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vovih1 про Бутырская: Сага о Кае Эрлингссоне. Трилогия (Самиздат, сетевая литература)

Будем ждать пока напишут 4 том, а может и более

Рейтинг: 0 ( 0 за, 0 против).
vovih1 про Кори: Падение Левиафана (Боевая фантастика)

Galina_cool, зачем заливать эти огрызки, на литрес есть полная версия. залейте ее

Рейтинг: +1 ( 1 за, 0 против).
Влад и мир про Шарапов: На той стороне (Приключения)

Сюжет в принципе мог быть интересным, но не раскрывается. ГГ движется по течению, ведёт себя очень глупо, особенно в бою. Автор во время остроты ситуации и когда мгновение решает всё, начинает описывать как ГГ требует оплаты, а потом автор только и пишет, там не успеваю, тут не успеваю. В общем глупость ГГ и хаос ситуаций. Например ГГ выгнали силой из города и долго преследовали, чуть не убив и после этого он на полном серьёзе собирается

подробнее ...

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Serg55 про Берг: Танкистка (Попаданцы)

похоже на Поселягина произведение, почитаем продолжение про 14 год, когда автор напишет. А так, фантази оно и есть фантази...

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Влад и мир про Михайлов: Трещина (Альтернативная история)

Я такие доклады не читаю.

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Stribog73 про Гиндикин: Рассказы о физиках и математиках (Физика)

Не ставьте галочку "Добавить в список OCR" если есть слой. Галочка означает "Требуется OCR".

Рейтинг: +1 ( 1 за, 0 против).
lopotun про Гиндикин: Рассказы о физиках и математиках (Физика)

Благодаря советам и помощи Stribog73 заменил кривой OCR-слой в книге на правильный. За это ему огромное спасибо.

Рейтинг: +2 ( 2 за, 0 против).

Интересно почитать: Как использовать VPN для TikTok?

ЕГЭ 2020. 100 баллов. Математика. Профильный уровень. Тригонометрические уравнения [Юрий Садовничий] (pdf) читать онлайн

-  ЕГЭ 2020. 100 баллов. Математика. Профильный уровень. Тригонометрические уравнения  (и.с. ЕГЭ. 100 баллов) 1.68 Мб (скачать pdf) (скачать pdf+fbd)  (читать)  (читать постранично) - Юрий Викторович Садовничий

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0cTB0 «3K3aMeH», 2020. — 110, [2] c. (CepHH «Er3.
100 OajuioB»)
ISBN 978-5-377-15020-6
XlaHHaa KHHra nocBBiijeHa 3aAanaM, aHajioruHHBiM 3 aflane 13 E r 3
no MaTeMaraice (TpHroHOMeTpuqecKoe ypaBHeime). PaccMaTpHBaioTCH
pa3HHHHbie MeTOflbi pémeHiw TaKHx ypaBHeHHH, a Tandee pa 3JiHHHi»ie
cn o co S ti OT6opa KopHen. Kmcra 6yaeT nojie3Ha ynainHMCH cTapnmx
KJiaccoB, yHHTejwM MaTeMaTHKH, peneTHTopaM.

ripnKa30M Na 699 MnmiCTepcTBa o6pa30Bamia h Hayicn PocchhOe^epaimn yne6Hí»ie noco6iw H3flaTejiLCTBa «3K3aMeH» AonymeHLi k Hcnojn»30BaHHK) b o6meo6pa30BaTejn>Hbix opraHH3am«ix.

ckoh

YAK 372.8:51
EEK 74.262.21

OopMaT 60x90/16. TapHHTypa «’IIlKOJibHaa».
EyMara ra3eTHaa. Y h.- h3a . ji. 3,04.
Ycji. nen. ji. 7. Tapare 5000 3K3. 3aica3 N° 1514.

ISBN 978-5-377-15020-6

© CaaoBHHHHH K). B., 2020
© H3£aTejibCTBo «3K3AMEH», 2020

OrJIABJIEHHE

B B e,zi;eH H e....................................................................................................................................5

rJIABA 1. ÜPE0BPA30BAH H E
TPHrOHOMETPHHECKHX BblPAHCEHHÍÍ......................... 7

§ 1.1 . OCHOBHbie $OpMyJIbI TpnrOHOMeTpHH.......................... 7
§1.2 . flO K a 3 a T e JIb C T B O TOtfCfleCTB
h y n p o m ¡ e H H e B b i p a ^ c e H H H .................................................................. 10
3adanu djui caMocmosimejibHOZo peiuem ui ...............16
§ 1.3. 3a,n;a*m H a B b n m c jie H H e b T p H r o H O M e T p n n ......................... 18
3adanu djui caMOcmoamejibHozo peuiem ui ...............23
rJIABA 2. OCHOBHblE METOAbI PEIHEHHH
TPHrOHOMETPHHECKHX yP A B H E H H Ü ........................ 25

§2.1 . IIpocTeHiime TpnroHOMeTpnHecKHe
ypaBHeHHH......................................................................25

§ 2.2. CBeflemie T p H r o H O M e T p ir c e c K o r o
ypaB H eH H H

k

K B a,n ;p a T H O M y ................................................................ 26

3adauu Ojul caMOcmosimejibHOZO pemenusi ...............30

§ 2.3.

P a 3 JiO H c e H H e H a

............................................................ 31

3adanu djui caMOcmosimejibHozo pemenusi ...............35

§ 2.4.

Ü O H H JK e H H e

.................................................................. 36

3adanu djm caMOcmosimejibnozo peuiem iñ ...............39

§ 2.5. BBe^eHHe flonojiHHTejibHoro yrjia ............................... 39
3adanu Ojm caMocmoamejibHOZo peiuenun ...............44
rJIABA 3. OTBOP KOPHEñ
B TPHrOHOMETPHHECKHX yPA BH EH H H X ................. 45

§ 3.1. Ot6op

K opH eñ n p n noM om H

T p H r o H O M e T p ir q e c K o r o H e p a B e H C T B a ............................45

3adanu Ojisi caMocmojimejibHOzo peuiem ui ...............57

3

§ 3 . 2 . Or6op KopHeñ

b

npoMencyTOK

Ha HHCJIOBOH npHMOH................................................................... 58

3adan.u djui caMocmosimejibHoeo peuiemix............... 69
§ 3.3.

HaxottCAeHHe o6 id,h x KopH eii
AByx TpHroHOMeTpHnecKHx ypaBHeHHH............................71

3adanu djia caMOcmosimejibHOZo pewenua............... 61
rJIABA 4. PEÜIEHHE CHCTEM
TPHrOHOMETPHHECKHX yPABHEHHÍl .................... 83
3adanu djm caMOcmosimejibHOZO pemenusi............... 91
TJIABA 5. PEmEHHE TPHrOHOMETPHHECKHX
HEPABEHCTB..........................................................................93
3adav,u Ojia caMOcmosimejibHoeo peuienusi............. 102
ÜTBeTM k 3aAanaM
AJiH caMOCTOHTejn>Horo pemeHHH............................................ 103

4

BseA eH H e

flaHHafl KHHra nocBHiijeHa 3aflanaM, aHajiormnibiM 3aaane
npot})HJibHoro E r 3 no MaTeMaTHKe (TpnroHOMeTpHnecKoe
ypaBHeHne). KHHra pa30HTa Ha rjiaBbi no TeMaM, MaTepnaji b
Kanc^oH rjiaBe no^aeTCH « ot npocToro k cjiohchomy ».
13

3a,n¡aHa 13 CHHTaeTCH caMOH npocTOH 3 a ^ a n e ñ b tom 6noK e
E r 3 n o MaTeMaTHKe, b kotopom npe^nojiaraeT C H pa3BepH yToe
pem eH H e. 0,n;HaKO cjiohchocth MoryT B03HHKHyTb KaK n p n p emeHHH TpnroHOMeTpHHecKoro ypaBHeHHH, b tom n n c jie H3-3a
ó o jib m o ro n n c jia Heoóxo^HM bix (})opMyji, T an h n p n n o c n e a y io m;eM oTÓope K opH eñ. B flaHHOH KHHre CHCTeMaTH3HpoBaHbi M e TOflbi pemeHHH TpHroHOMeTpiraecKHX ypaBHeHHH, aaH bi Bce H e oóxoflHM bie (JjopMyjibi, a TaKHce noKa3aH bi pa3JiHHHbie cn o co ó b i,
K O T O p b IM H MOHCHO B f la jI b H e H i n e M O T Ó H p a T b K O p H H .
I l e p B a H r j i a B a H B jin e T C H n o ,n ¡ro T O B H T e jib H O H , b H e ñ a a H b i Bce
H e o ó x o f l H M b i e < } )o p M y jib i h n o K a 3 a H o , K a K c h x n o M o n ^ b i o mohcho n p e o 6 p a 3 0 B b i B a T b
T o p b ie 3 a fla n H

T p n ro H O M e T p H H e c K H e

B b ip a n c e H H H . H e K o -

stoh r j i a B b i n o c B H m e H b i B b n m c jie H H H M

b T p n ro -

H O M eT pH H .
B o B T o p o ii r jia B e n o K a 3 a H O p e m e H H e n p o c T e H n iH X T p n ro H O M eT pnnecK H X ypaB H eH H H ,
f lb i ( c B e ^ e H H e
T .f l.)

pem eH H H

3a^ann

3 toü

a

T a K H ce n p H B e f l e H b i o c H O B H b ie M eT O -

k K B a^paT H O M y,

p a 3 J io H c e H H e

T p H r o H O M e T p ir a e c K H x

H a mhohchtcjih h

ypaB H eH H H .

H e K O T o p b ie

r j i a B b i ,n;ocTaTO H H O cjiohchbi, o c o 6 e H H O T e , b koto -

p b i x H c n o ji b 3 y e T C H M eT O fl B B e ^ e H H H f l o n o j i H H T e j i b H o r o y r n a .

TpeTbH rjiaB a HBjineTCH ochobhoh b a hh h o h K H H r e . B H e ñ
noKa3aHbi pa3JiHHHbie cnoco6bi oTÓopa KopHeá b TpnroHOMeTpnnecKHX ypaBHeHHHx, TaKHe KaK ot 6 o p HepaBeHCTBOM, ot 6 o p
b npoMencyTOK, HaxoncfleHHe o óiijh x KopHeñ flByx TpnroHOMeTpnnecKHX ypaBHeHHH. M ohcho 6e3 npeyBejiHneHHH CKa3aTb, hto HHTaTejib, ocBOHBmHH flaHHyio rjiaB y, He flOJiHceH
HMeTb HHKaKHx npoóneM n p n pemeHHH 3aaanH 13 E r 3 n o Ma­
TeMaTHKe.

5

HaKOHen,, ^eTBepTan h nnTan rjiaBbi nocBHm¡eHBi Gojiee
3a,u:aHaM. B neTBepToií rjiaBe paccMaTpHBaiOTCH pa 3 jiHHHBie cnocoóbi pemeHHH chctcm TpHroHOMeTpHnecKHx ypaBHeHHH, b nHTOH rnaBe noKa3aHbi HeKOTopbie npHMepbi pemeHHH
TpHrOHOMeTpHHeCKHX HepaBeHCTB.
cjiohchbim

R anchan rjiaB a KHHrH coflepncHT TeopeTHnecKHH MaTepnaji,
HecKOJibKO pa3oópaHHbix npHMepoB, b kotopm x fleMOHCTpnpyiotch pa3JiOTHbie MeTOflbi pemeHHH 3aflan n o flaHHOH TeMe, a
Tandee 3a,n;anH ,n¡jiH caMOCTOHTejibHoro pemeHHH, CHaónceHHbie
OTBeTaMH.
A btop Ha,n¡eeTCH, hto f la H H a n K H H ra 6 y # e T n o j i e 3 H a y n a iijhmch C T a p m n x K J ia c c o B ¿jjih caM O C TO H TejibH O H n o flro T O B K H k
Er3 ,

a

TaK H ce

y n H T e jiH M

M aT eM aT H K H ,

T eM , KTO H H T e p e e y e T C H flaH H O H T eM O H .

TKejiaeM yenexoe!

6

peneT H T opaM

h

B ceM

rJIABA 1
Ü P E 0B P A 30B A H H E
TPHrO H O M ETPHM EC KHX BbIPAJKEHH H

§ 1 . 1 . O C H O B H B ie (JjO p M y jIB I
T p H rO H O M G T p H H

B flaHHOM pa3Aejie

m m npHBeAeM npaKTHnecKH Bce $opM yTpnroHOMeTpHH, BCTpenaiomHecH n p n pemeHHH 3 a a a n Ha
npeo6pa30B aH H e TpHroHOMeTpHnecKHx BbipaHceHHÍi, a Tandee
n p n pemeHHH TpnroHOMeTpHHecKHX ypaBHeHHH. KpoM e T oro,
npHBOflHTCH TaÓJIHI^a 3HaneHHH TpnroHOMeTpHHecKHX (JjyHKIJHH
OCHOBHBIX yrjIOB.

jibi

1) Ta6jrau;a 3HaneHHH TpnroHOMeTpHHecKHx (JjyHKi^HH cjieAyiom¡HX yrjioB nepBOH neTBepTH:

cosa

CD
O
00

sin a
1
2

4 5 °í—1
UJ

i

60t f )

s

&
2
1

&
1
2

2

ctg a

tg a
i

s
1

1

s

i

2) O opM yjibi npHBefleHHn:

n ±l a | =--------s in | —
cosa;
s i n 0 ^ ± a j = -co sa ;
sin( 7i+ a ) = -sin a ;

— —- a
eos

12

J

= sin a; eos —+ a = - sin a;

. a;
c o sÍ|^Sn
-^ -aa ^|| =
= -s
-s in
in
——
a;

K2

J

371 ^
eos f —
cosí
+ a l = sm a;

sin(Tr-a) = sin a; cos(7t±a) = -co sa ;

tg^-|TaJ = ±tga: c tg ^ + a j = ±ctga.
7

3) PaBeHCTBO sin2 a + cos2 a = 1 , cnpaBeAJiHBoe

ajih

Bcex

aHa^eHHH a , Ha3MBaeTCH ocHoenbiM mpuzoHOMempunecKUM
mowdecmeoM. Ü3 otoh #opMyjiw cjie^yioT eme ABe #opMyjii>i:

1 + tg2 a = — \ - y a * —+ nn;
cos2a
2

1

2

1 + ctg a =
4)

r—, a * n n ;
sin a

neZ;
neZ.

OopMyjibi cjiOMceHH^:
cos(a + p) = eos a eos p - sin a sin p;
cos(a - P) = eos a eos p + sin a sin P;
sin(a + P) = sin a eos p + eos a sin P;
'sin(a - p) = sin a eos p - eos a sin P;
tg(a + p)
tg (a -P )

tg a + tgp
a ,p ,a + p * —+ nn; n e Z ;
1 -tg a tg p ’
2
tg a - tg P
a, p, a + p * ~ + nn; n e Z .
1 + tg a tg p ’

5) OopMyjibi ABOHHoro

h

T p otaoro apryMeHTOB:

sin 2a = 2 sin a eos a;
eos 2a = eos2 a - sin2 a = 2 eos2 a - 1 = 1 - 2 sin2 a;
tg2a =

2 tg a
I

. . n . nn
n
a * 4 + T ’ a ^ 2 +7r7i;7lGZ;

sin3a - 3 s i n a - 4 s i n 3 a = sin a (3 -4 sin 2 a);
eos 3a = 4 eos3 a - 3 cosa = cosa(4cos2 a - 3 ) .
6) OopMyjibi noHHHceHHH CTeneHn:

sin2a = í r c° s 2 a ; cos2a = l± c o s 2 a .
2

tg2a = l - c o s 2 a .
1 + eos 2 a ’

sin3 a = ~ sin a ~ sin3a . cog3 a = eos 3 a + 3 eos a
4
A

8

7 ) OopMyjibi npeo 6 pa 30 BaHHH cyMMbi b npoH3BefleHHe:

cosa + cosp = 2cos^——eos——
2
2
eos a - eos B = -2 sin ^ + ^ sin ——
2
2
sin a + sinB = 2 sin —+—eos——
2
2
• o o . cc-B
a+B
sin a - s in B = 2sm --- -e o s -----

2

2

8 ) OopMyjiti npeo 6pa 30BaHHH npoH 3BefleHHH

b

cyMMy:

eos a eos P = —(cos(a - p) + cos(a + P));

2

sin a sin P = —(cos(a - P) - cos(a + P));

2

sin a eos p = —(sin(a + P) + sin(a - P)).

2

9)

d>opMyjibi, Hcnojib3yiomHe TaHreHC nojioBHHHoro apry-

MeHTa:

to |P

2 tg |
l - t g 2|
sin a = -------— ; cosa = ---------- —, a * 7r + 27 t7i; u g Z;
i + t g 2|
i + t g 2|

_ s in a _ = Id eo sa. ^
1+co sa
sina

a .I d e o s a
2 1 + cosa

a * n+2m. n 6^

9

§ 1 .2 . ,HoKa3aTeJii>cTBO toska ^ c t b
h ynpomemie BwpasKBHHH
npHBefleM HecKOJiBKO npHMepoB «oKasaTeabCTsa tphfoho MeTpHHeCKHX «WWeCTB H ynpomeHHfl Bbipa*eHHH C HCn0JIb30BaHHeM paccMOTpeHHbix B npeflbiflymeM naparpa4>e «opMjui.
rtpHMep 1. fl 0 Ka3aTb,
ax

a

hto

npn Bcex flonycTHMbix 3HaaeHH-

cnpaBeflJiHBO paBeHCTBO
1 - sin 2a
1 + sin 2a

Peuienue. IIpeo6pa3yeM jieByio nacTB flam o ro paBeHCTBa
cjieayiomHM o6pa30M:

l-c o s (j-2 a j

hx

l+ c o s |j ^ -2 a

1 - sin 2a
1 + sin 2a

IIpHMep 2. floicasaTb, nTO npn Bcex AonycTHMbix 3HaneHHp cnpaBe,n¡jinBO paBeHCTBO
2 sin
= -V2 ctg p.
2 eos

Peuiemie. IIpeo6pa3yeM jieByio nacTb flaHHoro paseHCTBa
cjieflyioiijHM o6pa30M:
2 sin ^ + pj - V2 sin p

2 ^sin ^ eos p + eos ^ sin pj - yÍ2 sin p

2 eos ^ + pj - 73 eos p

2 ^cos ^ eos p - sin ^ sin pj - yfs eos p

_ (V2 cosp + >/2 sin P )-V 2 sinP _ V2 cosP _
(V 3cosp -sinp )-> /3cosp
-sin p
10

/77

IIpuMep 3. floKa3aTt,

hto

npn Bcex 3HaneHHHx a cnpaBefl-

JIHBO p a B e H C T B O

sin2(30° + a ) - sin2(30° - a ) = ^

- s iV2 a .

Peuienue. IIpeo6pa3yeM jieByio nacTt flaHHoro paBeHCTBa
cjie^yiomHM o6pa30M:

sin2(30° + a ) - sin2 (30° - a ) =
_ 1 - cos(60° + 2a)
2

1 - cos(60° - 2a) __

2

~

_ cos(60° -2a)-cos(6Q ° + 2a) _

2
=

. (60° - 2 a ) + (60° + 2a)

Sm

2

Sm

(60° - 2a ) - (60° + 2a) =

2

= - sin 60° sin (-2 a ) =
IIpHMep 4. floKa3aTb, hto npn Bcex flonycTHMMX 3HaneHHíix a cnpaBeAJiHBo paBeHCTBO

1 +sin 2a-feos 2a = ctga.
.
--------------------1+sin 2 a -e o s 2a
Peuienue. IIpeo6pa3yeM jieByio nacTb flamoro paBeHCTBa
cjieAyiomHM o6pa30M:

l + sin 2a + cos2g _
1+sin 2 a -e o s 2a
_ ( l + cos2a) + sin 2a _
(l- c o s 2 a ) + sin2a
_ 2cos2 a + 2 sin a c o sa _
2sin2 a + 2 sin a c o sa
2cosa(cosa + sin a ) _ cosa _ c^.
2 sin a (s in a + cosa) sin a

11

n pHMep 5. floKaaaTb,
hx

hto

npn Bcex flonycTHMbix sHaneHH-

x cnpaBeflJiHBO paBeHCTBO

sin(B+ » ) c o s ( ^ - x ) t g ( x - ^ = ^

^

C0S( 2 + * ) C° S( ^ f + ^ 2 ( 71+ * )

Peuienue. IIpeo6pa3yeM JieByio ^acTb aairaoro paBeHCTBa
cjieflyiomHM o6pa30M:
sin(7t + x) eos ^

- x j ■gt [x - 1 j

cos| ^ + x ) cos[ ^ + x J t g ^ + x )
= (~Sin x ) • ( - sin x) • ( - c tg x) =
( - s in jc ) ( s in ;t)( - tg jc )

ctgx = _ ctg2 ^
tgx

IIpHMep 6. IIpH Bcex flonycTHMLix sHa^emwix q ynpocTHTB
BtipaxceHHe
l + cosq + cos2q + cos3q
cosa + 2cos2 a - 1
Peuienue. IIpeo6pa3yeM #aHHoe Bbipa^ceHHe cjie^yiomHM
o6pa30M:
l + cosq + cos2q + cos3q _
eos q + 2 eos2 q - 1
_ (1 + eos 2q) + (eos q + eos 3q) _
cosq + cos2q
= 2cos2cc+ 2 eos q eos 2q _
cosq + cos2q
= 2 eos q(cosq +eos 2 q^
coSa + cog2a
O tb e t : 2cosq.

12

2cosq.

ripiiMep 7. ripH Bcex ^¡onycTHMbix SHaneiniflx a ynpocTHTb
BbipanceHHe

_2_____ 2
sin a sin 3a

4cos2a
sin 3 a *

Peuienue. IIpeo6pa3yeM flamioe BbipaHceirae cjie^yiomHM
o6pa30M:
2_____ 2
sin a sin 3a
_

4cos2a _
sin 3a

2__________ 2____________4 eos 2a
sin a s in a (3 -4 s in 2 a) s in a (3 -4 s in 2 a)
2 ( 3 -4 s in 2 a ) - 2 - 4 c o s 2 a
sin 3a
4 - 8 s i n 2 a - 4 ( l - 2 s i n 2a) _
sin 3a

q

O t b e t : 0.
IIpHMep 8. Ilp n Bcex flonycTHMbix 3HaneHHHx x ynpocTHTb
B b ip a H c e H H e

l +t g x t g 2 x
t g x +c t g x *
Peuiemie. IIpeo6pa3yeM aairaoe BbipaHceHHe cjie^yiomHM
o6pa30M :

l + t g x t g 2 x _ í ^ [ suia: sin 2 * ^ .^ sinA: ^ cosa:^
t g x +c t g x
v cosa: cos2 a:J I cosa: sinA:J
_ eos x eos 2 a: + sin x sin 2a: . sin2 x + eos2 x _
eos x eos 2a:
sin a:eos a:
cosa:
. 2 _ tg2A:
eos x eos 2a: sin 2a:
2

O tb e t : —t g 2 x .
2
13

IIpHMep 9. rip n Bcex flonycTHMtix 3 Ha^eHHHX a ynpocTHTb
BbipanceHHe

y¡2eos a - 2 cos(45° - a ) + ^ tg a
2 sin(30° +a)-\ÍS sin a
Peuienue. IIpeo6pa3yeM ,n¡aHHoe BbipaHceinie cjieflyiomHM
o6pa30M:

y¡2c o s a -2 c o s (4 5 ° - a )
2 sin(30° + a ) - y¡3sin a

y¡2tga =

V 2 c o sa -2 (c o s4 5 ° cosan-sin 45° sin a ) | J ñ ^ a _
2(sin 30° eos a + eos 30° sin a ) - V3 sin a
V2 cosa-2|— cosa + — sina j
l 2

2

r

:— J------ J=---- v------- - + v2 tg a =
2 — cosa + ^— sina ->/3 sina
= ^ s i n a +^ t g a = 0 _
O t b e t : 0.
IIpHMep 10. Ilp n Bcex flonycTHMbix 3HaneHHHx a ynpoBbipaHceHHe

cthtb

tg3a-tg|j^-ajtg|j^ + ajtga + l.

Peuienue. IIpeo6pa3yeM flairaoe BmpajKeHHe cjieflyiomHM
o6pa30M:
tg3a-tg^-ajtgí^ + ajtga + l =

si*1! ? ~ a ]sin [-^ + a ]s in a

■tgsa—

i;

J

3

--- +1.

C°S 13 ~ a J C°S 1 3 + a )C0S a
14

~

í eos 2a - eos — ]sin a

= t g 3 a " i r -------------i } —

+1=

— cos2a + cos— Icosa
2\
3)
í cos2a + i Jsina
= tg 3 a - ^ -----------------+ 1 =
( eos 2 a - - I c o s a
tg 3 a

(2(1- 2sin2a) + l ) sina , x
(2(2 eos2 a - 1) - 1) eos a

= t g 3 a _ M 5 « z 4 s i n ! a +1 =
4cos a - 3 c o s a
= tg 3 a - ^ a + l = l.
eos 3a
O t b e t : 1.
IIpHMep 11. Ilpn Beex 0 < a < 90° ynpocTHTb BBipatfcemie
Vl + sin a - V l- s in a
4 sin—
2

Peiuenue. IIpeo6pa3yeM aaimoe Bbipa^ceHHe cjieayiomuM
o6pa30M:

Vl + s in a - V l- s in a _
4 sin—
2

4 sin—
2

.a
a
sin—+ eos— -sin— eos—
2

2

2

2

4 sin—
15

s in c|l + c o sa- Ji + ^í s•i na- - c o s a-

1
2’

4 sin “
a ^

. a

TBK KaK eos — > sin

2

2

npn 0 < — < 4 5 ° .

2

O t b e t : 0 ,5 .

S a^am i fljia caMOCTOHTejibHoro peuieHHH
1.

floKa3aTb, hto npn Bcex ^onycTHMBix 3HaneHHHX a cnpaBeflJIHBO paBeHCTBO

sinq+cosa = tg 2a +^ — .
eos a - s i n a

2.

eos 2a

floKasaTb, hto npn Bcex ,n¡onycTHMBix 3HaneHHHX a cnpaBeflJIHBO paBeHCTBO
l + s i n a - 2 s i n 2 Í45° - —1
---------------------- ^-------- — = sin —.
4cos —

3.

^

floK aaaT b , hto n p n B cex #onycTH M bix 3HaneHHHx a cn p a BeflJIHBO paBeHCTBO

sin a
, sin a + cosa
--------------+ --------- 2------ Slna = cosa.
s in a - c o s a
1 -tg a
4.

,Zi;oKa3aTb, hto npn Bcex flonycTHMbix 3HaneHHHx a cnpaBeflJIHBO paBeHCTBO
sin a + s in —

_________ 2_
1 + cosa + eos—

2

16

5.

floKa3aTb, hto npn Bcex flonycTHMbix 3HaneHHax a cnpaBeflJIHBO paBeHCTBO
t g r n +J = l ± sin2a_

v4
6.

J

cos2a

¿JoKaaaTb, hto npH Bcex ,n;onycTHMbix 3HaneHHHx a cnpaBeflJIHBO paBeHCTBO

tg q + t g 2 q - t g 3 q = - t g q - tg 2 a -tg 3 a .
7.

IIpn Bcex AonycTHMbix 3HaHeHHHX a ynpocTHTb BbipanceHHe
1 -s in 42 a -c o s4 2a | ^
2 sin 4 2q

8.

Ilpn Bcex flonycTHMbix 3HaneHHHx q
HceHHe

C°S[ 2 _ a J C° Sv 2 + y _ C0S^TC~

h

p ynpocTHTb Bbipa-

cos(2rc “ P)

sin | —+ q + p
9. Ilpn Bcex flonycTHMbix 3HaneHHHX q ynpocTHTb BbipanceHHe
(sin q + co sq )2 - 1

2tg2 ^

tg[ —- q |- s in q c o s q

10.

n p n Bcex ^onycTHMbix 3HaneHHHx q ynpocTHTb BbipanceHHe

2(sin 2q + 2 eos2 a - 1)
eos q - sin a - eos 3q + sin 3q
11.

ü p H Bcex aonycTHMbix 3HaneHHHx q ynpocTHTb BbipanceHHe

2 s in q c o s q -

sin q - sin(7r + 3 q ) + sin 2q
2c o sq + l

17

12. IIpn Bcex

f lo n y c T H M b ix

3HaneHHHX x ynpocTHTb BbipaaceHHe

sin 2x sin ^ + x j + sin x cos(tt- 2x)

13. Ilpn Bcex #onycTHMbix 3HaneHHHX a ynpocTHTb Bbipaaceime
tg(180° -a )c o s(1 8 0 ° - a )tg ( 9 0 ° - a)
sin(90° + a)c tg (9 0 ° - a )tg ( 9 0 ° + a)

§ 1 .3 . 3 aflaH H H a B M ra c jie H H e

b

TpH roH OM eTpH H

PaCCMOTpHM HeCKOJIbKO 3a#aH, CBH3aHHbIX C BbinHCJieHHeM
3HaHeHHH TpHroHOMeTpnnecKHx BbipaaceHHH. Tan ace na k h
npn pemeHHH 3a^an npeflbiflyin;ero naparpanmcjiHTb
tg9° -tg 6 3 ° + tg81° -tg 2 7 °.

Pemenue. IIpeo6pa3yeM flairaoe BBipaHcemie cjieflyiomHM
o6pa30M:
tg 9o-tg 6 3 ° + tg81° - tg 2 7 ° =
sin 9° | sin 81°
eos 9° eos 81°
sin 9° eos 81° + eos 9° sin 8 i°
cos9°cos81°
_

sin 27° t sin 63°
eos 2 7° eos63°
sin 2 7 °co s6 3 °+ cos27° sin63°
cos27°cos63°

_
sin 90°_______ sin 90°
cos9°cos81° cos27°cos63°
1____________ 1
cos90cos81° cos27°cos63°

2____________ 2
eos 72° + eos 90° eos 36° + eos 90°

2
eos 72°

2
eos 36°

_ 2(cos36° -co s72°) _ 4sin54° sin l8 ° _
eos 36° eos 72°
eos 36° eos 7 2°
4cos36° cos72° _ ^
eos 36° eos 72°
O t b e t : 4.

19

IIpHMep 4. B bihhcjihtb

sin 70° sin 50° sin 10°.
Peuienue. IIpeo6pa3yeM aainioe BBipaaEcemie cjieflyiomHM
o6pa30M:

sin 70° sin 50° sin 10° =
= —sin 70° eos 40° - —sin 70° =
2
4
= - ( s in ll0 °+ s in 3 0 °) - - s in 7 0 ° =
4
4
= -jsinllCT + i - - s i n 7 0 ° =
4
8 4
4sin70°
4

sin70° =
8

4

8

O t b e t: 1 / 8 .
IIpHMep 5. H3BecTH0, HTO sin a = - - ^ ,

7i

me: (eos 1 + sin 1) hjih
4 9 /3 6 ?
Peuienue. IIpeo6pa3yeM flaHHoe b ycjiobhh 3a^aHH TpnroHOMeTpimecKoe BMpaHcemie cjie^yiomiiM o6pa30M:

cosl + s i n l - yÍ2\— co sl+ — s in ll =

l2

= y¡2 ^cos eos 1 + sin

22

2 J

sin 1j = >¡2 eos | l -

.

Tan Kan (J)yHKu;HH y = cosx Ha mrrepBajie 0 < x < ^
BaeT

h

1 < ^ nojiynaeM,

y6bi-

hto:

^ c o s ( l- g > V 2 c o s ( |- j) =
= >/2 í eos—eos—+ sin —sin —1 =

[

3

flonaHceM T e n e p t,

4

hto

3

4)

2

f l ji a 3Toro cpaBHHM 3 th

^

HHCJia. ÜMeeM:
— V
36

o 98 V 3 6 ^ + 3 6 /3 • » 31 v 18%/3 961 v 972.
TaK KaK 961 < 9 7 2 ,

to h


36

2

TaKHM o6pa30M, H3
49
36

nojiyneHHbix flByx HepaBeHCTB cjieflyeT, hto c o s l + s i n l > — .

O t b e t : IlepBoe

hhcjio

Sojibine.

3aAa*iH ^ jih caMOCTOHTejibHoro pemeHHH
1.

B b lH H C JIH T b C t g — - t g —.

2.

BbnracjiHTb 8 eos 260° sin 130° eos 160°.

3.

B b lH H C JIH T b

20sin80° sin 65° sin 35°
sin 20° + sin 50° + sin 110o

4.

B b IH H C JIH T b

16sin251° -1 0 c o sl6 1 o
eos 19°

8

8

23

5.

BbIHHCJIHTb c o s lO °

cos50° cos70°

6.

HañTH tg 2 a , ecjiH sin a = —, a s i n 4 a > 0 .

7.

HaiiTH c o s ^ 2 a - |j , ecjiH tg a =

8.

1
HañTH t g 2 a , ecjin s in a = -¡-

5

4

h

7
9. HañTH sin—, ecjra eos 2a < —
2
8

JI
0 < a < -.

¿i

h

1
c o s a < —-.
4

10. BbiHHCJiHTb (sin a -c o s a )(s in p -c o s P ), ecjin sin(a + P) = 0,8
h cos(a-P) = 0,3 .
11. BtnmcjiHTb tg 3 a , ecjiH H3BecTHO,
12. HañTH sin^-^ + 2 a j,

hto

s in a = 2 c o sa .

eCJIH H 3B eC T H O , hto

13. BbiHHCJiHTb t g ^ - ^ + 2 a j,
Y

tg a = — .
4

3
ecjin cosa = — h
5

71



2

sin x = -1 x = ~ —+2nk,

¿i

s in a: = 1 x = —+ 2nk.

2

3#ecb k — Jiioóoe n¡ejioe h hcjio . B aajiBHeHmeM mm 6yaeM
nncaTi» k e Z .
ypaBHeHHe BH^a cosa: = a paBHOCHJiBHO cjie^yiomeii cobo KynHocTH:
cosa : =

a

A: = arccosa+27ifc,
x = ± árceos a + 2nk
x = - árceos a + 2nk

npn a e ( - l , 0 ) u (0 ,1 ). E cjih |a |> l , to ypaBHeHHe cosA: = a penieHHH He HMeeT. E cjih a = 0 hjih a —± 1 , HMeeM:

25

cosa: = - l o x = n +2nk,

co s* = O o * = - + **.
di

e o s* = 1 x = 27T&.
ypaBHeHHH BH^a tg x = a

h

c tg x —a npH juo 6 m x fleñcTBH-

TejiBHMx a peinaioTCfl cneflyiomHM o6pa30M:

t g x = a x = a r c t g a + 7iAj,
c t g x = a * = a rc c tg a + nk.

§ 2.2. CBeAeHHe TpHroHOMeTpiraecKoro ypaBHeHHH
KBaAparaoMy

k

OflHHM H3 ochobhmx MeTOflOB pemeHHH TpHroHOMeTpiraeypaBHeHHH HBJineTCH CBefleHHe ypaBHeHHe k KBa^paTHOMy
OTHOCHTeJIBHO HOBOH HepeMeHHOH. IIpH 3TOM, eCJIH Mbl B KaneCTBe hoboh nepeMeHHOH 6epeM CHHyc hjih KOCHHyc KaKoro-Jinóo
apryMeHTa, to HeoGxoflHMo OToópaTi» tojibko Te k o ph h nojiyneHHoro ypaBHeHHH, KOTopwe no Moayjiio He npeBocxo^HT e^HHHH¡y.

ckhx

IIpHMep 1 . PemHTB ypaBHeHHe
eos 2* = 1 - s i n * .

Peuienue. IIpeo6pa3yeM flaHHoe ypaBHeHHe cjieflyiomHM 06pa30M:
cos2* = l-sin*
1 - 2 s in 2 x = 1 - s i n * o
s i n * ( 2 s i n x - l ) = 0
sin* = 0,

x = nk,

1 ;
sm x = —
2

O

O t b e t : x = nk, x = ( - \) k^-+nk-, k e Z .

6

26

IIpHMep 2. PernuTb ypaBHeHHe

sin3;csin;t =

8

.

Pemenue. IIpeo 6pa 3yeM flairaoe ypaBHeHHe cjieayioiHHM 06pa30M :

s in 3 x s in ^ = - —
8

—(eos 2x - eos áx) = - —
2

8

cos 4 jc-

cos 2 x

- — = 0

4

(2 eos2 2x - l ) - c o s 2j c - i = 0
4
2 eos 2 2jco cos2x =

5

cos2x —

(thk

ksk

= 0
|cos2x| < 1) o

4
cos2x

_l-y/ñ

2x —±árceos-—
4
x

=

± — árceos^

2

+ 2rcfe; f t e Z o
' f ^ +nk.
4

O t b e ti x = ±~árceos——^f——+Tiki
2
4

Hg Z .

IIpHMep 3. PemHTb ypaBHeHHe

5 sin2 x - 4 sin x eos x - eos2 * = 4 .

Pemenue. IIpeo6pa3yeM flaHHoe ypaBHeHHe cjieayiomHM 06pa 30M:

27

5sin 2 * - 4 s i n * c o s * - c o s 2 x = 4
5 sin2 x - 4 sin x eos x - eos2 x = 4 sin 2 x + 4 eos2 x
sin2 x - 4 s i n o c o s * - 5 eos2 x = 0.
Tan Kan Te x , npn
hhhmh

kotopbix

c o s * = 0, He

hbjihiotch

ypaBHeHHa, pa3,n¡ejiHB o6e nacTH paBeHCTBa Ha eos2*

noJiynHM:
tg 2 * - 4 t g * - 5 = 0

tg x = - l
tg x = 5;

* = ~ —+ nk,
keZ .
4
* = arctg5 + 7i&,

O t b e t: x = - —+ nk9 * = arctg5 + 7ufc; k e Z .
4

IIpHMep 4. PemHTb ypaBHeHue
4 - eos |2 ti (l 3 * + 9)2j = 5 sin ^7i (l 3 * + 9)2j .

Peuienue. IlycTb y =

tc(1 3 *

+ 9)2 , y > 0 . ÜMeeM:

4 -c o s2 z/= 5sinz/
°

4 - ( l - 2 s i n 2 i/) = 5sini/

2sin2 z/-5sinz/ + 3 = 0 o

sinz/ = l ,
3

smi/ = —;
T .K .

|sin i/|< l,

o

“ o*

sin^ = ly = £ + 2jtft; k e Z < ¿ >

,-i+a*
[*>0;

28

peine

f,{13x +9f ^
[ft>0;

+2nk><

j(13* + 9)2 =Í+2fc,o

[k>0;
)= ±Ji+ 2k,
V
2
'<
V2

13x + 9 =

j

[k> 0 ;
-9±, - + 2fc
2

x=-

13

k>0.
-9±, -+2k

O t b et : x = -

keZ, k>0.

13

üpiiMep 5. PeniHTb ypaBHeHHe
2sin2(j: + 7 j

'

3-64

v

4'-392-88in2x+16 = 0.

Peiuemie. IIpeo6pa3yeM flaHHoe ypaBHeHHe cjieayioinHM 06pa30M:
2sin2(x + -j]

3-64

v

4;-392-88in2x+16 = 0 o

o 3 •641‘c°^2*+2^- 392 •88in2x +16 = 0 o
o 3 •641+8ü,2x - 392 •88in2x +16 = 0 o
o 192 •648in2x - 392 •8811121+16 = 0 o
o 24-64

c-49-88in2x+2 = 0

IlycTL 88in2x = y, ye I1 ’8
8 (tbk

kbk

sin2:c e [ - l , l ] ) .

ÜMeeM:
29

y = 2,
Í24i/2 -49i/ + 2 = O,

,8

\yz

1

y = ü



«

i ’8

z/ = 2 o 89in2x = 2 s in 2 * = -
o
2;c = (-1)Aarcsin—+ nk, k e Z
(-1)* arcsin ^ + kA:
JC =

(-1)* arcsin i + 7ift
O

t b c t

:

jc

= -

keZ .

3aAaHH ,h¡jih caMOCToaTejibHoro pemeHHa
1. PemHTb ypaBHeHHe 2 eos2 jc + 3 sin x = 0 .
2. PeinHTB ypaBHeHHe 2sin 3x sin jc + (3 V2 - 1)eos2jc = 3 .
3. PemHTb y p aB H e H H e 4 eos 4x + 6 sin2 2jc + 5 eos 2jc = 0 .
4. PemHTb y p aB H e H H e 5 eos 2jc+14 eos x + 7 = 0 .
5. PemHTb ypaBHeHHe eos2 4x - 2 eos 4x - 3 = 0 .
6. PemHTb ypaBHeHHe 3 eos 2x +11 sin x = 7 .
7. PemHTb ypaBHeHHe 5 eos 4x + 5 =? 22(cos x + sin jc) 2 .
8. PemHTb ypaBHeHHe eos 4 jc = eos4 x - sin4 x .
9. PemHTb ypaBHeHHe 4 eos2 3 jc- 4 eos ^3jc10. PemHTb ypaBHeHHe cos2jc + 3sinx + l = 0 .
30

- 1= 0 .

§ 2.3. Pa3JioHceHHe Ha MHoacHTejra
flp y r H M
ckh x

TOA

b

8.h c h b i m

ypaB H eH H H
CO C TO H T

/ ( * ) = ()

B

TOM ,

K a K H M - ji H 6 o

M H O H C H TeJIH

M eT O flO M

pem eH H H

T p n ro H O M eT p H H e-

Me-

H B JiH e T C H p a 3 J io H c e H H e H a M H O H C H T e jra .
H T O Ó bl

J ie B y iO

H aCTb

o 6 p a 3 0 M p a 3 ji o jK H T b

H npH paB H H T b

K

H y jIIO

ypaB H eH H H
Ha ABa

K aH C A b lH

h jih

B H fla
6 o jie e

M H O H C H T eJIb

HO

O T A e JIb H O C T H .

npHM ep 1. PeinHTb ypaBHeHHe

s in * + sin 5 * = 0.
Peuienue. IIp eo 6p a 3yeM AaHHoe ypaBHeHHe cneAyiomHM 06pa30M:

s in * + sin 5 * = 0 o
2 sin 3 *c o s2 * = 0
sin 3 * = 0,

eos 2 * = 0;

0t

b

et:

x

=

71

71K

—+ —

,

IIpHMep 2. PemHTb

ypaBHeHHe

eos x + eos 5x = eos 2x + eos 4x.

Peiuenue. IIpeo6pa3yeM

a &hh oe

ypaBHeHHe cjieAyiomHM 06-

pa30M :

eos x + eos 5x = eos 2x + eos 4x
o 2 c o s3 *c o s2 * = 2 c o s3 *c o s*
c o s3 *(c o s2 *-c o s*) = 0
Qy

y

2

2

2 c o s3 *sin — sin —= 0

cos3jc =

n , nk
x = —+ — ,

O,

6

n nk
x = - +— ,

3

6

2nk
sin — = O, x = 2
x = 2nk,
sin —= 0;
2

3

2nk
x =-

keZ;

«
ti nk
2nk » / 7
O t b e t : x = —+ — , jt = ----- ; k s Z .

6

3

3

IIpHMep 3. PeniHTb ypaBHeHne
cos(jc2 +x)

+ COS

+

+ eos

+

j=

Peuienue. IIpeo6pa3yeM flaHHoe ypaBHeHne cjieayiomHM 06pa30M:
cos(;t2 + a:) + cos^jc + ^ j + cos^JC + - ^ j = 0 o

cos(a:2 +x) + 2 cos(x + n) eos —= 0
3
cosí#2 + je )-e o s* = 0
^
o
.
* 2 + 2* . x 2 n
- 2 s in ----------s in — = 0
2
2

s in í-i^ -o ,

sin — = 0;
2
x 2 + 2 x - 2 n k = 0,
2

x = 2nkt k e Z

^ x = - l ± Vi + 2nk, x = ±yf2nk; k 0 .
O t Be t :
32

X=

-l±s¡l+2nk , x = ±j2ñk;

k0.

IIpHMep 4. PeiHHTB ypaBHeHHe
j + s in ^ “

8 * J + cos6jt = 1.

P euienue. IIpeo6pa3yeM flaimoe ypaBHeHHe cjie^¡yiomHM 06pa30M:

(%x

—l + s i n í - ^ - 8* l + cos6;t = 1

2;

l 2

eos 2 X - eos Sx + eos 6 jc- 1 = 0
2 sin 5:t sin 3 * - 2 sin2 Sx = 0
sin3^(sin5^: -sin 3 j:) = 0
sin 3* = 0,
sin * = 0,
eos 4 * = 0;

sin 3* sin* eos 4 jc = 0
nk

nk

*=T ’
x = nk,

sinlf-fJ=0’
c o s ( f - ¡ ) = °;

6 2
8 * _ n , n +^
. 2 3 2
O t b e t : x = ^ + 2nk y * = ^

x = - + 2 nk,
3

ksZ
+

5 n , 2 tcA:

* 9

3 *

fe e Z .

IIpHMep 6 . PemHTb ypaBHemie

(eos x - 1)(2 sin x - eos 2x - 2 ) = 2 sin 2 x.
Pem enue. IIpeo 6pa 3yeM flaHHoe ypaBHemie cjieAyioro¡HM 06pa30M:
(c o s Jc -l)( 2 s in o :-c o s 2 j c - 2 ) = 2 sin 2 x
(eos x - 1)(2 sin x - eos 2x - 2 ) = 2(1 - eos 2 x)
( c o s x - l ) ( 2 s in x - c o s 2 :x;- 2 ) = 2(1 - e o s jc)( 1 h- eosx )
(c o s jc -l)( 2 s in jc -c o s 2x - 2 + 2 + 2 cosjc) = 0
o (eos jc—1)(2 eos x + 2 sin jc - eos 2x ) = 0
(eos x - 1)(2 eos x + 2 sin * - (eos2 x - sin 2 jc)) = 0
(eos x - l ) ( 2 (cos x + sin x) - (eos jc + sin jc)(cos x - sin jc)) = 0

34

(eos x - 1 )(cos x + sin A:)(sin x - eos x + 2) = 0
cosjc- 1
cosa: +

= 0,
sin x = 0,

sin a; - eos a: + 2 = 0;
c o sjc

=

x

1,

tg * = - 1 ;'

=

2nk,

x = ~ —+ nky k e Z y
4

nocKOJibKy ypaBHeHHe cosa: —s i n # = 2 pemeHHH He HMeeT
jiy orpaHHHeHHocTH (JjyH ^H H y = cosa: h y = s i n x .
O

t b

e

t:

b ch -

x = 2nk, x = - - + n k ; k e Z .
4

3a,ija*m ¿jjih caMOCTOHTejibHoro pemeHHH
1.

P e m H T b ypaBHeHH e

2 sin 2 a: + yfs eos x = 0 .

2.

P e n iH T b ypaBH eH H e

sin 3 a: - sin x + eos 2 a: = 1 .

3.

P e n iH T b

ypaBHeHHe sin x - sin 3 a: = eos 2 a: sin 3 a: .

4.

P e n iH T b

ypaBHeHHe sin x + eos ^5# -

5.

P e n iH T b y p aB H eH H e

6.

PeniHTb ypaBHeHHe eos 7x + eos x = 2 eos 3#(sin 2 a: - 1) .

7.

P e n iH T b

ypaBHeHHe 2 eos 2x eos 7 a: - eos 4 a: = 1 .

8.

Pem H T b

ypaBHeHHe cos0 ,2 a: - cos0 ,8 a: + cos0 ,6 a: = 1 .

9.

PeniHTb ypaBHeHHe 2 sin — eos— = sin x .

= y fs sin(3 x + 7t ) .

4(sin4A:-sin2A:) = sinA:(4cos2 3 a: + 3 ) .

2

2

10. PeniHTb ypaBHeHHe sin x sin 3 x = eos 2 x eos 4 a: .
X

X

X

11. PemHTb ypaBHeHHe 5 s in —= eos—eos— .

35

12. HañTH HeoTpimaTejibHbie pemeHHH ypaBHeHHH
l + sin7jc = ^ c o s - ^ - s in - ^ j .

13. PeniHTb ypaBHeHHe sin5;*;-sin;*; = \/8 cos3* .
4 tt
14. PeniHTb ypaBHeHHe eos 3* = eos 5* + — -sin ^ .
15. PeniHTb ypaBHeHHe sin 4#+ 2 s i n o c o s

=0.

16. PeniHTb ypaBHeHHe eos3 * - 2 eos6 * - e o s 9* = - 2 .
17. PeniHTb ypaBHeHHe eos4 jc = 4 c o s a c o s 2 * - l .

§ 2.4. IIoHHHceHHe CTeneHH
HeKOTopbie

TpnroHOMeTpHHecKHe

ypaBHeHHH

pemaiOTCH

nonuMenueM cmenenu. fljin 3Toro Hcnojib3yiOTCH (jiopMyjibi
nyHKTa 6 nepBOH rjiaBbi nepBoro n a p a r p a ^ a . 3 th (JjopMyjibi
npHMeHHiOTCH b cjiynae, K or^a mbi xothm noHH3HTb CTeneHb
ypaBHeHHH 3a cneT H3MeHeHHH apryMeHTOB TpnroHOMeTpHHeCKHX (JíyHKHiHH.

IIpHMep 1. PeniHTb ypaBHeHHe
sin2 * + sin2 Sx = 1.
Pemenue. IIpeo6pa3yeM ,n;aHHoe ypaBHeHHe cjieflyioin¡HM 06pa30M:
sin2 x + sin2 Sx = 1
^ 1 - cos 2 jc , 1 - cos 6 jc h
------------1------------- -- 1

2

2

eos 2x + eos 6x = 0
2 cos 4 x c o s 2 jc = 0 o

36

cos4;t = 0,
cos 2x = 0; <

O t b e t: * - £ + «
8 4

*= 5 + **,

4

8

x = —+ — , k eZ .
4 2

* = * + *ÍL; * e Z .

4

2

IIpHMep 2. PemHTt ypaBHeHHe
8eos 8* - 4 eos22 x + 5 = 0.

Peuienue. IIpeo6pa3yeM ^amioe ypaBHeHHe cjie,zjyiomHM 06pa30M:

8cos8;e-4cos 22x + 5 = 0 o
8(2 eos24jc- 1) - 4 1 + eos 4x + 5 = 0
eos 4x = — ,

2<

16cos2 4 x - 2 eos 4 * -5 = 0

cos4;c = —;
8

4x = ± — + 2nk,
3
<
5
4x = tarecos—+ 2nk, k eZ ;
8

* = ± í+ **,
6 2

x = ± —árceos—+ — .
4
8 2

O t b e t : x = ± —+ — , .r = ± —árceos—
6 2
4
8 2

keZ.

IIpHMep 3. PemHTb ypaBHeHHe
.

2 3

*.

2

♦2f K
U

5 x ) - c i n2llXj-*ÍT,2 ( n

2J

2

U

2 )

Peiueuue. IIpeo6pa3yeM flaHHoe ypaBHeHHe cjieayiomHM o6pa30M:

37

sin

4
1
o
l-c o s í^ -5 * )
lc
o
s
3
x
,
y2
J
o
2
2

1 - eos11*
2

2

1 -c o s í—-1 3 x
\2
O
2

c o s 3 x - c o s lljt- ( s in l3 ;t- s in 5 ;e ) = 0
o 2 sin 4 x sin 7x - 2 sin 4 x eos 9x = 0
sin 4 jc(sin 7 x - eos 9x) = 0
sin 4x ^sin 7x - sin ^ - 9x j j = 0

2 sin 4x sin

7 x - ^ + 9*
7x + - - 9 x
2
__
2
-cos= 0

>sin 4x s in ^8x sin4jc = 0,

j eos ^x -

=0<

4x = nk,

¡in^8jc--jj = 0, <
sin

8 x - ~ = nk,
4

{* -3 -0

x ~ = —+ nk, k
eos a + sin a = —5—

22 + b2

= 1.

39

HMeeM flajiee
eos a eos x + sin a sin * = ,

~r

yja2 + b2

o c o s ( x -a ) =

C —
yj d +b

o x - a = ta re c o s ,—-—= + 2nk; keZ

Va2+fc2

o * = a ± á rc e o s—p c .:— + 27i&,
4 a 2 +b2
b

cjiy^ae, ecjm - ¡ = £ = € [ - 1 , 1 ] ,

HeT peineHHH

h

b hpothbhom

Va2 +62
cjiynae.
rip n pemeHHH ypaBHeHHH ¿jairaoro Tnna moncho nojn>30BaTbCH Tandee ,n¡pyrHMH (JjopMyjiaMH cjioHcemiH, npeflCTaBJieHHbiMH b nyHKTe 4 nepBOH rjiaBti nepBoro naparpa(|>a.

IIpHMep 1. PemHTb ypaBHemie

sinx + V3cos;c = 1.
Peuienue. IIpeo6pa3yeM flaHHoe ypaBHemie cjie^yiomHM 06pa30M:

sin:r +V3 e o s* = 1 -^sin* + —

2
eos ^ sin x + sin ^ eos x =

O

40

t b

e t:

6

6

x = -^ + 2 n k ,

o

x = — + 2nk,

6

x + ~ = - - + 271^ k e Z ;
o

2

sin ^ x + j

* +£ = -£ +2tc/í5,
3

cosjc = —

2

x = Z+2nk;

2

x = — + 2nk.

2

ksZ.

üpmviep 2. PeniHTL ypaBHemie
eos x + sin X + eos 3 * + sin 3x = ~y¡6 eos x.
Peuiemie. IIpeo6pa 3yeM aaimoe ypaBHeime cjieflyiomHM 06pa30M:

eos x + sin x + eos Sx + sin Sx = -V 6 eos x
o (eos x + eos 3x) + (sin x + sin Sx) = -yÍ6 eos x
2cosjccos2jc + 2cosxsin2x = -V6cos^
c o s*( 2 cos 2;c + 2 sin 2;t +V 6 ) = 0 .
ypaBHeHua

cos;c = 0

hbjihiotch

x

=

K | 03

PemeimeM
keZ.

PeniHM Tenept ypaBHemie 2cos2je + 2sin2;e + V6 = 0 .
ÜMeeM:

2cos2* + 2sin2;e + V6 = 0 o


2

c o s 2 jc +



2

sin 2 * + — = 0
2

eos—eos 2x + sin—sin 2x =
4
4
cos^2x —

2

2 x ~ —= — + 2nk,
4 6
2 x - £ = -!™ + 2nk, k e Z ;

4

6

13tt .
24
7tí+ .71«.u
X = ----24
X = ----- +71«,

Ot b e t:

x

= —+ nk, x = - - ^ + nk , x - ^ y - + n k ; k e Z .
2
24
24
41

IIpHMep 3 . PeinHTt ypaBHeHHe
V3 sin 2 * + 2 sin 2 jc- 1 = 2 cosjc .
P euienue. IIpeo6pa3yeM flaHHoe ypaBHeHHe cjieAyiomHM 06pa30M:
-Js sin2jc + 2sin 2 x - 1 = 2 eos* o
y¡3 s in 2 * - c o s 2 * = 2 eo s*

/3

1

—— sin 2o:— cos2 * = e o s*

2

2

-c o sí 2* + — I = eo s*

o cos^2* + -^j + c o s* = 0

o2co8( f +5H f+í)-00
3*
2

7C
2

n

,

x _ 27i | 2 nk

— + — = — + nk,

6

| + | = | +7tft>
O t b e t: X = ^ - + ^ ,

9

3

IIpHMep 4. ÜMeeT

jih

rsZ.

9

3 ’

x = — + 2 nk.

x = — + 2 nk; k e Z .

3

ypaBHeHHe

1 2 c o s | ^ + * j = |4 - 5 c o s * |
xoth

6u OflHy napy KopHeñ, paccTOHHHe Menc^y kotopmmh He

npeBocxo^HT —?

2

Pew enue. IIpeo6pa3yeM aaHHoe ypaBHeHHe cjieffyioin¡HM 06pa30M:

12c° S( ^ +:r) H 4 ~ 5 c o s * | o 1 2 sin * = |4 - 5 c o s * |
42

sin* > 0,
sin* > 0,
5cos;t
+ 12sin:x: = 4,
12sin;t = 4 -5 co s* ,
5
eos*-1
2 sin* = 4.
12sin* = 5 c o sx -4 ;
PaccMOTpHM oT^ejibHO Kanc^oe ypaBHemie. HMeeM:

5
12
4
5 eos x + 12sin x = 4 o — eos*+— sin x = -—
13
13
13
eos x - árceos— = —

l

IB) 13

5 = ± árceos—
4 +2nk; k e Z
x - árceos—
13
13
5
4
x = árceos— ± árceos— +2nk,
13
13
5
4
ycjioBHK) sin * > 0 yflOBjieTBopneT x = árceos— + árceos— +2nk.
13

lo

HMeeM flajiee:

5 eos* -1 2 sin * = 4 — c o s * -^ § s in * = - 7-
13
13
13
eos x +árceos-^- I= - 7-
^
13j 13
5—= ± árceos--4 \-2nn; n e Z
x + árceos—
13

13

5 ± árceos—
4 +2nn.
x = - árceos—
13

13

5
4
ycjiOBHio sin x > 0 y^OBjieTBopneT x = - árceos— + árceos— + 2nn.
13
lo
HanMeHtmee paccToaHHe Mexcffy nojiyneHHMMH kophhmh ecTb

(

5
4W
5
4^1
árceos— + árceos— - -árceos— + árceos— —

V

13

13/ v

13

13/

= 2 árceos— > 5 ,
13

2

43

TaK KaK á r c e o s n o c K O J i b i c y
13 4

— < — = e o s-,
13
2
4

3HaHHT

ncxoflH oe ypaBHeHHe He HMeeT KopHeü, paccTOHHHe MeyKpy

ko-

TOpblMH MeHbUie JIHÓO paBHO —.

2

0

t b e t:

He HMeeT.

3aAaHH ,zi¡.jih caMocToaTejibHoro pemeHHH
1.

PeiHHTb ypaBHeHHe 2cos3* = >/3 cosrc-sin* .

2.

PeniHTb ypaBHeHHe cosí 2 x - — l- s in x = V
3)
2

3.

PeniHTb ypaBHeHHe V3 sin 2* + 2 sin2 x = 1 .

4.

PeniHTb ypaBHeHHe cos x - V3 sin x = V2 .

5.

PeniHTb ypaBHeHHe cos x - sin x = 1 .

6.

PeniHTb ypaBHeHHe 2 co s|* + ^ + 4 s in |* + ^ j = - .

7. PeniHTb ypaBHeHHe sin 2x = 1+ V¡2 cos x + cos 2x .
8.

HañTH Bce A, npn
HMeeT pemeHHe.

9.

PeniHTb ypaBHeHHe 2 sin x + 7 cos x =

44

kotopm x

ypaBHeHHe 2 sin re+ 3 cos x = A

2

.

rJIABA 3
OTBOP KOPHEft B TPHrOHOMETPHHECKHX
yPABHEHHHX

§ 3.1. Ot6op KopHeH npH noMOiu¡H
TpHroHOMeTpH^iecKoro HepaBeH CTBa
HacTO b 3a,n¡aHe TpeóyeTCH He tojibko peniH Tb TpnroHOMeTpH necK oe ypaBHeHHe, ho Taioice H3 nojiyneH H bix K opH eü oto 6paT b Te, KOTopbie y^oBjieTBopaioT HeKOTOpOMy ( n a n npaB H Jio,
npocT eñ m eM y ) TpnroHOMeTpHHecKOMy HepaBeHCTBy. I I p n stom
HepaBeHCTBO MonceT 6biTb 3a,n¡aHO h b hbhom bha©» a MonceT,
HanpH M ep, B03HHKHyTb npH HaxoHCfleHHH oÓJiacTH onpeAeJieHHH
^aH H oro ypaBHeHHH h jih npoBe^eHHH KaKHx-jiH6o n p e o 6 p a3 0 B a hhh.

n p H pemeHHH

hoao 6 hoh

3aflanH nojie3H o H3o6pa3HTb p n -

CyHOK, Ha KOTOPOM Ha TpHrOHOMeTpHHeCKyiO OKpy^KHOCTb Heo6 xoahmo HaHecTH Bce nojiyH HBnraecH k o p h h ypaBHeHHH h
pem eH H e AaHHoro n p o c T e ñ n ie ro HepaBeHCTBa.

IIpHMep 1. Pem H Tb ypaBHeHHe
■v/l + simc + cosj¡: = 0.

Peuienue. ¿JaHHoe ypaBHeHHe paBHOCHJibHO cJieA yiom eñ

chc -

TeMe:
-v/l + sin x + eos x = 0

Vi + sin je: = - e o s x
- cosje:> 0 ,
1 + sin jé = eos2 x ;

eos jé < 0,
1 + sin je: = 1 - s i n 2 je:;

eos jé < 0,
sin 2 jé+ sin jé = 0;
45

cosjc

i

x

=

0

< ,

nkt

<

x = ^ - +2iiky k e Z ;

x = n + 2nk,
x = — + 2tik .

2

Ot6op KopHeii TpnroHOMeTpHHecKoro ypaBHeHHH H3o6pa
^ceH Ha pHcyHKe 1.

2
Phc. 1
O t b e t:

x = K+2nk, x = — +2nk ; k e Z .

IIpHMep

2 . PemHTt ypaBHemie

2

logsin(- x ) [s in | + s i n ^ j = 1 .

Peuienue. flaHHoe ypaBHeHHe paBHocHJibHo cjie^yiomeíi chc

TeMe:

lo g s in (-r )ís i n f + S Í n ^ l = 1 O

sin(-x) > 0,
sin(-;t) * 1,
sin(-x) = s in - + sin— ;

sinjc 0;

x +—= —+ nk,
4

2

x = -+ 2 n k,
6

x = - + nk,

x = — +2nk, k e Z ;

x = —+2nk;
6

6

cos^:e+-^j> 0;

Tan Kan ecjiH x = —+2nkt
6
s^ + ^ =

571
6

a ecjiH x = — + 2 tcA5,

to

co s^

+ ^ + 2 tt^ =

co s|

|> 0 ;

to

c o s ^ ; c + - ^ j = COS ^ - ^ + - ^ + 2 7 1 ^ = C 0 S ^ j ^ < 0 .

Otbct:

x

= —+2nk, x = —+ nk; k e Z .
4

6

IIpHMep 6. PemHTb ypaBHeHHe

>/2cos^Jc + ^ j - sinjc=
50

cosjc

.

Pemenue. IIpeo6pa 3yeM flaHHoe ypaBHeHHe cjieayiomuM 06pa30M:
y¡2 eos ^ * + ^ j - sin * H eos * Io
eos * e o s - sin * s i n s i n * =| eo s* |

^ (^ 2 C° SX

^2

SÍn * ] “ sin * =lcos * I ^

o eos jc - sin * - sin * =| eos * | «> eos * - 2 sin * =| eos * |
fc o s*> 0 ,
1e o s * - 2 sin * = eos*;
1

e o s* < 0,
eos * - 2 sin * = - eos * ;

{

o

fc o s * > 0 ,
[sin * = 0 ;
J c o s * < 0,
[ t g * = l;

íc o s*> 0 ,

[*

= nk, k e Z ;

feos* < 0,
| * = —+ TC&, k £ Z\

x = 2nk,
x =:!™+2nk.
4

Ot6op KopHeñ TpHroHOMeTpnHecKoro ypaBHeHHH H3o6paHceH Ha pncyHKax 6 h 7.
sin *

sin *

P hc . 7

P hc. 6
5n

O t b e t : x = 2nk, * = — + 2nk; k e Z .
4
51

ITpitM ep 7 . PeniHTb ypaBHeHHe

log2( 3 s in x - c o s *) + log2(cosjc) = 0.
Pemenue. flaimoe ypaBHeHHe paBHocHjiBHo cjieayiomeH
TeMe:
1°S2(3sin x -

cosa:) + log2(eos jc) =

chc -

0o

,

Ícosjc > 0,

[(3 sin x - eos x) eos x = 1 ;

cosjc

> 0,

3 sin x c o s x - eos2 x

=

sin2 x + eos2 x; ^

C O S X > 0,
dejiuM na cos^x
sin2 x - 3 sin x eos x + 2 eos2 x = 0 ;
^

Ícosjc > 0 ,

l t g 2jc -3 tg * + 2 = 0 ; ^
cosjc

Ot

b

e t:

cosjc

> 0,

tgJC = l ,
tgjc = 2;

o

> 0,

X — ---1- Tlk,

4
x = arctg2 + 7ife, k e Z ;

x = —+ 2nk,
4
x = arctg2 + 2nk.

x = ^ + 2nkt x = arctg2 + 27u/e; k e Z .

HpHMep 8. PeniHTi» ypaBHeHHe

\Í5sinx + cos2x + 2 cosjc = 0.
Peuienue. flaHHoe ypaBHeHHe paBHocHjibHo cjie^yiomen
TeMe:
>/5 sin jc + eos 2 jc + 2 eos jc = 0 o

o Í cosjc < 0,
[5 s in x + cos2jc = ( - 2 cosjc)2;

52

^^

ch c -

icosx < 0,
| 5 sin* + ( l - 2 sin2 *) = 4 - 4 sin2 *;
eos* < 0,
2 sin2 * + 5sin * - 3

T .K .

|sinx|£l

0;

cos* < 0,

I

1

x = — + 2nky k e Z .

sin* = —;

6

Ot6op KopHeñ TpHroHOMeTpn^ecKoro ypaBHeHHH H3o6pa-

MceH Ha pHcyHKe 8.

O

t b c t

: * = —

6

+ 2nk; k e Z .

IIpHMep 9. PeiHHTb ypaBHeHHe
J —-e o s* = J —-c o s 3*.
V4
V4
Peiuenue. ÍJaHHoe ypaBHeHHe paBHOCHjibHO cjieflyiorqeH

chc -

TeMe:

J —-e o s* = J —-c o s 3* 4
4

V

V

53

--co sa: > O,

L » * á i.

3
o
«
— cos;t = — cos3;c;

4

4

j

4

1

cos 3jc- cosjc

= 0;

,3
COSX keZ;
^3
COSJC(2cos2x +3)(2cos2x - l ) = 0

x

eos 2* = O 2x

= —+

2

T .K .

2COS2X>0
= 1

nk, fe e Z x = —+ — .
4

2

E cjih fe = O, TO X = — s 3 ,1

.4 .

eCJIH fe = 1, TO * = —+ —= — £ . i 1
4

eCJIH

fe = - 1 ,

2

4

TO Jt = —

4

CjieflOBaTejibHo, tojibko

2
x

= - —£

,1

4

= — yAOBJieTBopneT ycjiOBHio 3aA¡aHH.

4

O t b e t : * = —.

IlpHMep 5: HaÜTH Bce :t, yflOBjieTBopjiiomHe ypaBHeHHio
V 4 9 -4 #2 •^sin 7cx + 3 eos ^ j = O.

Peiuenue. flaHHoe ypaBHeHHe paBHocHjibHo cjieayiom eH coBOKynHOCTHI

V 4 9 -4 * 2 •[ sin7iJC + 3 co s^-1 = 0
62

*-4
4«4

49-4 *2 = O,
Í49-4 jc2 > O,
| sin 71* + 3 eos— = 0:

X

t .k .

o o . S ( 2 si » S t 3 |.0i

2

sin -y- < 1

*-4

*-±I,
2

7

7

7

7

2

2

2

2

eos— = 0;



2

*-4

= - + 7ifc, fteZ;

* = ±-,
2
* = ±1,

j 2
2
|* = l + 2fc;

x = ±3.

O t b e t : o: = ± —, x = ±l,x = ±3.

2

IIpHMep 6. P e m H T b ypaBHemie

2\/3 sin (tu;+ 37i) - tg ^tzx - ^ j j log2(4 - x2) = 0.

Peiuenue. flam oe ypaBHeHHe paBHOcmibHo cjieayiomeñ coBOKynHOCTH:

jj

2\¡3 sin (nx + 37c) - tg ^tzx- — log2(4- jc2) = 0 o
= 0,
*0,

Í4-*2 >0,
12>/3 sin(7ijc + 3 ti) - tg^Tcx ~ j = 0;
63

W -l,

+
2 2
-2/3cos2(7t^) +

cos ( ttjc)

-2 > /3

sin(7tjc)
X = ±yfSt
-2 0,

X > ~ 10 y

<

icos o: = - 1 ,
\jc-hl0 < 0;

°

x = n + 2nny
J C < - 10 ,

k y J l y SZ;

x = 2nkt
6 = - 1 , 0 , 1 , 2 ,...;
* = n + 2nnt
n = - 3 ,- 4 ,- 5 ,...,

O t b e t : x = 2nky x = n+2nn; kt ne Zy k > - l , rc 0 ,

fx + s i n x > 0 ,

[x + s in x = x - s i n 2x;

[sin x + sin 2 x = 0 ;
x + s in x > 0 ,

o

x + s in x > 0,
s in x ( l + 2 cosx) = 0 ;

s in x = 0 ,

1

o

c o sx = — ;

1

2

x + s in x > 0 ,
x = nky
x = ±— + 2 nk,
3

keZ.

CpeflH pemeHHH ypaBHeHHH ycjiOBHio - 2 n < x < 2 n y^OBjieTBOPHIOT 3HaneHHH X = 0 , X = ±7U, x = ± — , x = + — . CpeflH
3
3
3 t h x 3HaneHHH x ycjiOBHio x + s in x >0 yflOBJieTBopnioT x = 0 ,
* -* ,
Ot

3
b

3

e t : X = 0, X =

71,

x =— , x =— .
3
3

IIpHMep 9 . HanTH Bce pemeHHH ypaBHeHHH
3>/2 eos— - c o s x = 3 ,
2
npHHafljiencamHe OTpe3Ky [ - 2 ; 10 , 99] .
P e iu e u u e . IIpeo 6pa 3yeM flaHHoe ypaBHeHHe cjieayiomHM 06pa30M :

66

3%/2 eos—- e o s * = 3 o

2

3V2 eos—- 2 eos2 —+ 1 = 3
2
2
2 eos2 —-3 V 2 c o s—+ 2 = 0 eos—=

2
2
2 72
- = ± - + 27tfc;
2
4

k e Z x = ± —+ 4nk.
2

IlyCTB X = —+ 47t/? .

2

E cjih
qaeM ,

k = 0,
*=

h to

to

2

jc =

— g [ - 2; 1 0 , 9 9 ] . I l p n

2

£ [-2; 1 0 ,9 9 ].

E cjih

k

nce

=

-1 n o jiy k = 1,

to

971

x = — £ [-2; 10,99]. CjieflOBaTejibHo, H3 3 to h cepHH noflxoflH T

2

71
2

TOJIbKO JC = —.

nycTb Tenepb ;t = ~ + 4nk .
E cjih k = 0, to jc = - —e [ - 2 ; 1 0 ,9 9 ]. I l p n

2

k = - 1 HMeeM

971
----- £ [-2 ; 1 0,99]. PaccMOTpHM noflpodHee cjiynaH, Koiyja
2
7k
k = 1, TO eCTb JC = --- . CpaBHHM 3TO HHCJIO c hhcjiom 10,99:
2
jc =

— v l0 ,9 9 « 7 itv 2 1 ,9 8 o jtv 3 ,1 4 .

2

771
771
— > 1 0 ,9 9 . CjieflOBaTejibHO, x = —
2
2
He yflOBjieTBopneT ycjioBHio 3aflaHH. TaKHM o6pa30M, OTBeTOM k
Tan Kan

ti

> 3,14,

to

3a#ane OyflyT ejiyHCHTb x = ± ^ .

O t b e t : x = ±I —,
2

67

ITpiiiviep 10 . HañTH Bce

kophh

ypaBHeHHH

2cosr + 5 •2 “cosx = 2>/6,
yAOBjieTBopniomHe HepaBeHCTBaM —< x < — .

3

Peiuenue. IlycTB 2cosx = y,

3

( tbk Kan |c o s * |< 1 ) .

ÜMeeivr:

(/ + - = 2 ^ o y2 -2V6i/ + 5 = 0
y

y = \Í6±1

m.K. y

G[£,2]

z/ = V6-l

2cos:r = V6 - 1 eos x = log2(V6 - 1)
x = ± árceos log2(V 6-1) + 27ü&; keZ.
CpaBHHM HHCJia
árceos lo g 2(V6 - 1 ) v
cos-^ v lo g 2(V6 - 1 )

ó

~V\og2(y/6-l)aKT,

to h

á r c e o s 1) <

mu

3

Hcnojib-

(JjyHKi^na y = árceos x — y6wBaioin;aH. Ta-

3yeM

tot

khm

o6pa30M (pHcyHOK 13), HHTepBajiy —< x < — npHHa^jie-

hto

7K&T cjieAyiomne
h

68

3

kophh

3

ypaBHeHHa: x = 2n - árceo slo g 2( V 6 - l )

x = 2n + árceoslog2( V 6 - l ) .

log2 ( V 6 - l )

log2 (>/6 - 1)

O t b e t : x = 27t±arccoslog2('/®- l) •

3 a,n¡aHH fljia caMOCToaTejibHoro peineHHa
1.

PeniHTb ypaBHeHHe e o s

2x + s in 2 x = 0 , 2 5 .

yK a3aTb Te H3 KopHeñ 3Toro ypaBHeHHH, KOTopbie npHHaajieacaT OTpe3Ky
2.

3 » ,|

PeniHTb ypaBHeHHe e o s 2 * - s i n * = 0 .
yKa3aTB Te H3 KopHeñ stoto ypaBHeHHH, KOToptie npHHa#jieacaT 0Tpe3Ky

*T

PeniHTb ypaBHeHHe c t g 2 x - 2 c tg * - 3 = 0 .
y n a3aT b Te H3 KopHeñ 3Toro ypaBHeHHH, KOTopbie npHHafljiencaT 0Tpe3Ky

7n~

271,-

PeniHTb ypaBHeHHe 5 s in 2 x - 4 sin x eos x - eos2 x

—0 .

yica3aTb Te H3 KopHen 3Toro ypaBHeHHH, KOTopbie npHHa#371 n
jieacaT 0 T p e3 K y ——, 0

69

5.

PeuiHTb ypaBHeHHe 2 sin 2 * + cos* + 4 s in * + l = O .
YKa3aTb Te H3 KopHeñ

stoto

ypaBHeHHH, KOTopwe npHHaa-

5 jc 77i

JiencaT OTpe3Ky

2

2

6.

HaHTH Bce pemeHHH ypaBHeHHH sin 2 * + cos* + 2 sin * = - 1 ,
yAOBJieTBopHK)Hi;He ycjiOBHio 0 < * < 5.

7.

HaHTH Bce pemeHHH ypaBHeHHH 3 tg 2^7t*--^j = 1 , yAOBJieTBOpHKDipHe yCJIOBHK) 1,5 < * < 3.

8.

HaHTH Bce * e (-7 i,7 r ), HBjiHiomHecH pemeHHHMH ypaBHeHHH
■=

yj-

2 COS* .

V -2 sin *
9.

HaHTH Bce pemeHHH ypaBHeHHH |s in 2 * | + cos* = 0 , npnHaAJieHcanjHe OTpe3Ky

-*i

10. PeniHTb ypaBHeHHe V3cos2 * + 0 ,5 sin 2 * + cos* = 0 .
HaHTH cyMMy ero pa3 JiHHHbix KopHeñ, npHHaAJiencamHx
0 Tpe3 Ky [- 7C, 7l].

11. HaHTH Bce pemeHHH ypaBHeHHH
2 sin í * + — 1 •sin í 3* +
= eos 4 * +
A
25y
l
25 J
y¡2
npHHaAJieHcam¡He OTpe3Ky

71 471

10’ 5

12. HaHTH HaHMeHbniHH KopeHb ypaBHeHHH
Vcos2* + * - l l = % /*-15-5cos* .
13. HaHTH Bce pemeHHH ypaBHeHHH
5 sin2 2* + 8 eos3 * = 8 eos * ,
npHHaAJiencamHe OTpe3 Ky 2 * 2n
.2

70

§ 3.3. HaxoacfleHne o 6iu;h x KopHeñ
f l B y x T p H r o H O M e T p iiM e c K H X y p a B H e H H H

B HeKOTopbix 3 a fla n a x B03HHKaeT Heo6xoflHMOCTb H axoncAeHHH o 6 iii;h x K opH eíí ,n;Byx TpHroHOMeTpHHecKHx ypaBHeHHH.
H a m e B c ero sto npoHCxoflHT n p n Bbinojm eHHH 3a,zjaHHH, b
k o t o po m
TpHroHOMeTpHHecKan (JjyHiajHH npncyTCTByeT b
3HaM eHaTejie K a K o ro -Jin ó o BbipaxceHHH. I I ohck oóiijh x p e m e HHH fls y x TpHrOHOMeTpHHeCKHX ypaBHeHHH MOHCHO OCymeCTBJIHTb

HeCKOJIbKHMH

CIIOCOÓaMH.

MOHCHO p e n ia T b

KaHCflOe

ypaBHeHHe b OT,zi;ejibHOCTH ( n p n stom b pa3H bix ypaBHeHHHx
AOJiHCHbi 6biTb Bbi6paH bi pa3JiHHHbie 6yKBbi, o6o3HaHaioii];He
n;ejiOHHCJieHHyK) nepeM eH H yio), a 3aTeM Ha TpnroHOMeTpHHeOKpyncHocTH HaxoflHTb o ó m n e pem eHHH . M ohcho Taicnce
n p n nOMOIH¡H n O flC T aH O B K H B bIH C H H Tb, KaKHe KOpHH OflHOrO
ckoh

H3 ypaBHeHHH HBJIHIOTCH

B

TOM HHCJie

H

peiUeHHHMH BTOpOrO

ypaB H eH H H .

IIpHMep 1. PeniH T b ypaBHeHHe
eos 2 * _

q

1 - t gx

Peiuemie. flaH H oe ypaBHeHHe paBHOCHJibHO cneflyiom eH

chc -

T eM e:

co s2 x = 0,
eos 2*

_Q^

l-tgx
_

X

x*

tgx*l,

co s* * 0 ;

n , nk
4

2 ’

— + nn,

x = —

+ nk.

4

x * —+nm, k,n,meZ;
2
Otóop KopHeñ TpHroHOMeTpHHecicoro ypaBHeHHH H3o6paHceH Ha pncyHKe 14.

71

O t b e t : x = - —+ n k ,k eZ .
4
IIpHMep 2. HaftTH Bce peineHHH ypaBHeHHH
l+ 2 sin 2 *-3V 2sin:K + sin2jc _ ^
2 s in x c o s x - l
Peuienue. flaimoe ypaBHemie paBHOCHjitHO cjieayiomeñ
cHCTeMe:
l + 2sin2 X-%y¡2 s\TlX + S\Tl2x = 11
-------------------------------2sin ;ccos:c-l
l + 2sin2 je-3>/2sin;c + sin2jc = s in 2 x - l,
sin 2x - 1 * 0 ;
Í2 sin2 * - 3V2 sin;e + 2 = 0,
[sin2jc * 1;
sinjc = V2,
^
sm x = — ;
^
s in 2 :e *l;
72

,

,

f

¡—

T.K.|sinx|

<

4

2x * —+ 2nn, k,neZ;

2

x = — + 2nkf

4

x * — + nn;

4

x = — + 2nk.

4

Ot6 op KopHeñ TpHroHOMeTpHHecKoro ypaBHeHHH H3o6paHa pncymce 15

.

meH

= ---- 1-2nk;

keZ,

IIpHMep 3 HaÜTH Bce

kophh

Ot b et : x

.

4

ypaBHeHHH

(l+ tg 2 jc)sin jc-tg 2 ;c+ l = 0,
yflOBjieTBopniomHe HepaBeHCTBy t g x < 0 .
Peiuenue. IIpeo6pa3yeM ^aHHoe ypaBHeHHe cneAyioinHM o6pa30M:
( l + tg 2 x )s in ; c -tg 2 :x:+ l = 0 s in *
eos2*

sin2*
2 *.

eos *

eos2* - o ^
2



eos *

73

sin * - sin2 x + { l - sin2
°

^ ^ l + s in * - 2 s in 2 x =

e o s2*

q

cos2*

s in * = 1,
í l + s in * - 2 s in 2 x = 0,
[eos* * 0;

jc =

1
2

s i n * - - —,

e o s* * 0 ;

— + 2n:fe,
2

* = - —+ 2nkt

x = - * + 2nk,

6
x = - — + 2nk;

x = - ^ + 2nk.

6

6

6

* * —+ 7m,

2

HepaBeHCTBy t g * < 0

k ,n e Z ;

opean nojiyneHHMX KopHeñ yaoBJie-

TBopneT x = - —+ 2nk.

6

O t b e t : * = - —+ 2nk, k e Z .

npHMep 4. HañTH Bce * , yaoBjieTBopaioin;He ypaBHemno
c tg 3 * = ctg5 *.
Pemeuue. IIpeo6pa 3yeM aaHHoe ypaBHemie cjieayiomHM o6 pa 30M:
c tg 3 * = ctg5x o

sin 3 *

_ co s5 * ^
sin 5 *

_ ----------eos3 * sin5
* -e;—
o s5---------------U
* sin 3 * ^
;—
sin 3 * sin 5 *
o — f n2*
=0 o
sm 3 *sin 5 *
74

sin 2 * = 0,
s in 3 **0 ,
sin 5 **0 .

PemeHHeM ypaBHeHHH sin2jt = 0

hbjihiotch jc =

E cjih x = nky to sin3x = sin37i/ü = 0 npH

x = ^ + nk,

to

otjihh h bi ot

HyjiH npn

HneM ypaBHeHHH
Ot

b

e

t

s in Sx = s i n p ~ + 3 tü/?J

jiio Gom

Tzk

— ; keZ.

ijejiOM k . E cjih

sin5;c = s i n | ^ + 57t&j

h

n;ejiOM k . T hkhm o6pa30M, peme7C
x = —+ nk .

jik )6 om

hbjihctch

2

: x = —+ nk , k e Z .

2

ripwMep 5 . PeniHTb ypaBHeHHe
sin 3x

= -l.

cos| x - -

Peuiemie. IIpeo6pa3yeM flaHHoe ypaBHeHHe cjieayiomHM 06pa30M:
sin Sx

= -!

eos x - —
l
6

o _ s i n 3 ^ +1 = 0 o

l

eos

COSI

sin 3;c + cos x ~ — = 0 ,
eos x —

sin 3:c + cos x —

6 = 0o

X

sin S x + sin

^ j = 0,

* 0;

2 sin [2x + eos ^x -

i ( 2x +- ) = 0 ,

j = 0,

c o sH > 0 ,

cos^ jt-^ j*0;
75

2x + — = nk,

X = . J L + 5 *,

x - —* —+ nny k ,n e Z :

x*—

6

6

12

2

3

2

+ tc/z;

^ Xv- =
- ----- 1
^-----TtA; .

12

2

Or6op KopHeft TpHroHOMeTpH^ecKoro ypaBHeHHH H3o6pa-

HceH Ha pHcyHKe 16.

P hc. 16

Otbet :

x = — — +—
12

2

;

k&Z

.

IIpHMep 6. PemHTb ypaBHeHHe

eos 2* - eos 4jc- 4 sin 3 * - 2 sin * + 4 _
2sin;x:-l

q

Peiuenue. IIpeo6pa3yeM «aHHoe ypaBHeHHe cjieayiomHM 06pa30M:

co s2 ^-co s4 o :-4 sin 3 jc-2 sin jc + 4 ^
0
2 s in ;c -l

-------------------¡T-:------- :------------------- =

_ 2 sin ^ :sin 3 jc-4 sin 3 jc-2 sin x + 4 ^

2 s in ;c -l

----------------- — ------- -------------------=

76

sin 3*(sin x - 2 ) - (sin x - 2) _ n
---------------------------------- u
2 sin :c-l

(s in 3 *- l) ( s in :r - 2 )
--------------------------U w
2 sin *- l

{ sin3;t = 1,

ti , 2nk
* =- +
6
3

+ 2nn,
• * _* —
1 ; x * —
6
s in
ir

* * — +27m, k,neZ;

6

o * = — + 27lfe.
2
Ot 6op KopHeñ TpHroHOMeTpunecKoro ypaBHeHHH H3o6pa^ceH Ha pHcyHKe 17.

Phc. 17
0

t b

e

t

:

x = — + 2nk , k e Z .
2

IIpHMep 7 . PeniHTb ypaBHeHHe
V sin 3* • t g ^ 2 *- - ^ j = 0.

77

Pemenue. flaHHoe ypaBHeHHe paBHocmitHO

cobokyiihocth

A B y x CHCTeM :

\lsinSx - t g ^ 2 j c ~ j = 0
fsin3jt = 0,

Sx = nk,

71
1c o s ^2 jc- —

0,

f 'f - i ) - '

I

2 x - —* —+ nn,
6 2

\2x~ r Kk’

[sin3x > 0;

[2nn y * i;

o

3tgJC + 401og3Í/2 = 1 6 3 ,^
2\og3(21og9Z/-l) + 401og3Z/2 = 163
31oggZ/- 3 + 801og 2¡y = 163 o
831oggZ/ = 166 loggZ/ = 2 y = 9.

87

IloflCTaBJiHH HaH^eHHoe 3HaneHHe y
nocjieAHeñ CHCTeMbi, nojiyHaeM,
Ot b e t :

Tt/e, 9

hto

bo

tg x = l

BTopoe ypaBHeHHe
h x = —+

4

nk; k e Z .

keZ .

IIpiiMep 6. PernHTb cHCTeMy
|sinz/|sinz/ = ^
COSX
| c o s x - l ||2 + |s i n i / f = 4.
Pemenue. E cjih

cosx

h

sin y HMeioT pa3Hbie 3HaKH,

nepBoe ypaBHeHHe CHCTeMbi npeo6pa3yeTca k BHfly sin2 y = -1
pemeHHH He HMeeT. E cjih Hce cosx h sin y oflHoro 3HaKa,

to
h
to

sin2 z/ = l , OTKyfla sini/ = ± l . IIpeo6pa3yeM BTopoe ypaBHeHHe
c H C T e M b i. Ü M e e M :

|c o s x - l |2 + 1sin z/12= 4
(eos2* - 2 cosx+ 1)+1 = 4
cos2* - 2 cosx - 2 = 0 cosx = 1 - V 3
x = ± arc c o s(l-V 3 ) + 27tfc; k e Z ,
Kan BTopoá KopeHb KBa^paTHoro ypaBHeHHH Gojibine 1. IIoCKOJibKy cosx = l-> /3 < 0 , to h s in z / < 0 . CjieflOBaTejibHO,

tbk

sin y = -1

h

y = ~ ^ + 2nn; n e Z .

O t b e t : |^ ± a rc c o s(l-V 3 ) + 27rA;,-^-+27mj|; k , n e Z .

IIpHMep 7. PernHTb CHCTeMy
icos y

= 27*2+2:r-|cosi/|,

j^2sin i/ = logaX.
88

P eiu en u e. E cjih cosí/ < 0 , to nepBoe ypaBHeHHe CHCTeMbi

pemeHHH He HMeeT. rip n c o s i / > 0
Cfl K

bto

ypaBHeHHe npeo6pa3yeT-

BHAy

gx3+8 = 27*2+2 x o x 3 + s = S x 2 +6 x
(jc+ 2)(x2 - 2x + 4) = 3x(x + 2)
o (x + 2)(x2 - 5x + 4) = 0
(x + 2)(x - ! ) ( * - 4 ) =

0.

3HaneHHe x = - 2 He npHHaflJietfCHT o6jiacTH onpeflejieHHH
CHCTeMbi. E cjih x = 1, to sinz/ = 0 h (Tan Kan c o s z /> 0 )

y = 2nn; n e Z . Ilpn * = 4 nojiynaeM,

hto

sin i/ = l h ycjiOBHe

cosí/ > 0 He BbinojiHHeTCH.
IlycTb T enepb cosz/ = 0 . Tor,n;a B03M0JKHbi Asa cjiynaH. Ec­
jih

sin i/ = l ,

to e c T b

Hce sinz/ = - l ,

z/ = -^ + 2 7 m ,

npn 3tom

i/

=

to

lo g¿t = 2 h # — 4. E cjih

+ 2 tt/i , Tor/ja lo g¿x = - 2

h

k 3aA¡aHe nojiynaeM Kan oó'beAHHeHHe Bcex pa304
6paHHbix cjiynaeB.

x = —. Otbct

O t b c t : |( 1 ,2 tc/i) ;^ ,-—- + 27E/ij;^ 4 ,^ + 27ü/ij|;

neZ .

ripHMep 8. PeniHTb CHCTeMy
í2 sin(2;c + y) sin y = eos 2x,
[sin 2x - sin 2y = y¡2.

Peuienue. IIpeo6pa3yeM nepBoe ypaBHeHHe cncTeMbi cjieAyiom;HM o6 pa 30M:
2 sin(2x + y) sin y = eos 2x
eos 2x - cos(2x + 2y) = eos 2x

cos(2* + 2z/) = 0 o 2x + 2y = ~ + twi; m e Z.
¿i

89

PaccMOTpHM flBa cjiyHafl. ÜycTb cHanajia m
hhcjio ,

TO

m = 2l\

leZ.

IIoflCTaBHM

sto

ecTb

%y = —- 2 x + 2nl.

Toraa



HeTHoe

2x + 2u = - + 2nl

BbipaHceHHe

2

bo

BTopoe

h

ypaB-

HeHHe CHCTeMbi. ÜMeeM:

sin 2x -sin jj^-2;c + 27t/j = 7 2 o

o s in 2 x -c o s 2 x = 72 o s in ^ 2 x - ^ j = l o
^ 2 x ~ = %+2nk; k e Z < ^ x = — + nk.
4 2
8
flajiee
y = —- x + nl = ~ ^ - + n (l- k ) = - —+ nn; n e Z .

4
ÜycTb Tenepb m =

4

8

21 +

1 ; l e Z . Ilp n

8

2x+2i/ = ^+ jt(2í + l )

h

¿

B

stom

cjiynae nojiynaeM,

stom

2y = - - 2 x + 2nl.

2

hto

s in 2 * - s in ^ - 2 a : + 2 7 t/ j = 7 2 o
s in 2 * +cos2x = 7 2 o
O s i n f 2 * + 4 )= l o

\

4J

2 * + 4 = 4 +23ifc; k e Z < * x = £ + n k .

4 2

8

CjieflOBaTejibHo,

y = - ^ ‘- x + nl = ^ - - ^ + n ( l - k ) = ^ - + nn; n e Z .
OG'beflHHHH Bce pa3o6 paHHbie cjiynaH, nojiy*iaeM
3a^aHe.

otbgt k

O t b e t : | ^ + j i f c , - í + rtnj ; ^ + 3i f t , ^ + 7m j | ; k , n e Z .

90

3aAa*iH ajih caiviocTOHTejiBHoro peiueHHH
1.

P e u i H T b C H C T eM y y p a B H e H H H

Í49'tffx-14-7-tffx + 49 = 0,
{3\fy -tgx-5y¡2cosx = 0.
2.

P e i i i H T b C H C T eM y y p a B H e H H H

íi/ + sinx = 0,
|(2>/sinjc - 1)(2y + 5) = 0.
3.

P e r n H T b C H C T eM y y p a B H e H H H

J l 2 l cosx - 2 1 l cosx + 1 = 0,

[7^4 - s i n * = 0.
4.

P e r n H T b C H C T eM y y p a B H e H H H

x + sin(* + */) = —,
2

5

Sx - sin(* + y) = —.
2

5.

P e r n H T b C H C T eM y y p a B H e H H H

Íl2sin2 jc-sin2 z/ = 3,
[6 sin * + cosi/ = -2.
6.

P e r n H T b C H C T eM y y p a B H e H H H

fs in *- s in l = 0,
[cosjc7.

cos1 = 0.

P e r n H T b C H C T eM y y p a B H e H H H

Í2cos*-sin í/ = 0,
|4 sin * + 2cos¿/ = 5.
8.

P e r n H T b C H C TeM y y p a B H e H H H

[2sin*sini/ + co s* = 0,
|1+ sin y eos x = 2 eos2 y sin x.
91

9. HaHTH Bce pemeHHH cncTeMti ypaBHeHHH
t g 2( * - y) -

tg(jc - y) +1 = O,

sinx = ¡ '
y flO B J ie T B o p a io m H e y c jiO B H H M

10.

0 < jc< — ,

- n < y < 0 .

PeinHTb CHCTeMy ypaBHeHHH
3^ =

jc ,

X
2sin^ + sin2x = 2cos 2 —
.

2

11. PemHTb CHCTeMy ypaBHeHHH

cos4x + sin2z/ = -2,
x - y = 2n.

92

r«JIABA 5
PEUIEHHE TPHrOHOMETPHHECKHX
HEPABEHCTB

PaccMOTpHM CHanajia KaK pemaiOTCH npocTeHnme TpnroHOMeTpHnecKHe HepaBeHCTBa. HepaBeHCTBa BH^a s in j t> a pernaiotch cjie^yiomHM oópa30M:
sin x > a x e (arcsin a + 2nkyn - arcsin a + 2nk); k e Z
npn ae [-1 , 1). E cjih a > 1, to AaHHoe HepaBeHCTBO pemeHHH
He HMeeT. E cjih me a < - 1, to pememieM HepaBeHCTBa HBJiaeTch jno6oe x e R . HepaBeHCTBO s in jc < a HMeeT pememieM cjieAyiomne npoMencyTKH:
sin x < a x e (—7c- arcsin a + 2nkyarcsin a + 2nk)
npn a e [ - l , 1). HepaBeHCTBa c o s * > a ,

t gX a y

COOTBeTCTBeHHO paBHOCHJIBHbi:

cosjc>

a x e (-árceos a+ 2nk, árceos a+ 2nk), a e [ - l , l ) ;

cosx < a o x e (árceosa + 2nky2n -árceos a + 2nk)y a e (-1,1];

O SpaT H M
6yK B bi n n c a T b

kyn e Z

B H H M aH H e,
H e jib 3 H ,

H B jineT C H

B H yT pH

H e A o n y c T H M O Íí.

T p n ro H O M e T p H H e c K o e
K HeCKOJIbKHM

hto

H anpH M ep,

H epaB eH C T B O ,

npO C T eH H IH M .

npoM encyT K a

fljin

peniHTb
ero
c j i e f l y i o m H e npn-

T o ro

hto6m

H eo ó x o A H M O

PaC CM O TpH M

pa3Hbie

x e(2nkyn+2nn ) ;

3anncb

cbccth

M epw .

93

Ilpuiviep 1. HañTH Bce x H3 0Tpe3Ka O < x < ti, yflOBjieTBOprnomne HepaBeHCTBy

sin 2 * -eo s * + 72 sin* > - i .
42
Peuienue. üpeo6pa3yeM flaHHoe HepaBeHCTBo cjieflyiomHM
o6pa30M:

sin 2x - eos * + 72 sin x > -^
42
o

2 s ir u c c o s jc -c o s ^ + N / 2 s i n í : —

> 0

42
o

c o s a :(2 sin A :-l)+ -i(2 sin í:-l)> 0 o
72
| co s* + -^ = j(2 sin * -l) > 0

I

co s* + -i= > 0,
V2

2 s i n * - l > 0,

cos* + - ^ < 0,
72

I

eos* > —
72’
sin* > —,
2
eos* < -

sin* < —;
2

2 s i n * - l < 0;
—+2%k < x < — +2nk,
6
4

— + 2 n k < x < — +2nk, k e Z ;
6
4

PemeHHe H3o6paaceHo Ha pncyHKe 21.
94

72’

t .k .

xe[0,jrj

Phc. 21
O

t b

e

t

:

.

IIpHMep 2. HaÜTH Bce pemeHHH HepaBeHCTBa
yjsin2x < eo s* -sin * ,
yflOBJieTBopaiomHe ycjioBHio | * |< n .

Peiuenue. flamioe HepaBeHCTBo paBHocnjiBHO cjie^yiomen
CHCTeMe:
Vsin2* < e o s* -sin *
sin 2 * > 0 ,
o
c o s * -s in * > 0,
sin 2* < (eo s* -sin * )2;

sin2*> 0,
eos* > sin*,

sin 2* < l-sin 2 * ;

[ o < s in 2 * < -,
sin*;
2 n k < 2 x < - + 2nk>
6

— + 2nk 0,

<

11 +2 eos2 * - y¡3 sin 2* = (73 eos * - sin x f ;

7 3 eos jc- sin x > 0,

1 +2 eos2 x - y¡3 sin 2x = 3 eos2 * - 2V3 eos * sin * + sin2 x;
^ c o s * - s i n * > 0 , 0 j 3 C0SX_ sin x > 0 .

0 = 0;
E cjih x = 0, to nojiyneHHoe HepaBeHCTBO BepHo. E cjih
x = 7i, to 3TO HepaBeHCTBO He BbinoJiHHeTCH. E cjih * g (0, tc) , to
sin x > 0 h Mbi MOHceM pa3flejiHTb o6e nacTH HepaBeHCTBa Ha
sin x :
7 3 eos x - sin x > 0
y / S c t g x - l > 0 c t g x > 7 3

TaKHM o6pa30M,

oTBeTOM

k

T .K ..te ( 0 ,rc)

xe

3aflane CJiyJKHT OTpe30K

xe
O

t b

e

t

:

x

e

IIpHMep 4. HaÜTH Bce OTpnijaTejibHbie 3HaneHHH u , npn
KOTOpblX BbinOJIHeHO HepaBeHCTBO
------^ — + ------ ------- > 0 .

logacosuS log3C2iM
3
Pemenue. IIpeo6pa3yeM ^aHHoe HepaBeHCTBO cjieayiomHM
o6pa30M:

97

1
loffo
3
lu6 3co9uü

i
eos a
l0g33

- > 0

1

1

1
,
eosu
- --------Z----------lo g 3 —
\og33 c o s u
ó

■> 0 < = >

< = > ----------- |----------- + ------------- ---------- > 0 .

1
log3COSU-l
l + l o g 3cosi/
IlycTb log3cosu = x . HMeeM:
1

-+ —

1

> 0 o

x -1

1+x

x + l + — —> 0 ;

1

[x ± - l j

COS l¿ = 1 ,

jc =

X~ 1

0,

\og3c o s u = 0,

x > 1;

cosw > 1;

t . k .| cosu| 3;

íx * -l,

cosu = 1 u = 2 nk; k e Z .

Tan Kan Hy^cHo HaÜTH tojibko oTpimaTejibHbie pemeHHH
HepaBeHCTBa, HMeeM u = 2nk, k e Z, k < 0 .
O t b e t : u = 2nk; k e Z , k < 0 .

IIpHMep 5 . PeniHTB HepaBeHCTBO

logi |e o s * | log5(x2 - 9 ) < 0.
2

Peiuenue. Tan KaK |cosjc| 0 flocTaTOHHO noKa3aTb, hto

100

-V ÍÓ + 5 - n/ 3 > 0 o
o 5 > -JÍO + S o 25 > 13+2730 o
o 12 > 2V30 o 6 > V30 o 36 > 30.
TaKHM o6pa30M, oójiacTbio onpeflejieHHa #yHKii;HH f(x) 6y-

ReT CJiyHCHTb MHOHCeCTBO Bcex fleHCTBHTeJIbHblX HHCeJI.
0 t b e t : x g R.

IIpHMep 8. HañTH Bce 3HaneHHH

jcg [0,7t] ,

npn kotopmx

BbipaMCeHHH ctgx H
eos 2 a:-

1
2 eos 2 a:

HMeiOT p a 3 H b ie 3HaKH.

ko

P eiuem ie . H ncjia a h b HMeiOT pa3Hbie 3HaKH TOiyja h tojibToiyja, Koiyja ab < 0. B cbh3h c sthm 3aMenaHHeM HMeeM:

ct gxí cos2 a:--------— 1< 0
V
2cos2xJ
_ cosa: 2 cos2 2 a: - 1

sin a:

^

eos 2 a:

_ eos a: eos 4a: ^ n _
sin x eos 2 a:
sin a: eos a: eos 2x eos 4 x < 0
o sin2jccos2jccos4x < 0
sin4A:cos4A:

O K + 2 n k < 8 x < 2 n + 2nk; k e Z o

0
Tan Kan

a: g [0 , ti] ,

£ + JE* < * < ! + * * .

8
to

4

4

4

oKOHnaTejibHo nojiynaeM (pHcyHOK

101

sin x
5 tt

*

3 ti

K

8

eos X
O

P hc. 24

3aAa^iH ajih caiviocTOHTejibHoro pememiH
1.

PemHTb HepaBeHCTBO logcogxtg * + logcog;cl,5 > 1 .

2.

H elhth Bce 3HaneHHH x H3 npoMencyTKa - 1 < x < 4, y#OBjieTBopniomHe HepaBeHCTBy log0 75s in * > log_9.0,75 .


16

3.

PemHTb HepaBeHCTBO 2 sin 2 2x + 3 eos 2x < 0 .

4.

PemHTb HepaBeHCTBO log2( l + cos4*) < l + lo g ^ s in x .

5.

PemHTb HepaBeHCTBO 2 eos 8 x > 3 + 4 sin 4x .

7.

P e m H T b H epaB eH C T B O

11 sin x + eos 2 x - 6 < 0 .
eos2*

9.

PemHTb HepaBeHCTBO 2 sin * - 1 < V6sin2 * - 6 s i n * - 1 2 .

10. PemHTb HepaBeHCTBO

102

ctg *

OTBETM K 3AflAHAM
A Jia CAMOCTOHTEJIbHOrO PEfflEHHfl

rJIABA 1. ÜPE0 BPA3 0 BAHHE
TPHrOHOMETPHHECKHX BBIPA5KEHHÉE
§ 1.2. ^OKa3aTejii»CTBO ToacaecTB
h ynpomeHHe BBipaHcemiH
7. 1+ctg2 2 a . 8. 1. 9. 0. 10.

sin a

11. 0. 12. sin x . 13. -1 .

§ 1.3. 3aAa^iH Ha BbrmcjieHHe
1. 2. 2. 1. 3. 5. 4.

-6. 5. A

b

TpnroHOMeTpHH

6. 24. 7.

8

7
9. ± J - . 10. -0 ,5 . 11. — . 12. 7 + 24V3 13
50
'
V8
11
2J 2
4J 2
15. cosa = tg2a = —

VÍ5
8.
7
4
31 14 120
17*
119'

rJIABA 2. OCHOBHME METOflM PEIHEHHJI
TPHrOHOMETPHHECKHX yPABHEHHÍÍ
§ 2.2. CBefteHue TpnroHOMeTpHnecKoro ypasHeHHH

K KBaApaTHOMy
1. x = (-1)"+1—+ nn ; n e Z .
6

2.

jc= ±—+ 7m;

neZ.

3. x = ±—+ n n ; n e Z .
6

4. * = ±arccos^ ^ ——+ 27m; n e Z .
10

103

n nn
5. x = —
+—
4 2



neZ

;

.

6. x = (-1)" —+ nn ; n e Z .
6
7. x =

- + tcti ; n

e

Z

.

nn.
_
8. x = —
; neZ.
9. * = ( - l ) n— +— ; tieZ.
18 3
=
(
l
)
n+1+7i7i; n e Z .
10. o:
6

§ 2.3. Pa3Ji(meHHe Ha MHoxcHTejra
1. x = —+ nn, x = ( - l ) n+1 arcsin— + n n ; n e Z .

2

4

2. x = 7m ,

jc = --+ 27171, jc = (- 1 )” —+ 7tn

2

6

; neZ.

3. x = —+ — , jc = 7iTi; n e Z .
4 2
7171
.571
~
4. x = — , jc = ±— + 7i7z; n e Z .

3

12

=7cn, x = ± —+ - ^ ¾

5. * =

9

tíeZ .

3

71 , 7171, x = (/- i\n
• -------- +V17
7171. „
1) 1— arcsin
— ;- 1n e, Z
6. x ==—+—
6 3
2
4
2
27171
2nn

7. x = 7: , TCTi
4 2
5
9
57t , 57171
10nn

8. x =
4
2
3

9. x ==— ; n e Z .
2

7i ,
10. * = —+7i7i,
2

n
nn

+— ; n e Z .
10 5

jc = —

11 . * = 2 tc+47iti, # = (- l ) n-4 arcsin
104

—- +4 ti7í ; n e Z .
4

ho

TC71

71

Tí71

. «

n

12. x = — , x = —+ — ; n e Z , n > 0 .
5
4 2
717171

ry

13. * = —+ — ; n e Z .
6 3
14. x = nn ; n e Z
jc =

15.

TíTV

f-y

TíTI

-4 £y

ry

— ; n e Z . 16. x = — ; n e z .
3
3

17. x = —+ — , # = ± arCCos-^—— + 27ütz; n e Z .
4 2
2
§ 2.4. noHH^ceHHe CTeneHH
(-1)" arcsin(V 3-l) ---nn j
------------------------- .1

1.

* =

2.

* = =±

n e z7, .

—+ nn ; n e Z .

6

3 . x = n + 2nn ; n e Z .
A

TíTt

4. x = — ;

ti

ry

eZ .

± —+ 7C7i; n e Z .
3
71 7X77
7X 7X77
/7
6. x = ---- 1
---- , ;t = — h----; n e Z .
56 28
12 6
5.

jc =

7. * = — + —

10

; neZ.

5

8. * = + 7 7 + ^ .
15

5

neZ.

§ 2.5. BseAeHHe ¿jonojiHHTejiBHoro yrjia
1.

—-+ 7X71^ ^ =
* ==—
12

=7X77 ,

3. x

71 , 7X77 .
-------- 1------ ;

4.

6

2

2

77G¿?.

2

.

n„ e^Z7 .

jc ==— — + 27X/7,

12

+

x = —+ 2 n n y x = - —+ 2 n n ; n e Z

2. x

12

24

x = - — + 2nn; n e Z .
12

105

5 . x = 2nn , x = - —+ 2nn ; n e Z .
2

7. x =

+ un

2

x = — + 2nn, x = — + 2nn; n e Z .
12
12

rJIA B A 3. OTBOP KOPHEÍÍ
B TPHrOHOMETPHHECKHX YPABHEHHHX
§ 3.1. Ot 6 op KopHeif npn noMom,H
TpHroHOMeTpH^ecKoro HepaseHCTBa

1 . x = nn; n e Z .
2. x =

± —+ 2nn;

4

neZ.

3 . x = arctg5 + 2nn ; n e Z .
4. x = nn, x = —+ 2nn; n e Z .

6

5. x =

6. x

=

—+
2


6

2nn , x =


3

+

2nn; n e Z .

+nn; n e Z .

7. :r = —+ 2 tt7*; n e Z .

3

8. * = — + 2 tt7*; n—+2KTi luí —+2nn,— +2nn lu
4. xCTB0 «3K3AMEH»
rHrHeHHHeCKHH CepTH(J)HKaT

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npeACTaBJiHeT BameM y bhmmqhwo ceppm n o c o 6 nn
pj\a noAroTOBKM k EflMHOMy rocyAapcTBeHHOMy sioaM eH y.
l l o c o 6 m p a 3p a 6 o T d H b i n e f l a r o r a M M m M e T O A M c r a M W
C 6 o / lb L U M M C T a x e M A e f T e J l b H O C T M M M H O r O J i e T H M M y c n e i l l H b l M

k

onbiTOM noAroTOBKM yMaiAMxcíi
EAHHOMy rocyAapcTBeHHOMy 3 K3 aMeHy.

CepHH nOCOÓMM p a 3 p a 6 oTOHbl KOK flJlfl

3 4 >eKTHBHOrO

TpeHHHra ynaiAHxcji no Bbino/iHeHHio 3 K3 aMeHai4 HOHHOH
paÓOTbl, TOK H fl/lfl 3 aKpenJieHM 51 M AMarHOCTMKH 3 HQHMM
no HaHÓonee cno>KHbiM p a 3 AenaM 3 K3 aMeHa.
rioCOÓHJI peKOMeHAOBOHbl yHaLAHMCSI, yHHTe/lJIM, MeTOAMCTOM
M H/ieHOM npneMHblX KOMMCCMM AJI5I nOflrOTOBKM B paMKOX
yn eéH o ro n p o u e c c a ho y p o x a x , AonojiHMTe/ibHbix 3 aH 5UMH,
a T aK xe caMonoAroTOBKH k 3 K3 aMeHy.

ESRSIso
MATEMATMKA

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2020

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MATEMATHKA

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ISBN 978-5-377-15020-6

j| E r a 2 0 2 0
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