafiks aqvs
x=
cos x funqciis grafiki simetriulia
π
wertilis
2
mimarT, radgan nebismieri x –isaTvis
⎛π
⎞
⎛π
⎞
cos⎜ − x ⎟ = − cos⎜ + x ⎟ ;
⎝2
⎠
⎝2
⎠
marTlac Pπ
da Pπ
wertilebi simetriulia Oy RerZis
2
−x
2
+x
mimarT (nax. 56) da amitom maTi abscisebi gansxvavdebian
mxolod niSniT. aqedan gamomdinare, y = cos x funqciis
grafiks [0; π ] SualedSi aqvs saxe (nax. 63).
cos x funqciis grafiki simetriulia Oy RerZis mimarT,
radgan igi luwi funqciaa. amitom [− π ; π ] SualedSi am
funqciis grafiks aqvs saxe (nax. 64).
y
y
π
π
2
0
−π
π
−
x
π
2
π
2
0
x
nax. 63
nax. 64
cos x funqcia periodulia periodiT 2π , amitom sabolood mis grafiks eqneba saxe (nax. 65).
y
−3π
−5π −2π
2
−3π
2
3π
π
−π
−π
2
0
π
2
3π
2
2π
5π
2
7π
2
x
nax. 65
y = cos x
funqciis
grafiks
kosinusoida
ewodeba.
rogorc
funqciis
grafikis
agebis
sqemidan
Cans,
125
kosinusoidis
asagebad
sakmarisia
vicodeT
y = cos x
⎡ π⎤
funqciis grafiki ⎢0; ⎥ SualedSi.
⎣ 2⎦
$56. y = arccos x gvordjjt!Uwjtfcfcj!eb!hsbgjlj!
!
y = cos x funqcia klebadia [0; π ] SualedSi, amitom
arsebobs misi Seqceuli funqcia am SualedSi, romelsac
arkkosinusi ewodeba da arccos x simboloTi aRiniSneba.
radgan cos x funqcia [0; π ] SualedSi Rebulobs yvela
mniSvnelobas [-1;1] Sualedidan, amitom y = arccos x funqciis
gansazRvris are D(arccos x) = [−1;1] , xolo mniSvnelobaTa
simravle_ E (arccos x) = [0;π ] . cxadia, Tu
−1 ≤ a ≤ 1 , maSin
arccos a warmoadgens im kuTxis sidides [0; π ] Sualedidan,
romlis kosinusi udris a –s, e. i.
y
cos(arccos a) = a .
π
SeiZleba vaCvenoT, rom
arccos(− x) = π − arccos x .
π
2
y = arccos x funqcia klebadia, rogorc klebadi funqciis Seqceuli
funqcia da radgan urTierTSeqceul
funqciaTa grafikebi simetriulia
1
x
-1 0
y = x wrfis mimarT, amitom am funnax. 66
qciis grafiks aqvs saxe (nax. 66).
§57. y = tgx gvordjjt!Uwjtfcfcj!eb!hsbgjlj!
!
y = tgx funqciis gansazRvris area yvela namdvil
π
ricxvTa simravle, garda
+ πk , k ∈ Z , saxis ricxvebisa,
2
xolo mniSvnelobaTa simravlea R .
2. y = tgx funqcia kentia, radgan nebismieri x –isaTvis
gansazRvris aridan
sin( − x) − sin x
tg (− x) =
=
= −tgx .
cos(− x)
cos x
1.
126
3.
y = tgx funqcia periodulia da misi umciresi
π.
marTlac,
x -is
nebismieri
dadebiTi
periodia
x
da
x±π
mniSvnelobisaTvis gansazRvris aredan
ricxvebi gamoisaxebian erTeulovani wrewiris Px da Px± π
wertilebiT, romlebic simetriulia koordinatTa O
saTavis mimarT (nax. 67), amitom am
y
ricxvebis Sesabamisi wertilebi tangensebis RerZze erTi da igivea. e. i.
Px
tg ( x ± π) = tgx .
amrigad π warmoadgens erT-erT peri0
x ods. vaCvenoT, rom igi umciresi dadebiTi periodia. Tu davuSvebT, rom
Px ± π
l ∈ ]0; π [ aris periodi, maSin nebismieri
x –isaTvis
nax. 67
tg ( x + l ) = tgx .
kerZod, roca x = 0 , miviRebT
tgl = 0
tolobas, romelic ar aris marTebuli l –is arcerTi
mniSvnelobisaTvis ]0; π [ Sualedidan.
⎤ π π⎡
4. y = tgx funqcia zrdadia ⎥ − ; ⎢ SualedSi, radgan,
⎦ 2 2⎣
π
π
rodesac x izrdeba − –dan
–mde tangensebis RerZze
2
2
misi Sesabamisi wertilis ordinati izrdeba −∞ –dan +∞ –
mde. radgan tgx periodulia periodiT π , amitom, cxadia,
π
⎤ π
⎡
rom igi zrdadia nebismier ⎥ − + πk ; + πk ⎢ , k ∈ Z , saxis
2
⎦ 2
⎣
SualedSi.
y = tgx
ganxiluli Tvisebebis safuZvelze avagoT
funqciis grafiki.
⎡ π⎡
vinaidan ⎢0; ⎢ SualedSi tgx funqcia izrdeba 0-dan
⎣ 2⎣
π
π
π
+∞ –mde, xolo misi mniSvnelobebi
,
da
–ze
6
4
3
3
Sesabamisad aris
, 1 da
3 , amitom am SualedSi
3
funqciis grafiks aqvs saxe (nax. 68).
127
tgx
funqciis
grafiki
simetriulia
koordinatTa
⎤ π π⎡
saTavis mimarT, radgan igi kenti funqciaa. amitom ⎥ − ; ⎢
⎦ 2 2⎣
SualedSi am funqciis grafiks aqvs saxe (nax. 69).
y
y
−
3
3
0
3
π π π π
6 4 3 2
π
2
π
2
0
x
x
nax. 68
nax. 69
tgx funqcia periodulia periodiT π , amitom mis grafiks
aqvs saxe (nax. 70).
y = tgx funqciis grafiks tangensoida ewodeba.
y
−
5π
2
−2π
−
3π
2
−π
−
π
2
0
π
2
π
3π
2
2π
5π
2
x
nax. 70
§58. y = arctgx gvordjjt!Uwjtfcfcj!eb!hsbgjlj!
!
⎤ π π⎡
⎥ − 2 ; 2 ⎢ SualedSi, amitom
⎣
⎦
arsebobs misi Seqceuli funqcia am SualedSi, romelsac
arktangensi ewodeba da arctgx simboloTi aRiniSneba.
y = tgx
funqcia zrdadia
128
⎤ π π⎡
funqcia ⎥ − ; ⎢ SualedSi Rebulobs yvela
⎦ 2 2⎣
R –dan,
amitom
y = arctgx
funqciis
mniSvnelobas
gansazRvris
are
D (arctgx ) = R ,
xolo
mniSvnelobaTa
radgan tgx
⎤ π π⎡
simravle_ E (arctgx) = ⎥ − ; ⎢ . cxadia, rom nebismieri a
⎦ 2 2⎣
warmoadgens im kuTxis sidides
ricxvisaTvis arctga
⎤ π π⎡
⎥ − 2 ; 2 ⎢ Sualedidan, romlis tangensi udris a –s, e. i.
⎣
⎦
tg (arctga) = a .
SeiZleba vaCvenoT, rom
arctg (− x) = −arctgx ,
amitom y = arctgx funqcia kentia.
y = arctgx funqcia zrdadia, rogorc zrdadi funqciis
Seqceuli funqcia da radgan urTierTSeqceul funqciaTa
grafikebi simetriulia y = x wrfis mimarT, amitom am
funqciis grafiks aqvs saxe (nax. 71).
y
π
2
x
0
−
π
2
nax. 71
§59. y = ctgx gvordjjt!Uwjtfcfcj!eb!hsbgjlj!
!
1. y = ctgx funqciis gansazRvris area yvela namdvili
ricxvis simravle, garda πk , k ∈ Z , saxis ricxvebisa,
xolo mniSvnelobaTa simravlea R .
129
2. y = ctgx funqcia kentia, radgan nebismieri x –isaTvis
gansazRvris aridan
cos(− x)
cos x
ctg (− x) =
=
= −ctgx .
sin(− x) − sin x
3. y = ctgx funqcia periodulia da misi umciresi
π.
marTlac
x –is
nebismieri
dadebiTi
periodia
x
da
x±π
mniSvnelobisaTvis gansazRvris aridan
ricxvebi gamoisaxebian erTeulovani wrewiris Px da Px± π
wertilebiT, romlebic simetriulia koordinatTa O
saTavis mimarT (nax. 72), amitom am ricxvebis Sesabamisi
wertilebi kotangensebis RerZze erTidaigivea. e. i.
ctg ( x ± π) = ctgx .
amrigad, π warmoadgens erT-erT periods. vaCvenoT, rom
igi umciresi dadebiTi periodia. Tu
y
davuSvebT, rom l ∈ ] 0; π [ aris periodi, maSin nebismieri x –isaTvis
Px
ctg ( x + l ) = ctgx .
π
0
x
kerZod, roca x = , miviRebT tolo2
Px ± π
bas:
⎛π ⎞
ctg ⎜ + l ⎟ = 0 ,
nax. 72
⎝2 ⎠
romelic ar aris marTebuli l –is arcerTi mniSvnelobisaTvis ] 0; π [ Sualedidan.
4. y = ctgx funqcia klebadia ] 0; π [ SualedSi, radgan,
rodesac x izrdeba 0-dan π –mde, kotangensebis RerZze
misi Sesabamisi wertilis abscisa klebulobs +∞ –dan −∞ –
mde. radgan ctgx periodulia periodiT π , amitom cxadia,
rom igi klebadia nebismier ] πk ; π + πk [ , k ∈ Z , saxis
SualedSi.
ganxiluli Tvisebebis safuZvelze avagoT y = ctgx
funqciis grafiki.
⎤ π⎤
vinaidan ⎥ 0; ⎥ SualedSi ctgx funqcia klebulobs
⎦ 2⎦
π
π
π
+∞ –dan 0-mde, xolo misi mniSvnelobebi
,
da
–ze
6
4
3
130
3
,
3
funqciis grafiks aqvs saxe (nax. 73).
Sesabamisad
aris
3,
1
da
amitom
am
SualedSi
y
3
π
6
0
3
3
1
π
4
π
3
π
2
x
nax. 73
ctgx
funqciis grafiki simetriulia
x=
π
2
wertilis
π
+ πk , k ∈ Z , ricxvisaTvis
2
⎛π
⎞
⎛π
⎞
cos⎜ − x ⎟ − cos⎜ + x ⎟
2
2
⎛π
⎞
⎝
⎠
⎝
⎠ = −ctg ⎛ π + x ⎞ .
=
ctg ⎜ − x ⎟ =
⎜
⎟
π
π
2
⎛
⎞
⎛
⎞
⎝
⎠ sin − x
⎝2
⎠
sin ⎜ + x ⎟
⎜
⎟
⎝2
⎠
⎝2
⎠
aqedan gamomdinare,
y = ctgx
funqciis grafiks
]0; π [
SualedSi aqvs saxe (nax. 74).
ctgx funqcia periodulia periodiT π , amitom mis grafiks aqvs saxe (nax. 75).
mimarT, radgan nebismieri x ≠
y
0
π
2
nax. 74
131
π
x
y
−3π
−
5π
2
−2π
−
3π
2
−π
−
π
2
0
π
2
π
3π
2
2π
5π
2
3π
x
nax. 75
y = ctgx funqciis grafiks kotangensoida ewodeba.
§60. y = arcctgx gvordjjt!Uwjtfcfcj!eb!hsbgjlj!
!
y = ctgx funqcia klebadia ] 0; π [ SualedSi, amitom
arsebobs misi Seqceuli funqcia am SualedSi, romelsac
arkkotangensi ewodeba da arcctgx simboloTi aRiniSneba.
radgan ctgx funqcia ] 0; π [ SualedSi Rebulobs yvela
R –dan,
mniSvnelobas
amitom
y = arcctgx
funqciis
D(arcctg ) = R ,
xolo
mniSvnelobaTa
gansazRvris
are
simravle_ E (arcctg ) = ] 0;π [ .
cxadia,
nebismieri
a
ricxvisaTvis arcctga warmoadgens im kuTxis sidides ] 0; π [
Sualedidan, romlis kotangensi udris a –s, e. i.
ctg (arcctga) = a .
SeiZleba vaCvenoT, rom
arcctg (− x ) = π − arcctgx .
y = arcctgx
funqcia
y
klebadia, rogorc klebadi
funqciis
Seqceuli
funqcia da radgan urTierTSeqceul
funqciaTa
π
grafikebi
simetriulia
●2
y = x wrfis mimarT, amitom
am funqciis grafiks aqvs
x
0
saxe (nax. 76).
nax. 76
132
§61. ebnpljefcvmfcb!fsUj!eb!jhjwf!bshvnfoujt!
usjhpopnfusjpvm!gvordjfct!Tpsjt!!
!
1. vinaidan erTeulovani wrewiris nebismieri Pα (xα , yα )
wertilis koordinatebi akmayofileben pirobas xα2 + yα2 = 1 ,
xolo xα = cos α , yα = sin α , amitom gvaqvs:
cos 2 α + sin 2 α = 1 .
(1)
π
2. Tu α ≠ k , k ∈ Z , maSin
2
tgα ⋅ ctgα = 1 .
(2)
marTlac,
sin α cos α
⋅
= 1.
tgα ⋅ ctgα =
cos α sin α
π
3. Tu α ≠ + πk , k ∈ Z , maSin gvaqvs igiveoba
2
1
1 + tg 2 α =
.
(3)
cos 2 α
marTlac
sin 2 α cos 2 α + sin 2 α
1
.
1 + tg 2 α = 1 +
=
=
cos 2 α
cos 2 α
cos 2 α
4. Tu α ≠ πk , k ∈ Z , maSin gvaqvs igiveoba
1
1 + ctg 2 α =
.
(4)
sin 2 α
marTlac
cos 2 α sin 2 α + cos 2 α
1
.
1 + ctg 2 α = 1 +
=
=
2
2
sin α
sin α
sin 2 α
(1)_(4) formulebi saSualebas gvaZlevs erT-erTi
trigonometriuli
funqciis
mniSvnelobis
mixedviT
vipovoT amave argumentis danarCeni trigonometriuli
funqciebi, Tu cnobilia romel meoTxedSia argumenti.
nbhbmjUj/ vipovoT sin α , cos α da ctgα , Tu tgα = − 3 ,
4
⎤π ⎡
α ∈ ⎥ ; π ⎢ . (2) formulis Tanaxmad
⎦2 ⎣
1
4
ctgα =
=− .
tgα
3
133
Tu gaviTvaliswinebT, rom α moTavsebulia meore
meoTxedSi, sadac kosinusi uaryofiTia, (3) formulidan
gveqneba
1
1
4
cos α = −
=−
=− .
2
5
9
1 + tg α
1+
16
sin α gamoiTvleba Semdegnairad
⎛ 3⎞⎛ 4⎞ 3
sin α = tgα ⋅ cos α = ⎜ − ⎟ ⎜ − ⎟ = .
⎝ 4⎠⎝ 5⎠ 5
§62. psj!bshvnfoujt!kbnjtb!eb!tywbpcjt!!
usjhpopnfusjvmj!gvordjfcj!
!
mtkicdeba, rom nebismieri α da β –saTvis adgili aqvs
tolobebs:
sin (α + β ) = sin α cos β + cos α sin β ,
sin (α − β) = sin α cos β − cos α sin β ,
cos(α + β) = cos α cos β − sin α sin β ,
cos(α − β ) = cos α cos β + sin α sin β .
π
π
π
vaCvenoT, rom Tu α ≠ + πk , β ≠ + πn da α + β ≠ + πm ,
2
2
2
(k , n, m ∈ Z ) , maSin
tgα + tgβ
tg (α + β) =
.
1 − tgα ⋅ tgβ
marTlac
sin α sin β
+
tgα + tgβ
sin α cos β + cos α sin β
cos α cos β
=
=
=
sin
α
sin
β
1 − tgα ⋅ tgβ 1 −
cos α cos β − sin α sin β
⋅
cos α cos β
sin (α + β )
=
= tg (α + β) .
cos(α + β )
analogiurad SeiZleba vaCvenoT, rom:
tgα − tgβ
π
π
π
tg (α − β) =
, α ≠ + πk , β ≠ + πn , α − β ≠ + πm ,
1 + tgα ⋅ tgβ
2
2
2
(k , n, m ∈ Z ) .
134
jamisa da sxvaobis trigonometriul funqciaTa
gamosaTvlel formulebs Sekrebis formulebi ewodeba.
§63. ebzwbojt!gpsnvmfcj!
!
rogorc cnobilia, nebismieri x ricxvi SeiZleba
Caiweros x = 2πk + α saxiT, sadac 0 ≤ α < 2π , k ∈ Z . aqedan
gamomdinare, radgan sinus da kosinus funqciebis periodia
2π , am funqciebis mniSvnelobaTa gamoTvla nebismieri x
argumentisaTvis daiyvaneba sin α da cos α funqciebis
mniSvnelobebis gamoTvlamde, sadac α ∈ [0;2π [ . ufro metic
sin x
da
cos x
funqciebis mniSvnelobaTa gamoTvla
SeiZleba daviyvanoT sin α –s da cos α –s moZebnamde, sadac
⎡ π⎤
α ∈ ⎢0; ⎥ . amis saSualebas gvaZlevs e. w. dayvanis
⎣ 2⎦
formulebi, romelTac aqvT Semdegi saxe:
⎛π
⎞
⎛π
⎞
sin ⎜ + α ⎟ = cos α , cos⎜ + α ⎟ = − sin α ,
2
2
⎝
⎠
⎝
⎠
⎞
⎞
⎛π
⎛π
sin ⎜ − α ⎟ = cos α , cos⎜ − α ⎟ = sin α ,
2
2
⎝
⎠
⎝
⎠
sin (π + α ) = − sin α , cos(π + α ) = − cos α ,
sin (π − α ) = sin α , cos(π − α ) = − cos α ,
⎛3
⎞
⎛3
⎞
sin ⎜ π + α ⎟ = − cos α , cos⎜ π + α ⎟ = sin α ,
2
2
⎝
⎠
⎝
⎠
⎛3
⎞
⎛3
⎞
sin ⎜ π − α ⎟ = − cos α , cos⎜ π − α ⎟ = − sin α .
⎝2
⎠
⎝2
⎠
es formulebi marTebulia nebismieri α –saTvis da
maTi
damtkiceba
SeiZleba
Sekrebis
formulebis
saSualebiT. magaliTad,
π
π
⎛π
⎞
sin ⎜ + α ⎟ = sin cos α + cos sin α = 1 ⋅ cos α + 0 ⋅ sin α = cos α .
2
2
⎝2
⎠
analogiurad, radgan tangensisa da kotangensis
periodia π , xolo nebismieri x ricxvi SeiZleba Caiweros
saxiT,
sadac
0≤α<π,
amitom
am
x = πk + α
k ∈Z ,
funqciebis
mniSvnelobaTa
gamoTvla
nebismieri
x
tgα
ctgα
da
funqciebis
argumentisaTvis daiyvaneba
mniSvnelobebis moZebnamde, sadac α ∈ [0; π [ . iseve rogorc
135
sinusisa da kosinusis SemTxvevaSi tgx da ctgx funqciebis
mniSvnelobaTa gamoTvla SeiZleba daviyvanoT tgα –s da
⎡ π⎤
ctgα –s moZebnamde, sadac α ∈ ⎢0; ⎥ , dayvanis formulebis
⎣ 2⎦
saSualebiT, romelTac aqvT saxe:
⎞
⎞
⎛π
⎛π
tg ⎜ + α ⎟ = −ctgα , ctg ⎜ + α ⎟ = −tgα ,
2
2
⎝
⎠
⎝
⎠
⎞
⎞
⎛π
⎛π
tg ⎜ − α ⎟ = ctgα , ctg ⎜ − α ⎟ = tgα .
⎝2
⎠
⎝2
⎠
davamtkicoT erT-erTi maTgani
⎛π
⎞
sin ⎜ + α ⎟
⎛π
⎞
⎝2
⎠ = cos α = −ctgα .
tg ⎜ + α ⎟ =
π
2
⎛
⎝
⎠ cos + α ⎞ − sin α
⎜
⎟
⎝2
⎠
dayvanis formulebis damaxsovreba advilad SeiZleba
Semdegi wesis mixedviT:
π
aRebulia kent ricxvjer
Tu dayvanis formulaSi
2
da mas akldeba, an emateba α , maSin dasayvani funqcia
π
aRebulia luw ricxvjer,
icvleba “kofunqciiT”∗, Tu
2
maSin dasayvani funqciis saxelwodeba ucvleli rCeba.
dayvanili funqciis win iwereba is niSani, ra niSanic aqvs
⎤ π⎡
dasayvan funqcias, Tu CavTvliT, rom α ∈ ⎥ 0; ⎢ .
⎦ 2⎣
ganvixiloT magaliTebi
1. sin 1935o = sin (360 o ⋅ 5 + 135o ) = sin 135o = sin (90o + 45o ) =
2
.
2
29
2 ⎞
2
⎛
⎛π π⎞
2. ctg
π = ctg ⎜ 9π + π ⎟ = ctg π = ctg ⎜ + ⎟ =
3
3 ⎠
3
⎝
⎝2 6⎠
= cos 45o =
= −tg
3
π
=−
.
6
3
∗ sinusis, kosinusis, tangensis da kotangensis kofunqciebi ewodeba
Sesabamisad kosinuss, sinuss, kotangenss da tangenss.
136
§64. psnbhj!eb!obyfwbsj!bshvnfoujt!!
!!usjhpopnfusjvmj!gvordjfcj!
Tu sin (α + β) = sin α ⋅ cos β + cos α ⋅ sin β formulaSi CavsvamT
β = α , miviRebT
sin 2α = sin α ⋅ cos α + cos α ⋅ sin α = 2 sin α ⋅ cos α .
amrigad,
sin 2α = 2 sin α ⋅ cos α .
analogiurad
cos 2α = cos α ⋅ cos α − sin α ⋅ sin α = cos 2 α − sin 2 α ,
2tgα
tgα + tgα
=
,
tg 2α =
1 − tgαtgα 1 − tg 2 α
e. i.
cos 2α = cos 2 α − sin 2 α ,
(1)
2tgα
π π
π
tg 2α =
, α ≠ + πk , α ≠ + n (k , n ∈ Z ) .
1 − tg 2 α
2
4 2
Tu (1) formulis marjvena mxares gamovsaxavT mxolod
sinusiT an mxolod kosinusiT, miviRebT
cos 2α = 1 − 2 sin 2 α ,
cos 2α = 2 cos 2 α − 1 .
aqedan
1 − cos 2α
1 + cos 2α
sin 2 α =
, cos 2 α =
.
2
2
α
Tu miRebul formulebSi α –s SevcvliT
–iT,
2
gveqneba
α 1 − cos α
,
sin 2 =
2
2
α 1 + cos α
,
cos 2 =
2
2
saidanac
sin
1 − cos α
α
,
=±
2
2
1 + cos α
α
.
(3)
=±
2
2
(k ∈ Z ) , (2) tolobis (3)-ze gayofiT
cos
Tu α ≠ π(2k + 1) ,
miviRebT
(2)
137
1 − cos α
α
.
(4)
=±
2
1 + cos α
(2)_(4) formulebSi niSani “+” an “_” unda aviRoT imis
α
.
mixedviT, Tu romel meoTxedSia
2
naxevari
argumentis
tangensi
mTeli
argumentis
sinusiTa da kosinusiT SeiZleba gamoisaxos formuliT
sin α
α
tg =
, α ≠ π(2k + 1) , (k ∈ Z )
2 1 + cos α
marTlac,
α
α
α
2 sin cos
sin
sin α
2
2
2 = tg α ,
=
=
α
2
1 + cos α
2 α
2 cos
cos
2
2
analogiurad SeiZleba vaCvenoT, rom
α 1 − cos α
tg =
, α ≠ πk , (k ∈ Z ) .
2
sin α
tg
§65. usjhpopnfusjvm!gvordjbUb!hbnptbywb!
obyfwbsj!bshvnfoujt!ubohfotjU!
!
xSirad mosaxerxebelia trigonometriul funqciaTa
gamosaxva
naxevari
argumentis
tangensiT
Semdegi
formulebis saSualebiT:
α
2tg
2 , α ≠ π(2k + 1) , (k ∈ Z )
sin α =
(1)
2 α
1 + tg
2
α
1 − tg 2
2 , α ≠ π(2k + 1) , (k ∈ Z )
cos α =
(2)
2 α
1 + tg
2
α
2tg
2 , α ≠ π + πk , α ≠ π(2n + 1) , (k , n ∈ Z )
(3)
tgα =
2
2 α
1 − tg
2
davamtkicoT es formulebi:
138
α
α
sin
2 =2
2 : 1 = 2 sin α ⋅ cos α = sin α ;
α
α
2
2
2 α
1 + tg
cos
cos 2
2
2
2
α
α
α
1 − tg 2
cos 2 − sin 2
2 =
2
2 : 1 =
α
α
α
1 + tg 2
cos 2
cos 2
2
2
2
2 α
2 α
= cos
− sin
= cos α .
2
2
(3) formula miiReba (1) tolobis gayofiT (2)_ze.
2tg
§66. usjhpopnfusjvm!gvordjbUb!obnsbwmjt!!
hbsebrnob!kbnbe!
!
gamoviyvanoT
sin α ⋅ cos β ,
sin α ⋅ sin β
da
cos α ⋅ cos β
namravlebis trigonometriul funqciaTa jamad gardaqmnis
formulebi.
Tu SevkrebT
sin (α + β ) = sin α ⋅ cos β + cos α ⋅ sin β
da
sin (α − β) = sin α ⋅ cos β − cos α ⋅ sin β
tolobebs, miviRebT:
sin (α + β ) + sin (α − β ) = 2 sin α ⋅ cos β ,
saidanac
1
sin α ⋅ cos β = [sin (α + β ) + sin (α − β )] .
2
analogiurad
cos(α − β ) = cos α ⋅ cos β + sin α ⋅ sin β
(1)
da
cos(α + β) = cos α ⋅ cos β − sin α ⋅ sin β
(2)
tolobebis SekrebiT miviRebT:
cos(α − β ) + cos(α + β ) = 2 cos α ⋅ cos β ,
saidanac
1
cos α ⋅ cos β = [cos(α − β) + cos(α + β)] .
2
aseve, Tu (1) tolobas gamovaklebT (2)-s, miviRebT
cos(α − β ) − cos(α + β) = 2 sin α ⋅ sin β ,
139
saidanac
sin α sin β =
1
[cos(α − β) − cos(α + β)] .
2
§67. usjhpopnfusjvm!gvordjbUb!kbnjt!
hbsebrnob!obnsbwmbe!
!
rogorc wina paragrafSi vnaxeT, marTebulia Semdegi
formulebi:
sin (x + y ) + sin (x − y ) = 2 sin x ⋅ cos y ,
(1)
cos(x + y ) + cos(x − y ) = 2 cos x ⋅ cos y ,
(2)
cos(x − y ) − cos(x + y ) = 2 sin x ⋅ sin y ,
(3)
advili saCvenebelia agreTve, rom
sin (x + y ) − sin (x − y ) = 2 cos x ⋅ sin y ,
(4)
SemoviRoT aRniSvnebi
x+ y =α, x− y =β,
(5)
saidanac
α+β
α −β
x=
, y=
,
(6)
2
2
(5)
da
(6)
tolobebis
gaTvaliswinebiT
(1)_(4)
formulebidan miviRebT:
α+β
α −β
,
sin α + sin β = 2 sin
cos
2
2
α −β
α+β
,
sin α − sin β = 2 sin
cos
2
2
α+β
α −β
,
cos α + cos β = 2 cos
cos
2
2
α+β
α −β
.
cos α − cos β = −2 sin
sin
2
2
ganvixiloT axla tangensebis jami tgα + tgβ .
sin α sin β sin α ⋅ cos β + cos α ⋅ sin β
sin (α + β)
tgα + tgβ =
+
=
=
.
cos α cos β
cos α ⋅ cos β
cos α ⋅ cos β
amrigad
sin (α + β)
π
π
tgα + tgβ =
, α ≠ + πk , β ≠ + nπ , (k , n ∈ Z ) .
cos α ⋅ cos β
2
2
analogiurad SeiZleba vaCvenoT, rom
sin (α − β )
π
π
tgα − tgβ =
, α ≠ + πk , β ≠ + nπ , (k , n ∈ Z ) ;
cos α ⋅ cos β
2
2
140
sin (α + β)
, α ≠ πk , β ≠ πn , (k , n ∈ Z ) ;
sin α ⋅ sin β
sin (β − α )
ctgα − ctgβ =
, α ≠ πk , β ≠ πn , (k , n ∈ Z ) .
sin α ⋅ sin β
ctgα + ctgβ =
§68. usjhpopnfusjvmj!hboupmfcfcj!
!
1. hbowjyjmpU! sin x = a ! hboupmfcb/ radgan nebismieri
x –saTvis −1 ≤ sin x ≤ 1 , amitom, roca a > 1 , am gantolebas
amonaxsni ara aqvs.
roca a ≤ 1 , maSin sin x = a gantolebis amosaxsnelad
sakmarisia vipovoT misi yvela amonaxsni nebismier 2π
sigrZis SualedSi da gamoviyenoT sinusis perioduloba.
⎡ π 3π ⎤
aseT Sualedad mosaxerxebelia aviRoT ⎢− ; ⎥ .
⎣ 2 2 ⎦
⎡ π π⎤
⎡ π 3π ⎤
da
SualedebSi
sin x
funqcia
⎢− 2 ; 2 ⎥
⎢2; 2 ⎥
⎣
⎣
⎦
⎦
monotonuria da TiToeul maTganze Rebulobs yvela
mniSvnelobas –1-dan 1-mde (nax. 77). amitom am Sualedebidan
TiToeulSi gantolebas eqneba erTaderTi amonaxsni.
⎡ π π⎤
SualedSi
arksinusis gansazRvris Tanaxmad
⎢− 2 ; 2 ⎥
⎣
⎦
moTavsebuli amonaxsni iqneba arcsin a .
⎡ π 3π ⎤
π − arcsin a
ekuTvnis
Sualeds da
radgan
⎢2; 2 ⎥
⎣
⎦
sin (π − arcsin a ) = sin (arcsin a ) = a , amitom π − arcsin a warmoadgens
am SualedSi moTavsebul amonaxsns (nax. 77).
y
y=a
1
a
−
π
arcsin a
2
-1
nax. 77
141
π π − arcsin a
3
2
2
π
x
imisaTvis, rom CavweroT sin x = a gantolebis yvela
amonaxsni, visargebloT periodulobiT, e. i. miRebuli ori
amonaxsnidan TiToeuls davumatoT 2πn , (n ∈ Z ) , saxis
ricxvi. miviRebT, rom amonaxsnTa simravle Sedgeba ori
usasrulo qvesimravlisagan, romlebic ganisazRvrebian
formulebiT:
x = arcsin a + 2πn , n ∈ Z ,
x = π − arcsin a + 2πn = − arcsin a + π(2n + 1) , n ∈ Z .
am simravleTa gaerTianeba mogvcems sin x = a gantolebis
amonaxsnTa simravles, romelic ganisazRvreba formuliT:
x = (− 1) arcsin a + πk , k ∈ Z .
k
im SemTxvevaSi, roca a udris 0-s, 1-s an –1-s, sin x = a
gantolebis amonaxsnTa simravle SeiZleba ufro martivi
formuliT ganisazRvros, kerZod:
sin x = 0 gantolebis amonaxsnTa simravlea
{x = πk , k ∈ Z } ;
sin x = 1 gantolebis amonaxsnTa simravlea
π
⎧
⎫
⎨ x = + 2πk , k ∈ Z ⎬ ;
2
⎩
⎭
sin x = −1 gantolebis amonaxsnTa simravlea
π
⎧
⎫
⎨ x = − + 2πk , k ∈ Z ⎬ .
2
⎩
⎭
2. hbowjyjmpU! cos x = a ! hboupmfcb/! radgan nebismieri
x –isaTvis −1 ≤ cos x ≤ 1 , amitom, roca a > 1 am gantolebas
amonaxsni ara aqvs.
roca a ≤ 1 , maSin cos x = a gantolebis amosaxsnelad
sakmarisia vipovoT misi yvela amonaxsni nebismieri 2π
sigrZis SualedSi da gamoviyenoT kosinusis perioduloba.
aseT Sualedad mosaxerxebelia aviRoT [− π ; π ] .
[− π ;0] da [0; π ] SualedebSi cos x funqcia monotonuria
da TiToeul maTganze Rebulobs yvela mniSvnelobas –1dan 1-mde (nax. 78). amitom am Sualedebidan TiToeulSi
gantolebas eqneba erTaderTi amonaxsni. arkkosinusis
[0; π ] SualedSi moTavsebuli
gansazRvrebis Tanaxmad
amonaxsni iqneba arccos a . radgan kosinusi luwi funqciaa,
amitom [− π ;0] SualedSi moTavsebuli amonaxsni iqneba
− arccos a .
142
y
1
y=a
a
−π
− arccos a
arccos a
π
x
nax. 78
imisaTvis, rom CavweroT cos x = a gantolebis yvela
amonaxsni, visargebloT kosinusis periodulobiT, e. i.
miRebuli ori amonaxsnidan TiToeuls davumatoT 2πk ,
(k ∈ Z ) , saxis ricxvi. miviRebT, rom amonaxsnTa simravle
Sedgeba Semdegi ori usasrulo qvesimravlisagan:
{x = arccos a + 2πk , k ∈ Z },
{x = − arccos a + 2πk , k ∈ Z }.
miRebuli simravleebis gaerTianeba mogvcems cos x = a
gantolebis amonaxsnTa simravles
{x = ± arccos a + 2πk , k ∈ Z }.
im SemTxvevaSi, roca a udris 0-s, 1-s an –1-s cos x = a
gantolebis amonaxsnTa simravle SeiZleba ufro martivi
formuliT ganisazRvros, kerZod:
cos x = 0 gantolebis amonaxsnTa simravlea
π
⎧
⎫
⎨ x = + πk , k ∈ Z ⎬ ;
2
⎩
⎭
cos x = 1 gantolebis amonaxsTa simravlea
{x = 2πk , k ∈ Z } ;
cos x = −1 gantolebis amonaxsTa simravlea
{x = π + 2πk , k ∈ Z } .
3. hbowjyjmpU! tgx = a ! hboupmfcb/ am gantolebis
amosaxnelad sakmarisia vipovoT misi yvela amonaxsni
nebismieri π sigrZis SualedSi da gamoviyenoT tangensis
perioduloba. aseT Sualedad mosaxerxebelia aviRoT
⎤ π π⎡
⎥ − 2 ; 2 ⎢ . am SualedSi tgx funqcia zrdadia da Rebulobs
⎦
⎣
yvela mniSvnelobas −∞ –dan +∞ –mde (nax. 79), amitom
143
⎤ π π⎡
SualedSi
gantolebas
eqneba
erTaderTi
⎥− 2 ; 2 ⎢
⎦
⎣
amonaxsni.
arktangensis
gansazRvrebis
Tanaxmad
es
amonaxsni iqneba arctga .
y
y=a
a
−
π
2
arctga π
2
x
nax. 79
imisaTvis, rom CavweroT tgx = a gantolebis yvela
amonaxsni, visrgebloT tangensis periodulobiT, e. i.
miRebul amonaxsns davumatoT πk , (k ∈ Z ) , saxis ricxvi.
miviRebT, rom tgx = a gantolebis amonaxsnTa simravlea
{x = arctga + πk , k ∈ Z } .
4. tgx = a gantolebis amoxsnis dros Catarebuli
msjelobis analogiurad SeiZleba vaCvenoT, rom ctgx = a
gantolebis amonaxsnTa simravlea
{x = arcctga + πk , k ∈ Z } .
zemoT
ganxiluli
gantolebebi
warmoadgenen
umartives
trigonometriul
gantolebebs.
sazogadod,
trigonometriuli
gantolebis
amosaxsnelad
saWiroa
igivuri gardaqmnebis saSualebiT igi Seicvalos misi
tolfasi umartivesi trigonometriuli gantolebiT an
aseT gantolebaTa gaerTianebiT.
§69. ybsjtyj!jsbdjpobmvsj!nbDwfofcmjU!
!
CvenTvis cnobilia, Tu rogor ganisazRvreba xarisxi
racionaluri maCvenebliT. racionalurmaCvenebliani xarisxebis gamoyenebiT ki ganisazRvreba iracionalur maCvenebliani xarisxi. Tu α raime iracionaluri ricxvia, maSin
yovelTvis arsebobs β1 da β2 racionaluri ricxvebi iseTi,
rom
144