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Author: Гнездовский Ю. | и др.

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cos ϕ = sin

π

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ctg ϕ = tg

π

2

:

−ϕ .

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F ? > ? P N ? ' F 3 K * O C = @

F

@ * F J?>@ D (AX2 AX/&D[ P %@ ? $

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2

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!#"

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u FNB>@ ) @ '?C?"H" R FMK3* F P?F 'O- ½ , u1 Z ¿=?b C? F`NB>@ ) @ ?' C?"H* ?P "3L1* F J?" >O¿ S " P?C?'?+ @*@(ab) >"AS NL%* F J?" > C b "3,@ CL ½ ,u 1 Z

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a

A

A

O

O B

B b

b

p

u

S.TEU V

cR F b=_?FK3* P?F " '?'?+TQ%' @P?C K @?=3?S b??" * K\%S +T_?NB>b +HVNBR F b= _?FK3* P?F G " '?'?+TQ&' @P?C K = '?" S +T_?NB>b +H K\?>?C?QUS +T_?NB> b +TQ&NBR F b&"K3* ,_?" P?"K3" J?" '?C?"AS NLM_?F b NB_b FK3>BFK3* " QO !@ )3b C?J?'?+T"b NBJ?C FA'?F Q C * F Q "`_?P HhF QKF C H ' @J @b FH' @G ) +TS@ &c* K\ CE1 1 Z /&D&4 ;Z 6/ =ID Z ,\]A3=ID Cb C CE1 1 Z /&D&4 ;Z 6/ =ID 1 Z , *][]= =ID yKWb C`K3* F P?F '?+#NBRb@?S b?! &c* K\AF _?F b '?C?* "3b '?+HhC,b NBJ @ HhCO= * Fb F " a a

A

O

B

A

O

B

b

b

p

u

S.TEU V

OF J?>?C_b FK3>BFK3* CO=_?P?C?' @A b "3,@ C?"yNBRb N(C'?"sb "3,@ C?"y' @c" R FcR P @G '?C?$?" =¡' @ ) +TS@&c* K\ /&,4@ HhC¥NBRb@? OF J?>?C _b FK3>BFK3* CO= '?"(_?P?C?' @A b "3,@ C?"( NBRb N =' @ ) +TS@&c* K\ / ; 7/&D=ID * F J?>@ HhC&NBRb@?

u

d

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$

B

A1

S.TEU V

S " P?* C?>@b '?+T" =@`*@ >"<S " P?* CG

A1 OB C 1 A1 OB AOB1 .

9 " 9

%

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N &c N & B "HhN`>?P?NBR F S

N &ANBR N yKWb C.>?P?NBR F S@ ANBR3@ b "3C?* ' @F >G S.TEU V P?N'?FK3* C P @AC?NK @ * F" " ' @ ) +TS@ &c* (+*,u1

13M

CE,u13M

r,D&,2 A *BACE

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b j b j |an OaV |an ¤ g Ec T> bdc T>f>f

$

u

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N &c*NB>@ )@ '?C?"@ > H * F_?F >@ )@ '?F(' @?S@ HhCO=R\A"c NB>?S F Q F F ) ' @J?" ' @'?" >BF * F P @`* F J?>@?= b "3,@ @' @ H * F Q ^ ANBAcB R " C&F *b C?J?' @cUF *,* F J?" > C AB %@A P?C K ?B.C?) F P @" '?+AS "(>?P?NBR F S +T"ANG R CK1>BF '?" J?'?+HhC * F J?>@ HhC C A Hh" '?B :^ C < b ? C ' @ A B N

R C "(AnB A b CG ^ A AmB. AnB @ '?+ ANBR C O"WAC?'?" '?C?"H ANBR ^ AmB. n C ^ ?S b??" * K\F >?P?N'?FK3* =f' @¥>BAnB F * F G O AmB m ? P

F ¾ Q

F ? ' Z C b 3 " , @ * B N 3 R @

" ) B >

F ? ' ? $

F S AF G B _?F b '?" *cANBR N AF%AnB F >?P?N'?FK3* CO= @ANBR3@ " )>BF '?$?AmB F S?P?NBR F S +T"ANBR CO BF * F P?N & F '?C

O

E

d <

S _?C K @ '?+%?%@P?C K @$?" '?* P ?S b??" * K\US '?" '?" Q&* F J?>BF Q&S _?C K @ '?'?F R F NBRb@ ' @P?C K $?" '?* O P b "3C?*`' @K3* F P?F '?" S _?C K @ '?'?F R F NBRb@ ABC; ' @1P?C K p SZ$?" '?* P O ?S b??" * K\ S '?NB* P?" '?'?BC " Q* F J?>BF Q S _?CG O NBR F b K @ '?'?F R ABC; F,NBRb@ s_?C K @ '?'?+TQ1 F _?C?P @ " * K\1' @,>?P?NBR F S N & ABC. ABC ANBR N AkC.

"

BF * F P?F Q P @ S ' @ 1 J @ K3* CMA b C?'?+ " "(F >?P?N'?FK3* CO s"3b 360 C?J?C?' @,NBRb@SFAC?'UR P @ANKcF F ) ' @G A r J @

"

* \ K O r 1◦ cRb F S@1HhC?'?NB*@ r B 601 J @ K3* `R P @ANK @? @ A' @NBRb F S@UHhC?'?NB*@,F F ) ' @J @ " * K\ 0 1

S.TEU VXW

10 =

cRb F S@¥K3" >?NB'BAD@ F F ) ' @J @ " * K\ 00

F

1 60

1

J @ K3* ZHhC?'?NB* +%

100 =

@

1 ◦ . 60

A' @&NBRb F S@¥K3" >?NB'BAD@

1 0 . 60

cRb FH S¾FAC?' *BACED&AX/ ' @ ) +TS@ " * K\ $?" '?* P @b '?+TQ NBR F b=pF _?C?P @&G C?Q K\M' @>?P?NBR F S N & A NBR N =A b C?' @>BF * F P?F QMP @ S ' @P @AC?NK3N" "F >?P?N'?F G K3* C P?C K OW[?Y [?Y ϕ πn [?Y ◦ ◦ ◦ 180 = π

;

n =

?[ Y [?Y

1◦ ≈ 0,017453 00

1 ≈ 0,000005

;

180 ; ;

=

ϕ

10 ≈ 0,000291 1

[?Y

0

0

π

[?Y

· 180 ;

;

00

≈ 57 17 44 ,8.

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c p p b i z } i l

100 =

1◦ . 602

E

d <

*@ b,_?P?C?S "WA" '?+.R P @ANK3'?+T",C K3F F * S " * K3* S N &cC?",P @AC @ '?'?+T" Hh" P?+ J @ K3* F,SK3* P?" J @&cCLBK\MNBRb F S Y4 9Y OP @ANK3+

0 15 30 45 60 75 90 120 135 150 165 180 225 270 315 360

!@AC @ '?+

0

π π π π 5π π 2π 3π 5π 11π π 12 6 4 3 12 2 3 4 6 12

5π 3π 7π 2π 4 2 4

!@AC @ '?'? N & Hh" P?NZNBRb F S +LZS "3b C?J?C?' K3J?C?*@&c*&FK3'?F S '?F QZC¥F F ) ' @G J?" '?C?"c[?YF _?NK3>@&c*=* " ?_?C?NB*?1◦ =

π ≈ 0,017453, 180

30◦ =

π 6

CMAPO

OF =?J?* F,S "3b C?J?C?' @NBRb@ AOB P @ S ' @ n◦ Cb C ϕ P @AC @ '?F S =S +TP @G ,@ " * K\U)@ _?C K3#& ◦ K3F F * S " * K3* S " '?'?F :%@ K3* F NBRb +∠' @ AOB ) +TS@&c=* nCL S "3b C?J?C?' @ HhCOs%@ _?∠P?AOB C Hh" PO=y=NBR ϕ Fb S "3b C?J?C?' @>BF * F P?F R FP @ S ' @ P @AC @ '?F S =B' @ ) +TS@&c* , Z 1>= AOB, ϕ ϕ cR F b S P @AC @ '?F SB(' @ ) +TS@ " * K\ 1 Z L ◦ Cb CO=pJ?* F¾* FZ" = / = ?(F b '?+T360 QUNBR F b&)@ _?F b '?" *SK &_2π b FK3>BFK3* s"3b C?J?C?' @P @ ) S " P?'?NB* F R F@b '?+T"(NBRb +P @ S '?+_?F,S "3b C?J?C?'?" S "&_?" P?"K3" >@ &cC?"K\#_?P Hh+T"1F P @ )

N &c* S " P?* C?>@b '?+T"&C K Hh"3,G '?+T"&NBRb +%ys"3b C?J?C?' @ Hh" '? " R F C?)1'?CL' @ ) +TS@ " * K\ , Z 1>= = ; - CE, *][]= =ID s"3b C?J?C?' @NBRb@ Cb C&P @ S '?F R F_?F,S "3b C?J?C?'?"("HhN`S " P?* C?>@b G '?F R FNBRb@ "K3AOB * TNBR F b@ &cC HhC K\_?P Hh+HhC A1 A C C?) AF 1 OB P @1 )" '?'?+HhCM' @,P?C K ?? B1 B,

! ! $ ]" 0!#"$ ! ! D&/&,251>= F K3* P?F R FTNBRb@T_?P HhF NBR F b '?F R F* ?P " NBR F b '?C?>@T' @ ) +TS@ " *G

K\UF * '?F " '?C?"%_?P?F * C?S F b "3,@ " R F>@ * " *@>&R C?_?F * " '?NB) "%C&F F ) ' @J @ " * K\ P?C K O WNKWb F S '?+H ) ' @ >BFH sin

sin A =

a . c

cos A =

b . c

1325D&/&,251>= FK3* P?F R F&NBRb@U_?P HhF NBR F b '?F R FU* P?" NBR F b '?C?>@U' @ ) +G S@ " * K\UF * ?' F " '?C?"%_?P?Cb "3,@ " R F>@ * " *@,>1R C?_?F * " '?NB) "%C1F F ) ' @J @ " * K\ N KWb F S '?+H ) ' @ >BFH P?C K O Wcos

AX/ ; /&251>= FK3* P?F R F NBRb@¾_?P HhF NBR F b '?F R F¾* P?" NBR F b '?C?>@¥' @ ) +G S@ " * K\ F * '?F " '?C?"_?P?F * C?S F b "3,@ " R F&>@ * " *@&> _?P?Cb "3,@ "HhN¥>@ * " * N CUF F ) ' @J @ " * K\`NKWb F S '?+H) ' @ >BFH P?C K O W-

tg

tg A =

a . b

1347AX/ ; /&251>= FK3* P?F R FNBRb@,_?P HhF NBR F b '?F R F,* ?P " NBR F b '?C?>@,' @G ) +TS@ " * K\MF * '?F " '?C?"(_?P?Cb "3,@ " R F>@ * " *@>&_?P?F * C?S F b "3,@ "HhN>@ * "3G * NMCUF F ) ' @J @ " * K\`NKWb F S '?+H) ' @ >BFH P?C K O Wctg

ctg A =

b . a

3d u )w u

; (AX/&251>= FK3* P?F R F,NBRb@_?P HhF NBR F b '?F R F* P?" NBR F b '?C?>@' @ ) +TS@G " * K\ F * '?F " '?C?"R C?_?F * " '?NB) + > _?P?Cb "3,@ "HhN¥>@ * " * N¾C F F ) ' @J @ " * K\ NKWb F S '?+H ) ' @ >BFH P?C K O Wsec

c . b

sec A =

132 ; (AX/&251>= FK3* P?F R FNBRb@@' @ ) +TS@ " * K\ F * '?F " '?C?"{R C?_?F * " '?NB) + > _?P?F * C?S F G b "3,@ "HhN>@ * " * N C F F ) ' @ J @ " * K\ NKWb F S '?+H ) ' @ >BFH P?C K O W-

B c a A

C

b

cosec

S.TEU VXW W

cosec A =

c . a

xyC?'?NK =>BFK3C?'?NK =*@ '?R " ' K =>BF *@ '?R " ' K =fK3" >@ ' KMC#>BFK3" >@ ' KMFK3* P?F R F NBR3b@(' @ ) +TS@&c* K\ 4= ; 4BF *@ '?R " ' K = K3" >@ ' KsC,>BFK3" >@ ' KyFK3* P?F R F NBRb@•* P?" NBR F b '?C?>@)@ S C K\?** F b >BFF *,S "3b C?J?C?'?+NBRb@?

" $ # &%

! !

(' %

$

!

! %

_?P HhF NBR F b '?F R F * P?" NBR F b '?C?>@ P?C K % `_?P?F _?C K3'?+HhC ACB NB>?S@ HhC F F ) ' @J?" '?+>@ > S " P? C?'?+%=O*@ > C S "3b C?J?C?'?+.S '?NB*G P?" '?'?CL`NBRA, b F S B, =_?P?CCIH * FHNBR F b _?P HhF QOV C >@ * " * +%=B_?P?F * CG C a b R C?_?F * " '?NB)@? S F b "3,@ C?"(K3F F * S " * K3* S " '?'?FS " CP? C?' @ H A B; c 037[¡0 "y_?FTAS NHZK3* F P?F ' @ H&' @LFA ?* K\AP?NBR C?"y" R FK3* F P?F '?+ CNBRb +% ,+

E

d

@ '?'?+T" A, c A, a a, c a, b

Y4 9Y

!h" " '?C?" B = 90◦ − A;

b = c cos A a B = 90◦ − A; b = a ctg A; c = sin A √ a sin A = ; B = 90◦ − A; b = c cos A; b = c2 − a2 c √ a a tg A = ; B = 90◦ − A; c = ; c = a2 + b 2 b sin A "

a = c sin A;

('

!

BF * F P?F QUC?)Hh" P! &c*FK3*@b '?+T">?P?NBR F S +T"ANBR CO (FAF '?F%NBRb@ H¥>?P?NBR F S +T"ANBR C`C?)Hh" P! &c*<SR P @ANK @LC`P @AC @ ' @L¡ (P?NBR F S F Q%ANBR F QS(FAC?' *BACE,2 ' @ ) +TS@ " * K\>?P?NBR F S@%ANBR3@?=A b C?' @ >BF * F P?F QMP @ S ' @ 1 A b C?'?+ " "%F >?P?N'?FK3* CO ! @AC @ '?' @¾360 Hh" P @M>?P?NBR F S F Q&ANBR C¾F _?P?"WA"3b??" * K\Z>@ >¾F * '?F " '?C?"" " A b C?'?+ >&P @AC?NK3NF >?P?N'?FK3* CO=?' @>BF * F P?F QUF ' @P @ K3_?F b F" ' @?

B1

B2

A

b

O

B1 O

A1

A2

B2 C

a

S.TEU VXW

B3

S.TEU VXW

(P?NBR F S F Q ANBR F QS FAC?' *BACED&AX/ ' @ ) +TS@ " * K\ ANBR3@¥F >?P?N'?FK3* CO= A b C?' @>BF * F P?F QUP @ S ' @%A b C?'?"%P @AC?NK @oH * F QUF >?P?N'?FK3* CO OF = J?* F%S "3b C?J?C?' @>?P?NBR F S F Q%ANBR C AB P @ S ' @ n◦ Cb C ϕ P @AC @ '?F S S +TP @,@ " * K\M)@ _?C K3# & ^ ◦ K3F F * S " * K3* S " '?'?F ^ AB = n

AB = ϕ

<

& )

! -

OP @ANK3' @&CZP @AC @ '?' @1Hh" P?+>?P?NBR F S +LÀNBRBFUF *&K3F F * S " * K3* S N &c" R F $?" '?* P @b '?F R FNBRb@? OP @ANK3' @, P @AC @ '?' @S "3b C?J?C?' @>?P?NBR F S F QANBR C(J?C KWb " '?'?FP @ S G ' @R P •@ANK3'?F Q P @AC @ '?'?F QTS "3b C?J?C?'?"BF R\AD@ C$?" '?* P @b '?+TQ#NBR F b r1xy* F P?F r' r1'?F@L¡_?" P?"K3" >@ " o * H * C1F >?P?N * F J?>@ (ab). _?" P?"K3" J?" '?C&KF a>?P?N'?FK3* # & P @AC?NK @ C * F J?>@_?A" 1P?" K3" J?" '?C ' @b F R C?rJ?1'?F&K3* AF 2P?F ' @ _?" P?"K3" >@ " K * H * C KF >?P?N'?FK3* # &.P @AC?NK @ r . 2 F >?P?N'?FK3* C1S`* F J?>@L¡* F J?>@_?" P?"K3" J?" '?C¾K?P?N'?FK3* # &P @G 1 _? " P?"K3" J?" '?C#K&F >?P?N'?FK3* # C * F J?B >@¥ &P @AC?NK @ AC?NK @ r B 1 2 (P?NBR F S@ANBR3@ b "3C?*' @%F >?P?N'?FK3* C`P @AC?NK @ @?P?NBR F Sr@2 . A1 B1 r , 1 ANBR3@ b "3C?* ' @¥F >?P?N'?FK3* C#P @AC?NK @ (A F2NBBRb 2 F S F Q S "3b C?J?C?'?7 " H * C ANBR CP @ S '?+%r2 . < b ? C ? ' + N'?CLP @ ) '?+T" -A b CG ^ ^ AA 1B r ' @< NBR 1C = A2 B2Hh. " '? "cA b C?'?+ ANBR C ϕ 1 S _?C K @ '?'?F R FNBRb@P @ S ' A O l s"3b C?AJ?1C?B' @ @2_?BF 2b .F G S C?'?"yNBRb F S F Q<S "3b C?J?C?'?+ ANBR CO=' @c>BF * F P? N & F ' F _?C?P @ " * K\O b CO=hJ?* F * F¥" =hS "3b C?J?C?' @¥S _?CG K @ '?'?F R FcNBRb@cP @ S ' @_?F b F S C?'?"sS "3b C?J?C?'?+$?" 'G S.TEU VXW * P @b '?F R F NBRb@?=>BF * F P?+TQF _?C?P @ " * K\' @* N "cANBR N =J?* FC&S _?C K @ '?'?+TQUNBR F b s_?C K @ '?'?+TQNBR F b=F _?C?P @ &cC?Q K\?P?NBR3@?=F R P @ '?C?J?" '?' @ *,u1 AS NHpU" R F,P @AC?NK @ HhCUCMANBR F QUF >?P?N'?FK3* CU>?P?NBR3@? fKWb F S '?+T"cF F ) ' @J?" '?CZ P?C K D BWP @AC?NKF >?P?N'?FK3* COV P @AC @ '?' @1Hh" P @`>?P?NBR F S F QMANBR CO=F R P @ '?rC?J? C?S@ &c" QZK3" >?* F P $?" '?* P ϕ@b G '?F R F&NBRb@WV ◦ R P @ANK3' @ Hh" P @&>?P?NBR F S F Q¾ANBR CO=F R P @ '?C?J?C?S@ &c" Q K3" >?* F P $?" '?* nP @b '?F R F,NBRb@WV A b C?' @,>?P?NBR F S F QÀNBR CO=?F R P @ '?C?J?C?S@G &c" Q&K3" >?* F POV _b F @AK3l " >? * F P @?

S

_ n l l l cXj g U b >i

p j zBg c p b nE j V

¨

kml i

El Ef b n+p i VBwrdq~t vz l in+p i Vwrdqdu l f+p } p b i7g c p dc>b~

l= !

πr n; 180

)

l = rϕ;

!

S=

πr2 n; 360

0!

S=

1 2 r ϕ. 2

!

!

%@ >USKWb NBJ @ "' @ _?P @ S b " '?'?+LMF * P?" ) >BF S& S " >?* F P?F SBfP @ )3b C?J @&c*,' @G J @b '?N & C`>BF '?" J?'?N &* F J?>?CO=B*@ >`C`SKWb NBJ @ "ANBR C`' @?P?N'?FK3* C`P @ )3G b C?J @&c* ' @J @b '?N &C>BF '?" J?'?N & * F J?>?C ANBR COT' @b F R C?J?'?F SKWb NBJ @ " NBRb@P @ )3b C?J @ &c*,' @J @b '? N & C&>BF '?" J?'?N &K3* F P?F '?+ NBRb@? _b FK3>BFK3* C)@ E(C?>K3C?P?N "H1_?F b F" '?C?"fb NBJ @ Ks' @J @b FH1S* F JG 13/&13M NBRb@?=@,* F J?>?N >B" P?C K p sC1' @ ) F S "H " R F /&AX\]A Z 6/&13M 254715*OA O /&AX\]A Z 6/&13M 4713\](13M >?P?NBR F S F QMANBR COsN A"H#S P @ @ * { H * F *b NBJ A AFU* "3LZ_?F PO=_?F >@`F 'Z'?"_?" P?" QBA" *USM_?F b F" '?C?" S F >?P?NBR?C P?C K @W(P?o C OH * FHZ* F J?>@ F _?C?" *>?P?NBR F S

N &ANBR N P?C K OB BW A AB (F '?" J?'?F "U_?F b F" '?C?" S P @ @ &c" R FK\ b NBJ @¥' @ ) F S "H (13/ ; \]/&13M (13/ ; \]/&13M¦4713\](13M >?P?NBR F S F QANBR CO 254715*13/&13M NBRb@?=?@* F J?OB >?N B ' 31 3P ; /&/&13M (+*,u1 13M CE,u13M P?C K B' @ ) +TS@ " * K\¾E(C?R NBP @?= F _?C K3+TS@ "H@* F J?>BF QO=sK3F S " P?@ &c" Q>?P?NBR F S F "MAS C" '?C?"¾K1NB>@ )@ '?CG "H ' @ _?P @ S b " '?CAS C" '?CM* F J?>?CO=@"c" "c' @J @b '?F R FM ' @J @b '?F Q * F J?>?CMANBR CsCU>BF '?" J?'?F R F& >BF '?" J?'?F QU* F J?>?CUANBR Cy_?F b F" '?CO @ F " '?'? N & >?P?NBR F S

N & ANBR N )@ _?C K3+TS@ &c*1AS NHp NB>?S@ HhC KÀNG R F Q,_?" P?"WA`'?C HhCO= _?P?C?J?"H¥' @(_?" P?S FH¥Hh"K3* "cK3* F C?*% NB>?S@?=F F ) ' @J @ &c@ ' @J @b FBF '?" J?'?F Q¾* F J?" >¾'?"WAFK3*@ * F J?'?F ¡%@AFM)@AD@ * M' @ _?P @ S b " '?C?"BF QO1F'?F`_?FK3* NB_?C?* UC?' @J?" )@AD@ * (F F " '?'? N & ANBR N* P?"Hp` NB>?S@ HhCO=' @ _?P?C Hh" P ^ R\A" ' @J @b ' @(* F J?>@sANBR CO= >BF '?" J?' @%" "y* F J?>@?=@ * AcB, F J?>@?=3b "3,@ A@ ' @ C *@ >BF "()@AD@ '?cC? "%K H?' @P?C K \W A

B

!

£

u

= , Z 1>= P?C K T ?CO=D>BF * F P @ ' @ ) +TS@ " * K\ ; *7D&/&13M F F " '?'?F R F(NBRb@?= _?P?C?J?"H NB>@ )@ '?F = >@ >BF QC?) H * CLb NBJ?" Q# K3J?C?*@ " * K\_?" P?S +H ' @J @b ' @K3* F P?F ' @Zb NBJ @C>@ >" R F S P @ @ * BFK3* C%S F >?P?NBRh* F J?>?C ST_?F b FCG yKWb C(b NBJ * 3" b '?FH' @ _?P @ OA S b " '?C?C¥ P?C K O @W=B* FANBR3@ F _?C K @ '?' @OM* F J?>BF Q 1XL C K3F F * S " * K3* S N &cC?QH * FHhN¾S P @ " '?C!&.NBR F b AB, ' @ ) +TS@&c* K\ 1 Z A, -0D&4 ;Z 6/ =ID yKWb CS P @ " '?C?"fb NBJ @ F AOB * P?C?$ @ * "3b '?F " P?C K B BW= * F K3F F * S " * K3OC * S N &cN & ANBR N CNBR F b 6/ =ID ' @ ) +TS@ &c* 134@ >1S R P @ANK @L¡=?*@ >1C1S`P @AC @ ' @L¡=BA" Q K3* S C?* "3b G B '?+H J?C KWb FH=p@ K3F b &c* ' @ S "3b C?J?C?' @&>BF * F G P?F R FP @ S ' @K3F F * S " * K3* S

N &c" Q%S "3b C?J?C?'?"ANG R C& NBRb@W=S )3?* F Q,K3F) ' @ >BFH1_b &K ="KWb C?P?NBR F S F Q%ANBR C& NBR3G b@TK3F S _ @AD@ &c*,C1_?P?C H * FH'?"K3C?P?N "H P?C K f4 3 R F P?C?) F '?*@b '? N & FK3 Ox, C S " >?* F P OA ' @'?" QO @ K3 _?P?C Hh"H)@,' @J @b '? N &K3* F P?F '?N`NBRb@ @,>BF '?" $ Ox AOB, S " >?* F P @ −→ )@' @J @b '? N & * F J?>?N>?P?NBR F S F QBF '?$?FH S " >?* F P @ * F J?>BF Q OB F _?C K3+TS@ "H>?P?NBR F S AOB +T"ANBR CUb &c F QUS "3b C?J?C?'?+% B) (F b F" '?C?"fS " >?* F P @ −→ '?"sF _?P?"WA"3b??" *FA'?F ) ' @J?'?FS "3b C?J?C?'?N%NBR3G OB b@ CMANBR C AOB AB. @ >O="KWb CS " >?* F P −→ S P @ @ K3S F >?P?NBRy* F J?>?C O Sc_?F b FC?* "3b G '?FH' @ _?P @ S b " '?C?CO='?"%F OB, _?C K3+TS@ " *_?F b '?F R F,F F P?F *@M P?C K O4 3 W=* F C ^ R\A" ◦ ∠AOB = n◦ AB = n◦ , 0 < n◦ < 360◦ Cb C&S,P @AC @ ' @L¡C ^ R\A"

yKWb C%S " >?* F P

∠AOB = ϕ

AB = ϕ,

0 < ϕ < 2π).

F _?C K @b%* P?C%F F P?F *@( _?P?F * C?SLFAD@sJ @ K3F S F QK3* P?"3b?G OB >?CsC&" "(_?F S F P?F * ' @,NBR F b ◦ P @AC @ '?F SBW=B* F −→

n (ϕ

∠AOB = 360◦ · 3 + n◦ = 1080◦ + n◦

C

AB = 360◦ · 3 + n◦ = 1080◦ + n◦

^

C ^ AB = 2π · 3 + ϕ = 6π + ϕ). yKWb C&F,J?C KWb "%F F P?F * F S'?C?J?" R F,'?"(C?) S "K3* '?F =?* F C ◦ ◦ ^ ◦ ◦

(∠AOB = 2π · 3 + ϕ = 6π + ϕ

Cb C

∠AOB = n + 360 · k

C

AB = n + 360 · k

^

b &c F "($?"3b F "(J?C KWb F k@ >?C H¾F P @ ) FH="KWb C'?"TF R F S F P?" '?F = * F%F F " '?' @%>?P?NBR F S@%ANBR3@ CZF F " '?'?+TQ&NBR F bZ)@AD@ &c* K\¾KBF * F P?F R F1K3F S _ @AD@ " *1K' @J @b G '?F QK3* F P?F '?F QBF "KWb F" '?C?"TF F " '?'?+L >?P?NBR F S +LMANBRFA'?F R F`C¾* F R F`"P @AC?NK @?= _?P?CH * FHANBR C¾F * >b@A+TS@G &c* K\&' @F >?P?N'?FK3* CO=P @AC?NK>BF * F P?F QUP @ S " 'UP @AC?NK3N,ANBRW

!! $

]"

! ! %! ! %@Z_b FK3>BFK3* CS S "WA"H _?P HhF NBR F b '?N & A" >@ P?* F S N K3C K3* "HhN >BF F PG AC?' @ * P?C K sWp(F b FC?* "3b '?N & _?F b N FK31@ K3$?C K K N A"H K3J?CG *@ * M' @Oxy J @b '?F QZK3* F P?F '?F Q1NBRb F S = >BF * F P?+T"F P @ ) N c & * K\1_?P?C¾S P @ " '?C?C S F >?P?NBR* F J?>?C s" >?* F P K3F ' @ _?P @ S b " 'UK(FK3#& @," R F?C >BF F P AC?' @ * + @MN >BF F P AAC?' @ * + F F ) ' @J A(r, @ "H 0), C * F J?>?C B x y, * " OF R\AD@F * '?F " '?C x y C x S.TEU VXW B(x, y). , '?" )@ S C K\?*F *&A b C?'?+ _?FAS Cr'?F r R FcP @yG AC?NK @GS " >?* F P @?=B@)@ S C K\?** F b >BF,F *S "3b C?J?C?'?+ NBrRb@ ϕ AOB. xfb "WAF S@ * "3b '?F =F * '?F " '?C x y C x HhF'?F P @ K K H@ * P?C?S@ * , >@ >ME(NB'?>?$?C?CO=@ P?R NHh" '?* FH >BF * F P?+rL,?Srb??" * K\yM_?P?F C?) S F b '?+TQNBR F b ϕ. D&/&,251>= NBRb@ F P @ ) F S@ '?'?F R F_?FAS C'?+HP @AC?NK3FHpGS " >?* F G ϕ, P?FH −→ Ks_?F b FC?* "3b '?F Q_?F b N FK3# &@ K3$?C K K P?C K W=' @ ) +TS@ " * K\?* F P @&" R FA b C?'?" sin ϕ =

y . r

cos ϕ =

x . r

NBRb@ F P @ ) F S@ '?'?F R FBF '?$ @_?FAS C'?F R FcP @AC?NK @GS " >?* F P @>" R FA b C?'?"

1325D&/&,251>= −→

* F P?FH

AX/

BN Rb@ F P @ ) F S@ '?'?F R F`_?FAS C'?+H P @AC?NK3FHpGS " >G ϕ, K_?F b FC?* 3" b '?F Q_?F b N FK3#& @ K3$?C K K( P?C K ?W= ' @ ) +TS@ " * K\

; /&251>=

−→

OB

u

F * '?F " '?C?"TF P AC?' @ * +>@ K3$?C K K3">BF '?$ @(_?FAS C'?F R F(P @AC?NK @GS " >?* F P @?tg ϕ =

y x

(x 6= 0).

x y

(y 6= 0).

r x

(x 6= 0).

NBRb@ >BF * F P?+TQ¾F P @ ) F S@ 'Z_?FAS C'?+H P @ACG ϕ, N K3FHpGS " >?* F P?FH −→ KZ_?F b FC?* "3b '?F Q _?F b N FK3#&@ K3$?C K K P?C K (W= ' @ ) +TS@ " * K\ F * '?F OB " '?C?"U@ K3$?C K K3+ >F P AC?' @ * "`>BF '?$ @Z_?FAS C'?F R F1P @G AC?NK @GS " >?* F P @?1347AX/

; /&251>=

ctg ϕ =

BN Rb@ F P @ ) F S@ '?'?F R F_?FAS C'?+HZP @AC?NK3FHpGS " >?* F G ϕ, K_?F b FC?* "3b '?F Q¥_?F b N FK3#& @ K3$?C K K& P?C K yW=O' @ ) +TS@ " * K\ P?FH F * '?F OB " '?C?"A b C?'?+ _?FAS C'?F R FP @AC?NK @GS " >?* F P @T>@ K3$?C K K3"y" R Fc>BF '?$ @?

; (AX/&251>=

−→

sec ϕ =

BN Rb@ F P @ ) F S@ '?'?F R F(_?FAS C'?+H P @AC?NK3FHpGS " >G ϕ, K_?F b FC?* "3b '?F Q(_?F b N FK3#& @ K3$?C K K P?C K =3' @ ) +TS@ " * K\F *G * F P?FH '?F " '?OB C?"cA b C?'?+ _?FAS C'?F R FP @AC?NK @GS " >?* F P @&F P AC?' @ * "(" R F,>BF '?$ @?

132 ; (AX/&251>= −→

cosec ϕ =

r y

(y 6= 0).

(FK3" >@ ' KcNBRb@ C?'?F R\AD@F F ) ' @J @&c* xyC?'?NK = >BFK3C?'?NK =3ϕ*@ '?R " ' K = >BF *@ '?R " ' K = K3" >csc @ ' Khϕ. C%>BFK3" >@ ' Kh_?P?F C?) S F b G '?F R F(NBRb@' @ ) +TS@ &c* K\ 4= ; 4?$?C?QU_?P?F C?) S F b '?F R FNBRb@? (P?F " >?$?C?C#P @AC?NK @GS " >?* F P @¥* F J?>?C ' @ >BF F P AC?' @ * '?+T"&FK3C P @ S '?+ K3F F * S " * K3* S

N &cC H&>BF F P AC?' @ *@ I H H * F Q* F J?>?CO @ F ) ' @J?C HM_?P?F " >?$?!C &P @G − → AC?NK @GS " >?* F P @ ' @ _?F b N FK3 F P AC?' @ * J?" P?" ) @ ' @ _?F b N FK3 ry , @ @ K3$?C K K J?" P?" ) OB OF R\AD@ xfb "WA F S@ * "3b '?rF x .=W* P?C?R F '?FrHhy" * =P?C?y,J?"K3>?rC?x"y= E(NBx.'?>?$?C?C(_?P?F C?) S F b '?F R FsNBR3G b@F _?P?"WA"3b?! &c* K\1KWb "WA N &cC HhC&F * '?F " '?C HhCOsin ϕ = ctg ϕ =

ry , r

rx , ry

cos ϕ = sec ϕ =

rx , r

r , rx

tg ϕ =

ry , rx

cosec ϕ =

r . ry

)

!

!#"$&%

C& +b@_?P?C?'?*@,S csS S @ >>@ >%* P?C?R F '?FHh" * P?C?J?"K3>?C?"sE(NB'?>?$?C?C?* F P @?= * Fc_?P?CS S "WA" '?C?C?P?N'?FK3* >BF F P AC?' @ * '?F Q_b FK3>BFK3* C _?FK3* P?F C HF >?P?N'?FK3* K&$?" 'G "WAC?'?Oxy C?J?'?F R F%P @AC?NK @ ; CED&/&D&\]/&,8 13(.L * P?FH¾S%' @J @b "T>BF F P AC?' @ * O *,.-0/&1325476 OF J?>?N_?" P?"K3" J?" '?C"WAC?'?C?J?'?F QF >?P?N'?FK3* C`K_?F b FC?* "3b '?F Q,_?F G .y* FU' @J @b ' @1* F J?>@M>?P?NBR F S +LMANBR b N FK3# &@ K3$?C K KBF '?$?FH¥P>?ϕP?NBR F S F QANBR C`S "3b C?J?C?'?+ B(P? C H * FH P @AC?NKWGS " >?* F P ϕ −−→ N A" *,F P?* FH>BF '?" J?'?F Q&K3* F P?F '?+NBRb@S "3b C?J?C?'?+ OPϕ ϕ

! %@U"WAC?'?C?J?'?F Q¥F >?P?N'?FK3* C¥* F J?>?C C _?P?C¾b &c FH A" Q K3* •S C?* "3b '?FH CMb &c FH $?"3b FH K3F S _ P@ϕAD@ &c*?P- ϕ+2πk ϕ

k Pϕ = Pϕ+2πk ∀ϕ ∈ R, ∀k ∈ Z. %@ _?P?C Hh" PO=_?P?Cb c FH $?"3b FH * F J?>?C K3F S _ @AD@ c*%Cb "3,@ ' @1_?" P?"K3" J?" '?C?C"WAC?'?C?J?'?F Q F >?P?Nk'?FK3* C K_?PF b 2πk FC?* "3b '?F Q _?F b N FK3

&

&

#&

*

@ K3$?C K K =B* F J?>?C K_?F b FC?* "3b '?F QM_?F b N FK3#&F P AC?' @ *=* F JG Pπ +2πk >?C KF * 2P?C?$ @ * "3b '?F Q`_?F b N FK3#&@ K3$?C K K = * F J?>?C Pπ+2πk P 3π 2 +2πk KF * P?C?$ @ * "3b '?F QU_?F b N FK3#& F P AC?' @ * x _?FHhF #&"WAC?'?C?J?'?F Q F >?P?N'?FK3* C * ? P C?R F '?FHh" * P?C?J?"K3>?C?" E(NB'?>?$?C?C _?P?F C?)3G y

S F b '?F R FNBRb@F _?P?"WA"3b?! &c* K\,KWb "WA N &cC H 1 Pϕ F P @ ) FH sin ϕ D&/&,2 NBRb@ P @ S " ' F P AC?' @ * "¥>BF 'G ϕ ϕ Pπ P0 $ @ P @AC?NK @GS " >?* F P @ −−→ "WAC?'?C?J?'?F Q −1 cos ϕ O 1 x F >?P?N'?FK3* CO=F P @ )

N &cOP " R F&ϕ NBR F b K_?F G b F C?* "3b '?F Q%_?F b N FK3# & @ K3$?C K K P?ϕC K W 1325D&/&,2 NBRb@ P @ S " 'U@ K3$?C K K3"c>BF 'G −1 ϕ $ @ P @AC?NK @GS " >?* F P @ −−→ "WAC?'?C?J?'?F Q F >?P?N'?FK3* CO=F P @ )

N &cOP " R F&ϕ NBR F b K_?F G S.TEU VXW b F C?* "3b '?F Q%_?F b N FK3# & @ K3$?C K K P?ϕC K W ')2568 47AX/ ; /&251 ' @ ) +TS@ " * K\ZFK3 =D>@ K @ &c@ K\Z"WAC?'?C?J?'?F Q¥F >G P?N'?FK3* CS* F J?>B"s_?" P?" K3" J?" '?C" "TKy_?F b FC?* "3b '?F Q_?F b N FK3# &@ K3$?C K K C&K3F S _ @AD@ &c@M_?F,' @ _?P @ S b " '?!C &K(FK3# & F P AC?' @ *1 P?C K W +

)

AX/ ; /&2 NBRb@ P @ S " 'F P AC?' @ * "* F J?>?C ?S b?!&c" Q K\* F J?>BF Q M, _?" P?"K3" J?" '?CMFK3CM*@ '?ϕR " ' K3F S,Kc_?P HhF QO=>BF * F P @M K3F S _ @AD@ " *KcP @AC?NK3FHpG S " >?* F P?FH −−→ "WAC?'?C?J?'?F QMF >?P?N'?FK3* CO=F P @ ) N &cC H Kc_?F b FC?* "3b G '?F QU_?F b N FK3OP # & ϕ@ K3$?C K KNBR F b ϕ ')2568 (1347AX/ ; /&251 ' @ ) +TS@ " * K\FK3 =>@ K @ &c@ K\ "WAC?'?C?J?'?F Q F >?P?N'?FK3* C¥SU* F J?>B",_?" P?" K3" J?" '?C¥" "K_?F b FC?* "3b '?F Q¾_?F b N FK3# &.F PG AC?' @ *,C&K3F S _ @AD@ &c" QM_?F' @ _?P @ S b " '?!C &KFK3# &@ K3$?C K K, P?C K W y

tg

tg ϕ

y

M

1

1 Pϕ Pπ −1

ctg

M Pϕ

ϕ

P0 1

O

ϕ

Pπ −1

P0 1 ctg ϕ

O

x

x −1

S.TEU VuW

−1

S.TEU V

y 1

y ϕ

1 Pϕ Pπ −1

ϕ O

P0 1

M sec ϕ

−1 Pϕ

O

1

x

x −1 M cosec ϕ

−1

S.TEU V

P0

Pπ

S.TEU V

1347AX/ ; /&2 NBRb@ P @ S " '@ K3$?C K K3"y* F J?>?C Sc>BF * F P?F QBF G M, *@ '?R " ' K3F S_?" P?"K3" >@ " * K\ϕKs_?P HhF QO= K3F S _ @AD@&c" QKs P @AC?NK3FHpGS " >?* F P?FH −−→ "WAC?'?C?J?'?F QF >?P?N'?FK3* CO=hF P @ ) N &cC HKM_?F b FC?* "3b '?F Q_?F b NG OP FK3# ϕ& @ K3$?C K KNBR F b ; (AX/&2 NBRb@ ϕ P @ S " '1@ K3$?C K K3"%* F J?>?C ?S b?! &c" Q K\1* F J?>BF Q _?" P?"K3" J?" '?CFK3C@ ϕK3$?C K K`K`>@ K @ * "3b '?F Q > "WAM, C?'?C?J?'?F QF >?P?N'?FK3* CS +

u w u

>BF '?$?"P @AC?NK @GS " >?* F P @ −−→ >BF * F P?+TQ¾F P @ ) N " *`NBR F b K_?F b FCG ϕ * 3" b '?F QU_?F b N FK3#& @ K3$?C OP K K, ϕP?,C K W 132 ; (AX/&2 NBRb@ P @ S " 'ZF P AC?' @ * "* F J?>?C ?S b?!&c" Q K\Z* F JG >BF Q@ K @ * "3b '?F Q"WAM, C?'?C?J?'?F QF >?P?N'?FK3* C KT_?F b FCG S%>BF '?$?"TP @AC?NK @GS " >?* F P @ −−→ >BF * F P?+TQF P @ ) N " *NBR F b OPϕ , ϕ * "3b '?F QU_?F b N FK3# & @ K3$?C K K, P?C K W K3F F * S " * K3* S C?CZKf>uUOT{U b 5pfEU S.TEU V

P0 1 x

P 3π −1

2

T>f>uUOT{ l U b 5pfEU

+ O

P 3π −1

2

y 1

` pf>| b fEU#T: l i pf>| b fEU

'

$

!

! %

sS "WA"H NKWb F S '?+T"cF F ) ' @J?" '?C¾ P?C K ,3 W K3* F P?F '?A,+.B,* P?" CNBR F b S '?'?C?NB>*@ G P?" '?'?C?"`NBRb + * P?" NBR F b '?C?>@ _?P?F * C?S F b "3,@ C?"¾K3F F ABC; * S " * K3* S " a,'?'?b,F cS " P?C?' @ H ABC, A, _?" P?C Hh" * POV _?F b NB_?" P?C Hh" * POV _b F @A V P @AB,C?NC; K<S _?C PK @ ' G p S r '?F Q`F >?P?N'?FK3* COV P @AC?NKTF _?C K @ '?'?F QF >?P?N'?FK3* COV S +K3F *@ R ha >&K3* F P?F '?" a.

¨B©uü1gpo(g hg k epopn

B a

P = a + b + c.

c ha A

C b

S.TEU V

+ +

¨B©u®±¡u kjhg k epopn

S=

1 aha . 2

S=

1 ab sin C. 2

u w u

c7[ D YD0[D72Y-

S=

p

p(p − a)(p − b)(p − c) .

F P HhNb@&' @ ) S@ ' @`S&J?"K3* UAP?" S '?" R P?" J?"K3>BF R FUH@ * "H@ * C?>@1B" P?F ' @ < Tb " >K @ 'BAP?C?Q K3>BF R F& F >BF b FsS W S= B

S = p2 tg

R O

A B C tg tg . 2 2 2

¨B©uwOugBkhg ZnOkm ope m k l

C

a2 = b2 + c2 − 2bc cos A;

A

S.TEU V

a2 sin B sin C · . 2 sin A

b2 = a2 + c2 − 2ac cos B; c2 = a2 + b2 − 2ab cos C.

O" F P?"HhNU>BFK3C?'?NK3F S) ' @b C&" "AP?" S '?C?"b CBAD@?=B>BF * F P @F * '?FK3C?* K\M> sS AF,'O H ¨B©u ©ugBkhg Zm ope m k l a b c = = = 2R. sin A sin B sin C

O" F P?"H@1K3C?'?NK3F SU +b@US _?" P?S +T"@ )@ ' @MS * C?>@ HhC1yb C'?" R F,C¾xyP?"WA'?" R FsFK3* F >@?

cS S ¡H@ * "H@G

¨B©u®¦¡ugBkhg e gehm k l

B

O

a−b = a+b

r C

A

S.TEU V

A−B 2 . A+B tg 2 tg

O" F P?"H@ *@ '?R " ' K3F S F * >?P?+T*@ S S '?"Hh" $?>?C Hz@ K3* P?F '?FHhFH{C H@ * "H@ * C?>BFH (F R3@ '?FH !h" R C?FHhF '?*@ '?FH /< +

u w u

¨B©uwªOu¥k k lDgqOjfg a+b = c

A−B 2 . C sin 2

cos

a−b = c

A−B 2 . C cos 2

sin

("Hh" $?>?C?Q`H@ * "H@ * C?>C@ K3* P?F '?FH¥%@ Pb 1F b S " QBA"H * CÈ(F P HhNb + _?F b NBJ?Cb&S,>BF '?$?" c ' @J @b "c S ¨B©u¡u ¾gepohg k lhg k epopn hghgi1g k m ?khk e A sin = 2

r

(p − b)(p − c) A . cos = bc 2 s A (p − b)(p − c) tg = . 2 p(p − a)

r

p(p − a) . bc

¨B©uw»Ou ¾gepohg jo m k pohm epehk q k n ehkm o

hghgi1m ?khk e o hg k epopn R= R=

a b c = = . 2 sin A 2 sin B 2 sin C

a+b+c . 2(sin A + sin B + sin C) R=

R=

p . A B C 4 cos cos cos 2 2 2

abc . 4S

¨B©uw¹Ou ¾gepohg jo m l pohm epehk q k n ehkm o

hghgi1m ?khk e o hg k epopn r=

r

(p − a)(p − b)(p − c) . p

B C sin 2 2 . A cos 2

a sin

+

r=

¨

r=

S . p

u w u

B¨ ©u¨¬Oucg gepohg&nOkm k k e hg k epopnOk l M* P?" NBR F b '?C?>B" -?@B_?Ff* P?"H K3* F P?F ' @ HVBB_?FhAS NH K3* F P?F ' @ HC&NBRb NDV SBy_?FAS NH NBRb@ HC1K3* F P?F '?"(' @LFA ?* K\AP?NBR C?"(" R FK3* F P?F '?+ CUNBRb +% Y4 9Y @ '?'?+T"

!h" " '?C?" A tg = 2

a, b, c

s

(p − b)(p − c) ; p(p − a) C tg = 2

a, B, C

A = 180◦ − (B + C); A+B C = 90◦ − ; 2 2

a, b, C

(F,' @ QBA" '?'?+H

sin B =

a, b, A

b sin A ; a

s

B tg = 2

s

(p − a)(p − c) ; p(p − b)

(p − a)(p − b) p(p − c) b=

tg

a sin B ; sin A

c=

a sin C sin A

A−B a−b A+B = tg . 2 a+b 2

A+B C A−B 2 2 a sin C c= sin A

' @LFAC H

C = 180◦ − (A + B);

c=

A

C

B.

a sin C sin A

, yKWb C *F C Hh" " *1FA'?F1) ' @J?" '?C?" = a > b, B Hh" '? " " ◦ . yKWb C a 90 '?F * F C Hh" " *BF R\ADc.@M* P?" NBR F b '?C?> K3F F * S " * K3* S " '?'?F FK3* P?F NBR F b '?+TQMC&* NB_?F NBR F b '?+TQO yKWb C C *F ◦ * P?" NBR F b '?C?> _?P HhF NBR aF b K3* 1, HhF'?F =aC&*BbF R Fa* P??P?NBR F S N & ANBR N F >?P?N'?FK3* CU

! !

"

(P?NBR F S +H K3" RHh" '?* FH ?S b??" * K\F @ J @ K3* Z>?P?NBR3@&C_?F b NB_b FKWG >BFK3* CO=R P @ '?C?$ @>BF * F P?F Q&K3FA" PC?*LF P ANH * F R F,>?P?NBR3@? %@ P?C K c _?FK3* P?F " '>?P?NBR F S F Q K3" R3G Hh" '?*K($?" '?* P @b '?+H NBRb FH F R P @G AOB,P @AC?NG ? ' ? C ? J

" ? ' ? ' T + Z Q ? > ? P B N

R

F

S

F M Q A B N

R

F Q C C1LF P AF Q S "3ACB b C?J?C?' @NBRb@ K @ l r S&P @AC @ ' @AB : ϕ h ¡ L V

S 3 " b C?J?C?' @UNBRb@ s ◦ A B AOB S¥R P @ANK @L¡V n A b C?' @¥>?P?NBR F S F Q D A b lC?' @sLF P A+ AAOB NBR C r r ACB; s ϕ _b F @Ac>?P?NBR F S F R FcK3" RHh" '?*@?V AB; S * F J?>@`_?" P?"K3" J?" '?C¥P @AC?NOC K @ ⊥ AB;C O D LF P A+ S +K3F *@>?P?NBR F S F R OC FK3" R3G AB; DC S.TEU V Hh" '?*@?V A b C?' @TS +K3F * + >?P?NBR F S F R FK3" R3G Hh" '?*@? h AD = DB,

^

AC = ^CB =

ϕ , 2

∠AOC = ∠COB =

¨?¦¡uü¡£ hope kj

ϕ . 2

√ s = 2 2hr − h2 . Cb C ϕ n◦ s = 2r sin s = 2r sin . 2 2

¨?¦¡u®±¡u¡£ hope¥n k lDk q j o Cb C

l = rϕ

l=

πr n. 180

¨?¦¡uwOu jo m1n r=

s2 + 4h2 . 8h

¨?¦¡u ©u1ge e ,q k

+

tg

ϕ 2h = 4 s

Cb C

tg

n◦ 2h = . 4 s

_ `bdc>e T>fy l f+p } p b ig l i c>bddl j zXg El n l Up j V

&

d +

¨?¦¡u®¦¡u m k kj h=r− ϕ 4 ϕ

h = 2r sin2 h = r 1 − cos h=

r

Cb C

Cb C

2

1 ϕ s tg 2 4

Cb C

r2 −

s2 . 4

h = 2r sin2

n◦ . 4

n◦ . h = r 1 − cos 2 h=

1 n◦ s tg . 2 4

?¨ ¦¡uwªOu kj¥n k lDk k m g (ge b F @A(>?P?NBR F S F R F(K3" RHh" '?*@?= '?"TP @ S '?F R F(_?F b NB>?P?NBR N =S +TJ?C KWb??" *G K\U_?FE(F P HhNb@ H1 2 r ϕ ± S4AOB 2

S=

Cb C

S=

πr2 n ± S4AOB , 360

R\A" − ϕBF F P AC?' @ * '?F Q1_b FK3>BFK3* CZK3* P?F C H P?C K ¡ "WAC?'?C?J?'?N & F >?P?N,G '?FK3* ZC ' @1FK3C F P AC?' @ *1F * >b@A+TS@ "H * F J?>?N K`>BF F P AC?' @ *@ HhC :(" P?" ))H * NM* F J?>?N`_?P?F S FAC H_?P HhN & =?_ @ P @b?b "3b '?N & FK3C&@ K3$?C K K (0, a). (P?C _?P H@ '?"y_?" P?"K3" J?" *"WAC?'?C?J?'?N & F >?P?N'?FK3* >1 y=a 1 ' @J?C?*=_?|a| P?C )@AD@J @'?"%C Hh" " *,P?" " '?C?QO |a| > 1

!#"$ !

KWb C a = y '?FK3* C S* F J?>B"

!§$

− 1, P 3π 2

* F_?P H@ Ks>BF F P AC?' @ *@ yHh=C

>@ K @ " * K\U"WAC?'?C?J?'?F Q&F >?P?N,G xy* P?F C H¾NBR F b

−1 (0, −1).

P0 OP 3π , 2

+ $

&

d +

R\A" * F J?>@_?" P?"K3" J?" '?C1"WAC?'?C?J?'?F Q1F >?P?N'?FK3* C1K%_?F b FC?* "3b G '?+H P' 0@ _? P @ S b " '?C?"HFK3C Ox. OF R\AD@_?P?C C K3>BFHh+HhCU N ANB*,NBRb + a= −1

ϕ=

(P?C SAS NL@L
BFHh+HhC NG |a| < 1, ANB*,NBRb +

a>1 a a=1

1

Pπ 2

C

a |a| < 1 Pϕ2

P ϕ1 a

−1

P0 1

O

ϕ = ϕ2 + 2πk ∀k ∈ Z.

x

P 3π 2

a = −1 −1 a

(P?C

_?P H@ 1 "WAC?'?C?JG > @ K a@ " =* K\ y'?F = Q F 1>?P?N'?FK3* C S1* F J?>B" K>BF F P AC?' @ *@ HhC Pπ (0, 1). xy* 2P?F C H NBR F b P0 OP π .

a < −1

2 xfb "WAF S@ * "3b '?F ={ "KWG b C * F C K3>BFHh+HhC N ANBa*=NBR1, b +

a

S.TEU V

ϕ = ϕ1 + 2πk ∀k ∈ Z

ϕ=

π + 2πk ∀k ∈ Z. 2

¨ ªOu®±¡u¡^75 [D74 ,7 6 7534 25BF F P AC?' @ * '?F Q1_b FK3>BFK3* CZK3* P?F C H P?C K D "WAC?'?C?J?a.'?N &F >?P?N,G '?FK3* ¥C' @¾FK3C@ K3$?C K KMF * >b@A+TS@ "H* F J?>?N KM>BF F P AC?' @ *@ HhC :(" P?" )) H * NM* F J?>?N`_?P?F S FAC H_?P Hh N &=?_ @ P @b?b "3b '?N & FK3CUF P AC?' @ *(a, 0). _?P H@ '?"f_?" P?"K3" J?" *T"WAC?'?C?J?'? N & F >?P?N'?FK3* (P?C 1 ' @J?C?*=_?|a| P?C > 1 )@AD@J x@='?a"%C Hh" " *,P?" " '?C?QO >1 yKWb C a =|a|−1, * F(_?P H@ >@ K @ " * K\BF F P AC?' @ *@ HhxC = −1 xy* P?F C HNBR F b R\A" +

Pπ

( − 1, 0).

P0 OPπ ,

&

d +

* F J?>@`_?" P?"K3" J?" '?CZ"WAC?'?C?J?'?F Q¾F >?P?N'?FK3* C¥KBF * F P?+L&P @ S '?+ xy* P?F C HAS@ , a. (P?: C H * FH ∠P0 OPϕ = ϕ2 . cos ϕ = cos ϕ = a. xfb "WAF S1@ * "3b '?F =2"KWb C * FC K3>BFHh+HhC NG < 1, A|a|NB* NBRb + ϕ = π + 2πk

P?C ( ?_ P H@ AS NL&* F J?>|a| @L < 1 C Pϕ Pϕ NBRb@ C 1

2

∠P0 OPϕ1 = ϕ1

a>1

|a| < 1

a = −1

1

a=1

2

y a < −1

∀k ∈ Z.

C

P ϕ1

ϕ = ϕ1 + 2πk ∀k ∈ Z ϕ = ϕ2 + 2πk ∀k ∈ Z.

(P?C

a

Pπ −1 a

O

a

P0 a 1

a

x

P ϕ2

−1

_?P H@ > @ aK @ " =* K\1&"WAC?'?C?JG x '?F =Q 1F >?P?N'?FK3* CS¾* F JG >B" K>BF F PBAC?' @ *@G HhC P0 xfb "WAF S@ * "3b G '?F ="(1, KWb 0). C * FC K3>BF G Hh+HhC1 N AaNB=*,NB1,Rb +

¨ªOuwOu ^75 [D74 ,7 S.TEU V Y2 W0325º6 7 7[D747 ϕ, [?Y 8 032 >BF F P AC?' @ * '?F Q1_b FK3>BFK3* CZK3* P?F C H P?C K D?"WAa. C?'?C?J?'? N &F >?P?N,G F * >b@A+TS@ "HM* F J?>?N K '?FK3* C%' @TFK3C%*@ '?R " ' K3F S ' @T_?P HhF Q >BF F P AC?' @ *@ HhC (" P?" )H * N,* F J?>?xN=C' 1)@J @b F%>BF F P AC?' @ *(_?P?F S FM : AC H _?P HhN &=f>BF * F P @(1, a). _?" P?"K3" J?" * "WAC?'?C?J?'? N & F >?P?N'?FK3* S¥AS NL * F J?>@L C Pϕ xy* P?PF ϕC H . C R\A" * F J?>@_?"3G ∠P OPϕ = ϕ1 ∠P OP = ϕ , P P?"K3" J?" '?C"WAC?'?C?0J?'?F Q F >?P?N'?FK3* C0K`_?F ϕb FC?* 2"3b '?+H0' @ _?P @ S b " '?C?"H 1 ' @J?C?*=C K3>BFHh+HhC` N ANB*NBRb + FK3C (P? C H * FH ϕ = 2πk ∀k ∈ Z.

1

2

1

Ox.

2

tg ϕ1 = tg ϕ2 = a. ϕ = ϕ1 + πk

∀k ∈ Z.

¨ ªOu ©u¡^75 [D74 ,7 6 7 Y2W0325%6 7 7[D747([?Y 8 032 >BF F P AC?' @ * '?F Q1_b FK3>Bϕ, FK3* CZK3* P?F C H P?C K D "WAC?'?C?J?'?N a.&F >?P?N,G '?FK3* ,C&' @FK3CU>BF *@ '?R " ' K3F S ' @,_?P HhF Q sF * >b@A+TS@ "H* F J?>?N y=1

+

&

d +

KT>BF F P AC?' @ *@ HhC :(" P?" )H * N,* F J?>?N,C`' @J @b F%>BF F P AC?' @ *%_?P?F G 1)._?" P?"K3" >@ " *¾"WAC?'?C?J?'?N &F >?P?N'?FK3* ¥SZAS NL _?P Hh N &=>BF * F (a, P @ * F J?>@L C

M S FAC H

P ϕ1

P ϕ2 . y

tg a 1 P ϕ1

y

M M

ctg

1

P ϕ1 −1

P0 1

O

P0 1

a

x

−1

O

x

P ϕ2

Pϕ2 −1

−1

S.TEU V W

S.TEU V

xy* P?F C H C R\A" * F J?>@_?"3G ϕ = ϕ1 P?"K3" J?" '?CU"WAC?∠P '?C?0J?OP '?F Q& F >?P?N'?FK3∠P * C&0KOP _?F ϕb F=C?ϕ* "32b , '?+H P0' @ _?P @ S b " '?C?"H >BF F P AC?' @ * '?F QFK3C (P?C Ox. 1 ' @G H * FH y J?C?*= C ctg K3>BFϕHh1+=Hhctg CU N ϕA2NB=*,a.NBRb + 1

2

a

1 a>

1

1

Pπ 2

P ϕ2

P ϕ1

−1

O

1

a

|a| < 1

a = −1

,

a>

a=1

ϕ = ϕ1 + πk ∀k ∈ Z. ¨ªOu®¦¡u^75 [D74 7

−1 P 3π 2

P0 x

6 75 036Y25s6 7 7[D747f[?Y 8 032 ϕ, >BF F P AC?' @ * '?F Q_b FK3>Ba. FKWG * CZK3* P?F C H P?C K s"WAC?'?C?JG '? N & F >?P?N'?FK3* C ' @FK3C F P AC?' @ *TF * >b@A+TS@ "HU* F J?>?N K&>BF F P AC?' @ *@ HhC :("3G P?" { ) H * NZ* F J?>?N1_?P?F (0, S FAa). C H >@G K @ * "3b '? N & >"WAC?'?C?J?'?F QF >G P?N'?FK3* CO (P?C HhF'?F_?FK3*G P?F C?* ¥AS |a|"&>>@ K 1@ * "3b '?+T" =h>BF G * F P?+T">@ K @ &c* K\F >?P?N'?FKWG * CMS%AS NL* F J?>@L C Pϕ Pϕ . xy* P?F C H C ∠P0 OPϕ = ϕ1 >BFK3" >@ ' K3+ 1

<

S.TEU V

1

∠P0 OPϕ2 = ϕ2 ,

2

&

d +

1 A"K3 >BF * F P?+L * F J?>@¾_?" P?"K3" J?" '?C cosec ϕ1 = cosec ϕ2 = a. "WAC?'?C?J?'?F Q& F >?P?N'?FK3* C&K(_?F b FC?* "3b '?+H' P@ 0_?P @ S b " '?C?"HFK3C Ox. xfb "WAF S@ * "3b '?F =B"KWb C * F,C K3>BFHh+HhC& N ANB*,NBRb + |a| > 1,

C

yKWb C

ϕ = ϕ1 + 2πk

∀k ∈ Z

ϕ = ϕ2 + 2πk

∀k ∈ Z.

* F%>@ K @ * "3b '?F Q?S b??" * K\_?P H@ >BF * F P @ a = − 1, y = − 1, >@ K @ " * K\"WAC?'?C?J?'?F QF >?P?N'?FK3* CS%* F J?>B" xy* P?F C H¾NBR F b P 3π .

OF R\AD@

y 1

1

N ANB*,NBRb + a=1

a = −1

2

ϕ=

P ϕ1

a = −1

3π + 2πk 2

C K3>BFHh+Hh2 C

∀k ∈ Z.

(P?C _?FK3* P?F C?* ¥>@ K @G * 3" b '?N & |a|> BF * F P @@G = F 1,>?P?N'?FK3* CZS S.TEU V K @ " * K\1"WAC?'?C?J?'?yF Q¾ * F J?>B" xy* P?F C HNBR F b Pπ . P0 OP π . 2 OF R\AD@_?P?C C K3>BFHh+HhC& N ANB*,NBRb + 2

Pπ −1

P0 1

|a| < 1

2

a=1

ϕ=

π + 2πk 2

∀k ∈ Z.

¨ªOuwªOu¡^75 [D7 0,7 5 036Y25%6 7 7[D747([?Y 8 032 ϕ, a. >BF F P AC?' @ * '?F Q1_b FK3>BFK3* CZK3* P?F C H P?C K D B"WAC?'?C?J?'?N &F >?P?N,G '?FK3* ¥C' @¾FK3C@ K3$?C K KMF * >b@A+TS@ "H* F J?>?N KM>BF F P AC?' @ *@ HhC :(" P?" )) H * NM* F J?>?N`_?P?F S FAC H>@ K @ * "3b '?N & >&"WAC?'?C?J?'?F Q&F >?P?N'?FK3(a,* CO0). (P?C HhF'?F&_?FK3* P?F C?* ÀS ",>@ K @ * "3b '?+T" =D>BF * F P?+T">@ K @G &c* K\F >?P?N|a|'?>FK3* 1CSAS NL@L Pϕ C Pϕ . xy* P?F C H ∠P0OPϕ = ϕ C R\A" * F J?>@_?" P?"K3" J?" '?C"WAC?'?C?J?'?F QF >?P?N'?F1K3* C ∠P0 OPϕ = ϕ2 , P0 K_?F b FC?* "3b '?+H' @ _?P @ S b " '?C?"HFK3C (P?C?J?"H Ox. sec ϕ = sec ϕ2 = a. xfb "WAF S@ * "3b '?F =B"KWb C * F,C K3>BFHh+HhC& N ANB*,NB1Rb + 1

2

1

2

|a| > 1,

@ K @ * "3b '?F Q?S b??" * K\(_?P H@ >BF * F P @ >@ K @ " * K\`"WAaC?='?C?−1, J?'?F QUF >?P?N'?FK3* CUS,* F J?>B" xy* P?F C xH=NBR −1, Fb P0 OPπ . OF R\AD@_?P?C C K3>BFHh+HhCU N ANB*,PNBRπb . + a= −1

P?C ( _?FK3* P?F C?* 1>@ K @ * "3b '?N & > "WAC?'?C?J?'?F Q F >?P?N'?FK3* C '?" S F )HhF'?|a|F < 1 xfb "WAF S@ * "3b '?F =B"KWb C * F,)@AD@J @P?" " '?C?Q&'?"%C Hh" " * yKWb C a = 1, * F1>@ K @ * |a| "3b G $?C?C sin cos tg ctg sec csc

!
@¾_?" P?"WA >BF P?'?"H)@ S C K3C?*¥F * * F R F =fS >@ >BF Q J?" * S " P?* C ' @LFAC?* K\MNBR F b x. ±¡uü :¥"OPT"O) sin x :

cos x = ± tg x =

sin x p ± 1 − sin2 x

ctg x =

±¡u®±¡u ¥ : "OPT"O)

tg x =

±

p

1 − sin2 x ;

π x 6= + πn ∀n ∈ Z ; 2

p

1 − sin2 x sin x

(x 6= πn ∀n ∈ Z).

cos x :

±

ctg x =

√

sin x = ± 1 − cos2 x cos x

±

√

√ 1 − cos2 x ;

cos x 1 − cos2 x

x 6=

π + πn ∀n ∈ Z ; 2

(x 6= πn ∀n ∈ Z).

¨

& w

±¡uwOu ¥ : "OPT"O)

sin x =

cos x =

tg x p ± 1 + tg2 x ±

±

"

n X k=0

n X

k=0

n X

¨

k=1

+

π 2

n;

π (n + 1) , ∀n ∈ Z. 2

1 p

(x 6= πn ∀n ∈ Z).

1 + ctg2 x

ctg x p ± 1 + ctg2 x

1 ctg x

tg x =

9

π + πn ∀n ∈ Z ; 2

π x 6= + πn ∀n ∈ Z ; 2

1 + tg2 x ∀x ∈

x 6=

ctg x :

cos x =

!

p

1 tg x

sin x =

1

±¡u ©u ¥ : "OPT"O)

tg x :

ctg x =

∀x ∈ $

π 2

(x 6= πn ∀n ∈ Z). n;

π (n + 1) , ∀n ∈ Z. 2

!! $ ]"

'

ns (n + 1)s sin x + sin 2 2 sin(x + ks) = . s sin 2

ns (n + 1)s cos x + sin 2 2 . cos(x + ks) = s sin 2 sin

sin kx =

nx (n + 1)x sin 2 2 x sin 2

(x 6= 2πm ∀m ∈ Z).

n X

sin cos kx =

k=1

+

nx (n + 1)x cos 2 2 x sin 2

(x 6= 2πm ∀m ∈ Z).

n(s + π) (n + 1)(s + π) sin x + sin k 2 2 . ( − 1) sin(x + ks) = s cos k=0 2 n X

n(s + π) (n + 1)(s + π) cos x + sin k 2 2 ( − 1) cos(x + ks) = . s cos k=0 2 n X n X

k sin(x + ks) =

k=1 n X

(n + 1) cos(x + ns) − n cos(x + ns + n) − cos x . s sin2 2

k cos(x + ks) =

k=1

(n + 1) sin(x + ns) − n sin(x + ns + n) − sin x . s sin2 2

!
BFH G P @LB@ HhFH 1F@ S P?FH 1N@ S P?FHfW ± ¹Ou®±¡u¥k U_?" P?"K3" >@ " *FK3@ K3$?C K KcS,* F J?>@L (πn, 0) ∀n ∈ Z. 7 sP @ K3_?F b F" 'sS +T"DFK3C@ K3$?C K K (@pC?'?* " P?S@b@L (π + 2πn; 2π + 2πn) ∀n ∈ Z

F * P?C?$ @ * 3" b ' @OVR P @ E(C?>UP @ K3_?F b F" 'M'?C"(FK3C1@ K3$?C K K 4 ) )y0[¡032 9 4[ 0 %75 ? (P?F C?) S FA' @ ◦ /&* 8 .

y 0 (x) = cos x ∀x ∈ R.

C E%E(" P?" '?$?C?P?N "H@M' @ (P?F C?) S FA' @`'?" _?P?" P?+TS ' @' @ 1 ' @J?" '?CU_?P?F C?) S FA'?F QUE(R.NB'?>?$?C?CUS'?Nb?LUE(NB'?>?$?C?C1K3C?'?NK -

R.

y 0 (2πn) = cos 2πn = 1 ∀n ∈ Z; y 0 (π + 2πn) = cos(π + 2πn) = − 1 ∀n ∈ Z.

* F J?>@L R P @ E(C?>¥_?" P?"K3" >@ " *&FK31@ K3$?C K K,_?FA (2πn, 0) ∀n ∈ Z NBR3b FHS , S *

F ? J > @ L _?FA1NBRb FHS ◦ 135◦. s* F P 45 @U;_?P?F C?) S FA' @ (π + 2πn, 0) ∀n ∈ Z y 00 (x) = − sin x ∀x ∈ R.

S@A+ AC E%E(" P?" '?$?C?P?N "H@&' @ s* F P @&_?P?F C?) S FA' @1E(NB'?>G R. $?C&'?" _?P?" P?+TS ' @' @ R. [D4 4 ?0 536 40 7 6 4 π x = + πn ∀n ∈ Z. 9◦ . 7 @L ◦ ^T[D ¨+¨

10 .

£

h

NB +TS@ " *' @F * P?" ) >@L

11◦ .

?7 6 4

i π π + 2πn; + 2πn 2 2

−

hπ

+ 2πn;

65 [¡0 < Y4 2

x= −

* F J?>?C1H@ >K3C HhNH@

i 3π + 2πn 2

∀n ∈ Z; ∀n ∈ Z.

65 [¡0 !OF J?>?C1HhC?'?C HhNH@

π + 2πn ∀n ∈ Z; 2

π + 2πn ∀n ∈ Z. 2 1C?'?C HhNHh+%H@ >K3C HhNHh+%ymin = − 1; ymax = 1. * F J?>@L π R P @ E(C?>>@ K @ " * K\R F P?C?) F '?*@b '?F Q +2πn, 1 ∀n ∈ Z 2 _?P HhF Q S(* F J?>@L π R F P?C?) F '?*@b '?F Q y = 1; − +2πn, −1 ∀n ∈ Z 2 _?P HhF Q y = − 1. i 4 %75 @ P @ * C H@' @(F * P?" ) >@L h π π ◦ [?Y 12 . − + πn; + πn , 2 2 R\A" b c F "($?"3b F "(J?C KWb F n

6 75 4¾87 2 75 s+T_?NB>b@U' @F * P?" ) >@L 13◦ . [π + 2πn; 2π + 2πn] ∀n ∈ Z; x=

&

4

S F R '?NB*@M' @F * P?" ) >@L

?7 6 4 0[¡0+4 3Y14 . _?" P?"3LFA"(J?" P?" )('?(πn, (P?CU Nb C1C?0))Hh∀n " '?∈" * Z.K\MLB@ P @ >?* " PUS +T_?NB>b FK3* CO :%53 4 7 '?" * 15◦ . 4 ?032 2+Y ,E(NB'?>?$?CO◦ [?Y2 [2πn; π + 2πn] ∀n ∈ Z.

◦

16 .

− 1 6 sin x 6 1 ∀x ∈ R.

O P @ E(C?>(P @ K3_?F b F" 'cSTR F P?C?) F '?*@b '?F Q_?F b FK3" =3F R P @ '?C?J?" '?'?F Q_?PG Hh+HhC C y = −1 y = 1. ] Y 4 7 0 0(4&2Y4 @L ' @ C Hh" '? G π max(sin x) = 1 x = + 2πn ∀n ∈ Z; R 2 AFK3* C?R3@ " * K\,S@L " "T) ' @J?" '?C?" π min(sin x) = − 1 x = − + 2πn, R 2 R\A"(J?C KWb F b &c F "($?"3b F " n

¨

!

£

X12 7 0 5 87 2Y ? 032 4 18◦ . [?Y f ) 4 6%_?FK3* P?F " 'U' @,P?C K E(sin x) = [ − 1; 1]. ◦

19 .

y 1 − π2

− 5π 2 −3π

−π

−2π

3π 2

O

π 2

π

2π

3π

x

−1

S.TEU V

OP @ E(C?>&E(NB'?>?$?C?C1K3C?'?NK' @ ) +TS@ " * K\ 25D&/&,2513DCE13M? OP @ E(C?> HhF'?F`_?FK3* P?F C?* M_ @ P @b?b "3b '?+H_?" P?" '?FK3FH_?F G Γ sin x SAF b B" 5◦ . (0, 1) : cos 0 = 1. π ◦ ] 4 6 . x = + πn ∀n ∈ Z. OP @ E(C?>U_?" P?"K3" 2>@ " *FK3@ K3$?C K KcS,* F J?>@L π + πn, 0 ∀n ∈ Z. 7 sP @ K3_?F b F" 'sS +T"DFK3C@ K3$?C K K (@pC?'?* " P?S@b@L _a`bdc>e > j kml in+p i V w sx>a l f+p } p b ig i b i T>up k n>55p j z>gT |dTE T f:g U~T>f>uU l T53p ) l f+p } p g 55n l dU i j z>g E5i T b j z U l }Eb i pf>T>T)U| c>b~}Eb Ud5 T e ~U n l l e b l f+p } p b i]g i ειδoς b i T> l T5Ef+p g T5 j V c d| T e TU~n l up e TBzwg ~U T>f>uU l T53p j Ebdc>b l & ¨

j z b i j V

£

π 2

+ 2πn;

3π + 2πn ∀n ∈ Z 2

F * P?C?$ @ * 3" b ' @OVR P @ E(C?>UP @ K3_?F b F" 'M'?C"(FK3C1@ K3$?C K K 4 ) )y0[¡032 9 4[ 0 %75 ? (P?F C?) S FA' @ ◦ /&* 8 .

y 0 (x) = − sin x

∀x ∈ R.

C E%E(" P?" '?$?C?P?N "H@M' @ (P?F C?) S FA' @`'?" _?P?" P?+TS ' @' @ 1 ' @J?" '?CU_?P?F C?) S FA'?F QUE(R.NB'?>?$?C?CUS'?Nb?LUE(NB'?>?$?C?C&>BFK3C?'?NR.K π π y 0 − + 2πn = − sin − + 2πn = 1 ∀n ∈ Z; 2 2 π π + 2πn = − sin + 2πn = − 1 ∀n ∈ Z. y0 2 2 * F J?>@L π R P @ E(C?>_?" P?"K3" >@ " *FK3(@ K3$?C K K − +2πn, 0 ∀n ∈ Z 2 _?FANBRb FH1S ◦ Sc* F J?>@L π _?FANBRb FH1S 45 ; +2πn, 0 ∀n ∈ Z 135◦. 2 s* F P @U_?P?F C?) S FA' @

y 00 (x) = − cos x

∀x ∈ R.

S@A+ AC E%E(" P?" '?$?C?P?N "H@&' @ s* F P @&_?P?F C?) S FA' @1E(NB'?>G R. $?C&'?" _?P?" P?+TS ' @' @ [D4 4 ?0 536 40 R. 7 6 4 x = πn ∀n ∈ Z. 9◦ . 7 @L ◦ ^T[D 10 .

NB +TS@ " *' @F * P?" ) >@L ◦

11 .

?7 6 4

* F J?>?C1H@ >K3C HhNH@

[ − π + 2πn; 2πn] ∀n ∈ Z;

65 ¡[ 0 < Y4

65 ¡[ 0 /OF J?>?CZHhC?'?C HhNH@

[2πn; π + 2πn] ∀n ∈ Z. x = π + 2πn ∀n ∈ Z; x = 2πn ∀n ∈ Z.

1C?'?C HhNHh+%H@ >K3C HhNHh+%ymin = − 1; y = 1. * F J?>@L R P @ E(C?>>@ K @ " * max K\R F P?C?) F '?*@b '?F Q@L 1) ∀n ∈ Z _?P HhF Q y = 1; (π + 2πn, − 1) ∀n ∈ Z y = − 1. @ ? [ Y 4

% 7 5

P @ * C H @ ' & @

F * ? P

"

) > @ L R\A" ◦ 12 . [πn; π(n + 1)], J?C KWb F b &c F "($?"3b F "

n

¨

$

13◦ .

£

6 75 4¾8742 7 5 s+T_?NB>b@U' @F * P?" ) >@L hπ

S F R '?NB*@M' @F * P?" ) >@L

2

+ 2πn;

i 3π + 2πn 2

∀n ∈ Z;

i π π + 2πn; + 2πn ∀n ∈ Z. 2 2 ?7 6 4 0[¡0 4 3Y- π 14◦ . + πn, 0 ∀n ∈ Z. (P?CU_?" P?"3LFA"(J?" P?" )('?Nb 2C1C?)Hh" '?" * K\MLB@ P @ >?* " PUS +T_?NB>b FK3* h

−

+

:%534 7 '?" * 15◦ . [?Y2 4?032 2Y+,E(NB'?>?$?CO◦

CO

16 .

− 1 6 cos x 6 1 ∀x ∈ R.

O P @ E(C?>(P @ K3_?F b F" 'cSTR F P?C?) F '?*@b '?F Q_?F b FK3" =3F R P @ '?C?J?" '?'?F Q_?PG C Hh+HhC y = −1 y = 1. 7 0 0(4&2Y4 @L ' @ C Hh" '? " " max(cos x) = 1 x = 2πn ∀n ∈ Z; R ) ' @J?" '?C?" AFK3* C?R3@ " * K\%S* F J?>@L min(cos x) = −1 x = π+2πn ∀n ∈ Z. R 1 X 2 7

0 5 8 7 2 Y ? 3 0 2 4 18◦ . E(cos x) = [ − 1; 1]. ◦ [?Y )f4 6%_?FK3* P?F " 'U' @,P?C K ? 19 .

y 1 −2π − 5π − 3π 2 2

π

−π − π2

O

π 2

2π 3π 2

5π 2

x

−1

S.TEU V

OP @ E(C?>&E(NB'?>?$?C?C&>BFK3C?'?NKc' @ ) +TS@ " * K\ (1325D&/&,2513DCE13M OP @ E(C?> HhF'?F_?FK3* P?F C?* _ @ P @b?b "3b '?+H_?" P?" '?FK3FH _?F G Γ cos x K3* P?F " '?'?F R F' @P?C K SAF b FK3CU@ K3$?C K Kc' @ π "WAC?'?C?$ Γ sin x

2

+ 2πk

A b C?'?+ S b " S F =?@,*@ >"%' @ 3π "WAC?'?C?$UA b C?'?+ S _?P @ S F =BR\A" + 2πk b &c F "($?"3b F "('?" F * P?C?$ @ * "3b '?2F "(J?C KWb F ¨

k

£

1◦ .

2◦ .

DY 5

¬OuwOu ¡epn po ³ y(x) = tg x 7[¡0 0 D032 4 @ _??P "WA"3b " ' @' @C?'?* " P?S@b@L

]0[¡0[f82 75

−

π π + πn; + πn ∀n ∈ Z. 2 2

?(" _?P?" P?+TS ' @' @C?'?* " P?S@b@L

π π + πn; + πn ∀n ∈ Z. 2 2 OF J?>?C1P @ ) P?+TS@?π x = + πn ∀n ∈ Z. 2 π ]0 ?0 2Y π 3◦ . tg(−x) = −tg x ∀x ∈ − +πn; +πn , ∀n ∈ Z. OP @ E(C?>&K3C HHh" * P?C?J?" '1F * '?FK3C?* "3b '?F,' @J @2b@>BF F P 2AC?' @ *

−

+

G_?" P?C?FAC?J?"KWG ◦ ^0[D4 7 4?0 536Y+`K%FK3'?F S '?+H _?" P?C?FAFH T0 = π (π >@W4?(. " P?C?FA+ T = πn ∀n ∈ Z\{0}. 75 7[ 4 2Y Sc' @J @b "sK3C K3* "Hh+>BF F P AC?' @ *?◦ [?Y )f4 60[¡0 5 036Y 0

5 . tg 0 = 0. ] 6◦ .

4 ∀n ∈ Z. OP @ E(C?>U_?" xP?"=K3" πn >@ " * FK3@ K3$?C K KcS,* F J?>@L (πn, 0) ∀n ∈ Z. T ^ D [ 7 < 0

6 4

2 Y6 77[¡0 0 D032 2 75 4 %@C?'?* " P?S@b@L ◦

7 .

πn;

π + πn ∀n ∈ Z 2

_?F b FC?* "3b ' @OVIR P @ E(C?>sP @ K3_?F b F" 'sS +T"DFK3C@ K3$?C K K (@pC?'?* " P?S@b@L

−

π + πn; πn ∀n ∈ Z 2

F * P?C?$ @ * 3" b ' @OVR P @ E(C?>UP @ K3_?F b F" 'M'?C"(FK3C1@ K3$?C K K 4 ) )y0[¡032 9 4[ 0 %75 ? (P?F C?) S FA' @ ◦ /&* 8 .

1 cos2 x " P?S@b@L

y 0 (x) =

π π + πn; + πn , ∀n ∈ Z. 2 2 AC E%E(" P?" '?$?C?P?N "H@?= π π − + πn; + πn ∀n ∈ Z 2 2 ∀x ∈

−

@(C?'?* % @_?" P?S@U_?P?F C?) S FA' @ME(NB'?>?$?C&'?" _?P?" P?+TS ' @? 1 ' @J?" '?CU_?P?F C?) S FA'?F QUE(NB'?>?$?C?CUS'?Nb?LUE(NB'?>?$?C?C&*@ '?R " ' K y 0 (πn) =

1 = 1 ∀n ∈ Z. cos2 πn ¨

£

* F J?>@L R\A" >@ " *FK3@ K3$?C K (πn, Kc_?F0), A1NBRb FHnS s* F P @U_?P?F C?) S FA' @ 2 sin x cos3 x " P?S@b@L

y 00 (x) =

b &c F "$?"3b F "J?C KWb F =DR P @ E(C?>¾_?" P?"K3"3G

45◦ .

π π + πn; + πn , ∀n ∈ Z. 2 2 AS@A+ AC E%E("3G π π − + πn; + πn ∀n ∈ Z 2 2 ∀x ∈

−

@`C?'?* % P?" '?$?C?P?N "H@O=?@S * F P @U_?P?F C?) S FA' @UE(NB'?>?$?CU'?" _?P?" P?+TS ' @? [D4 4 ?0 536 4 7 ?036%'?" * 9◦ . 7 b@U' @_?F b NBC?'?* " P?S@b@L 13◦ . h π πn; + πn ∀n ∈ Z; 2

S F R '?NB*@M' @_?F b NBC?'?* " P?S@b@L

−

i π + πn; πn 2

∀n ∈ Z.

?7 6 4 0[¡0+4 3Y (P?CU_?" P?"3LFA"(J?" P?" )('?(πn, Nb C10)C?)Hh∀n " '∈?" Z. * K\MLB@ P @ ?> * " PUS +T_?NB>b FK3* CO :%53 4 7 s" P?* C?>@b '?+T"@ K3C Hh_?* F * % + π 15◦ . x = +πn ∀n ∈ Z. 2 2%" S FK3* F P?F '?'?C?"_?P?"WA"3b +%14◦ .

tg x → + ∞

_?P?C

(P @ S FK3* F P?F '?'?C?"c_?P?"WA"3b +%tg x → − ∞

!

_?P?C

x→

π 2

+ πn − 0 ∀n ∈ Z.

x→

π

+ πn + 0 ∀n ∈ Z.

2

%@ >b F '?'?+L`C&R F P?C?) F '?*@b '?+LM@ K3C Hh_?* F *,'?" * 4 ?032 2 75 (" F R P @ '?C?J?" '?' @M'?C&K3'?C?) N =?'?C1K3S " PLN ◦ [?Y2

16 .

£

]Y4 7 4 4¾2Y4 ?$?C?C&*@ '?R " ' K' @ ) +TS@ " * K\ 47AX/ ; /&2513DCE13M

3

y

−2π

π

−π

2π x

O −

3π 2

−

π 2

π 2

3π 2

5π 2

S.TEU V

1◦ . ◦

2 .

DY 5

¬Ou ©u ¡epn po ³ y(x) = ctg x 7[¡0 0 D032 4 @ ?_ P?"WA"3b " ' @' @C?'?* " P?S@b@L

]0[¡0[f82 75

(πn; π(n + 1)) ∀n ∈ Z.

?(" _?P?" P?+TS ' @' @C?'?* " P?S@b@L

OF J?>?C1P @ ) P?+TS@ x = πn ∀n ∈ Z. ]0 ?0 2Y+ 3 . OP @ E(C?>&K3C HHh" * ctg( P?C?J?− " '1x)F * = '?FK3−C?* ctg "3b x'?F,∀x' @∈J @(πn; b@>Bπ(n F F P A+C?1)), ' @ * ∀n ∈ Z. ^0[D4 7 4?0 536Y+` K%FK3'?F S '?+H _?" P?C?FAFH T = π (π G_?" P?C?FAC?J?"KWG 4 . (πn; π(n + 1)) ∀n ∈ Z.

◦

◦

0 >@W?(" P?C?FA+ T = πn ∀n ∈ Z\{0}. [?Y )f4 6M200[¡0 5 036Y 0 7534Z7[ 4 2Y 5◦ . ] 4 π 6◦ . x = + πn ∀n ∈ Z. OP @ E(C?>U_?" P?"K3" 2>@ " *FK3@ K3$?C K KcS,* F J?>@L π + πn, 0 ∀n ∈ Z. 7 sP @ K3_?F b F" 'sS +T"DFK3C@ K3$?C K K (@pC?'?* " P?S@b@L !

£

π 2

+ πn; π(n + 1)

∀n ∈ Z

F * P?C?$ @ * 3" b ' @OVR P @ E(C?>UP @ K3_?F b F" 'M'?C"(FK3C1@ K3$?C K K 4 ) )y0[¡032 9 4[ 0 %75 ? (P?F C?) S FA' @ ◦ /&* 8 .

y 0 (x) = −

1 sin2 x

∀x ∈ (πn; π(n + 1)), ∀n ∈ Z.

%@@L π + πn, 0 , n _?" P?"K3" >@ " *FK3,@ 2K3$?C K K_?FA1NBRb FHS s* F P @U_?P?F C?) S FA' @ y 00 (x) =

2 cos x sin3 x

= − 1 ∀n ∈ Z.

b &c F "Z$?"3b F "ZJ?C KWb F =sR P @ E(C?> 135◦.

∀x ∈ (πn; π(n + 1)), ∀n ∈ Z.

%@C?'?* " P?S@b@L AS@A+ AC E%E(" P?" '?$?C?P?N "3G H@O=?@S * F P @U_?P?F C?) S (πn; FA' @π(n UE(+NB'?1)) >?$?C∀n U'?∈" _?ZP?" P?+TS ' @? [D4 4 ?0 536 4 7 ?036%'?" * 9◦ . T ^ [D7 b FK3* CO :%53 4 7 s" P?* C?>@b '?+T"%@ K3C Hh_?* F * +%15 x = πn ∀n ∈ Z. 2%" ◦S .FK3* F P?F '? '?C?" _?P?"WA"3b +%_?P?C ctg x → − ∞ x → πn − 0 ∀n ∈ Z. P @ S FK3* F P?F '?'?C?"c_?P?"WA"3b +%( _?P?C ctg x → + ∞ x → πn + 0 ∀n ∈ Z. %@ >b F '?'?+L`C&R F P?C?) F '?*@b '?+LM@ K3C Hh_?* F *,'?" * [?Y2 4 ?032 2 75 (" F R P @ '?C?J?" '?' @M'?C&K3'?C?) N =?'?C1K3S " PLN 16◦ . ] Y4 7 4 4¾2Y 4 ?$?C?C&>BF *@ '?R " ' Kc' @ ) +TS@ " * K\ (1347AX/ ; /&2513DCE13M 14◦ .

y

π

−π 3π 2 −2π

−

−

π 2

2π

Oπ 2

3π 2

3π 5π 2

x

S.TEU V

1◦ .

2◦ .

DY 5

¬Ou®¦¡u ¡epn po ³ y(x) = sec x 7[¡0 0 D032 4 @ ?_ P?"WA"3b " ' @' @C?'?* " P?S@b@L

]0[¡0[f82 75

OF J?>?C1P @ ) P?+TS@?-

−

π π + πn; + πn ∀n ∈ Z. 2 2

?(" _?P?" P?+TS ' @' @C?'?* " P?S@b@L π π + πn; + πn ∀n ∈ Z. 2 2 π x = + πn ∀n ∈ Z. 2

−

/< !

£

π h0 2Y+π sec( − x) = sec x ∀x ∈ − + πn; + πn , ∀n ∈ Z. OP @ E(C?>&K3C HHh" * P?C?J?" '1F * '?FK3C?* "3b '?F,FK3CU2F P AC?' @ *2 4 ?0 536Y+1K,FK3'?F S '?+H _?" P?C?FAFH G_?" P?C?FACG ◦ ^0[D4 7 T0 = 2π (2π J?"K3>4@.W?(" P?C?FA+ ∀n ∈ Z\{0}. [?Y )f4 6 0[¡T0 5 03=6Y 2πn 0 75 7[ 4 2Y S* F J?>B" (0, 1) : sec 0 = 1. 5◦ . '?" *OP @ E(C?>M'?"%_?" P?"K3" >@ " *FK3C&@ K3$?C K K ◦ ] D0 6 . 7 sP @ K3_?F b F" 'sS +T"DFK3C@ K3$?C K K (@pC?'?* " P?S@b@L π 2

3π + 2πn ∀n ∈ Z 2

+ 2πn;

F * P?C?$ @ * 3" b ' @OVR P @ E(C?>UP @ K3_?F b F" 'M'?C"(FK3C1@ K3$?C K K 4 ) )y0[¡032 9 4[ 0 %75 ? (P?F C?) S FA' @ ◦ /&* 8 .

π π + πn; + πn , ∀n ∈ Z. 2 2 π AC E%E(" P?" '?$?C?P?N "H@?= π − + πn; + πn ∀n ∈ Z 2 2

y 0 (x) = tg x sec x ∀x ∈

−

@(C?'?* " P?S@b@L % @_?" P?S@U_?P?F C?) S FA' @ME(NB'?>?$?C&'?" _?P?" P?+TS ' @? s* F P @U_?P?F C?) S FA' @

π π + πn; + πn , ∀n ∈ Z. 2 2 π AS@A+ AC E%E("3G π − + πn; + πn ∀n ∈ Z 2 2

y 00 (x) = sec3 x(1 + sin2 x) ∀x ∈

−

@`C?'?* " P?S@b@L % P?" '?$?C?P?N "H@O=?@S * F P @U_?P?F C?) S FA' @UE(NB'?>?$?CU'?" _?P?" P?+TS ' @? [D4 4 ?0 536 40 7 6 4 9◦ . x = πn ∀n ∈ Z. T ^ 7 ?CZHhC?'?C HhNH@

x = 2πn ∀n ∈ Z;

* F J?>?C1H@ >K3C HhNH@

x = π + 2πn ∀n ∈ Z.

1C?'?C HhNHh+ H@ >K3C HhNHh+ ymin = 1; y = − 1. * F J?>@L R P @ E(C?>>@ max K @ " * K\R F P?C?) F '?*@b '?F Q@L _?P HhF Q y = 1; (π + 2πn, − 1) ∀n ∈ Z y = − 1. %@C?'?* " P?S@b@L π π R P @ E(C?>1P @ K3_?F G

−

2

+ 2πn;

2

+ 2πn

∀n ∈ Z π 3π + 2πn; + 2πn 2 2 y = − 1.

b F" 'M'?"'?C"(_?P HhF Q ' @C?'?* " P?S@b@L y = 1; R P @ E(C?>UP @ K3_?F b F" 'M'?"(S +T"(_?P HhF Q ∀n ∈ Z 7 2 75 s+T_?NB>b@U' @C?'?* " P?S@b@L ◦ 6 75 4¾84 12 .

−

π π + 2πn; + 2πn ∀n ∈ Z; 2 2

S F R '?NB*@M' @C?'?* " P?S@b@L π

+ 2πn;

3π + 2πn ∀n ∈ Z. 2

?7 ?036 0[¡0+4 3Yc'?" * 13 . :%534 7 3s" P?* C?>@b '?+T"f@ K3C Hh_?* F * + 14◦ . 2%" S FK3* F P?F '?'?C?"_?P?"WA"3b +%2

◦

sec x → − ∞ sec x → + ∞

_?P?C _?P?C

x=

π +πn ∀n ∈ Z. 2

π + 2πn − 0 ∀n ∈ Z; 2 π x→ + 2πn − 0 ∀n ∈ Z. 2

x→

−

(P @ S FK3* F P?F '?'?C?"_?P?"WA"3b +%sec x → + ∞ sec x → − ∞

_?P?C _?P?C

π + 2πn + 0 ∀n ∈ Z; 2 π x→ + 2πn + 0 ∀n ∈ Z. 2

x→

−

%@ >b F '?'?+L`C&R F P?C?) F '?*@b '?+LM@ K3C Hh_?* F *,'?" * 4 ?032 2 75 (" F R P @ '?C?J?" '?' @M'?C&K3'?C?) N =?'?C1K3S " PLN ◦ [?Y2

15 .

!!

£

]Y4 7 4 ¾ 4 2Y4 "' @ 3π + 2πk b c F "($?"3b F "('?" F * P?C?$ @ * "3b '?2F "(J?C KWb F

&

1◦ . ◦

2 .

DY 5

+2πk

"WAC?'?C?$ZA b C?'?+S _?P @ S F =R\A"

k

¬OuwªOu ¡epn po ³ y(x) = cosec x 7[¡0 0 D032 4 @ ?_ P?"WA"3b " ' @' @C?'?* " P?S@b@L

]0[¡0[f82 75

OF J?>?C1P @ ) P?+TS@ ]0 ?0 2Y+ 3 . ◦

2

(πn; π(n + 1)) ∀n ∈ Z.

?(" _?P?" P?+TS ' @' @C?'?* " P?S@b@L (πn; π(n + 1)) ∀n ∈ Z.

x = πn ∀n ∈ Z.

cosec( − x) = − cosec x

∀x ∈ (πn; π(n + 1)), ∀n ∈ Z.

OP @ E(C?>&K3C HHh" * P?C?J?" '1F * '?FK3C?* "3b '?F,' @J @b@>BF F P AC?' @ * ^0[D4 7 4 ?0 536Y+1K,FK3'?F S '?+H _?" P?C?FAFH G_?" P?C?FACG 4◦ . T0 = 2π (2π J?"K3>@W?(" P?C?FA+ [?Y )f4 6M20 T0[¡=0 5 2πn 036Y 0 ∀n7∈534ZZ\{0}. 7[ 4 2Y 5◦ . ] D 0 ? '

" * O P

@ ( E ? C M > ? ' % " ? _

" ? P " K3" >@ " *FK3C&@ K3$?C K K 6◦ . T ^ D [ 7 < 0

6 4

2 Y

6 7 7 ¡ [ 0 0 D 032 2 75 4 %@C?'?* " P?S@b@L ◦ !

7 .

£

(2πn; π + 2πn) ∀n ∈ Z

_?F b FC?* "3b ' @OVIR P @ E(C?>sP @ K3_?F b F" 'sS +T"DFK3C@ K3$?C K K (@pC?'?* " P?S@b@L (π + 2πn; 2π(n + 1)) ∀n ∈ Z

F * P?C?$ @ * 3" b ' @OVR P @ E(C?>UP @ K3_?F b F" 'M'?C"(FK3C1@ K3$?C K K 4 ) )y0[¡032 9 4[ 0 %75 ? (P?F C?) S FA' @ ◦ /&* 8 .

y 0 (x) = − ctg x cosec x ∀x ∈ (πn; π(n + 1)), ∀n ∈ Z.

%@?$?Cπ(n &'?" +_?P?1)) " P?+T∀n S ' @?∈ Z s* F P @U_?P?F C?) S FA' @ y 00 (x) = cosec3 x(1 + cos2 x)

∀x ∈ (πn; π(n + 1)), ∀n ∈ Z.

@C?'?* " P?S@b@L % AS@A+ AC E%E(" P?" '?$?C?P?N "3G H@O=?@S * F P @U_?P?F C?) S (πn; FA' @π(n UE(+NB'?1)) >?$?C∀n U'?∈" _?ZP?" P?+TS ' @? [D4 4 ?0 536 40 7 6 4 π 9◦ . x = + πn ∀n ∈ Z. 7 ?C1H@ >K3C HhNH@

x=

1C?'?C HhNHh+ H@ ymin = 1; * F J?>@L π 2

i π 2πn; + 2πn 2

π + 2πn;

∀n ∈ Z;

i 3π + 2πn 2

∀n ∈ Z.

65 [¡0 /OF J?>?CZHhC?'?C HhNH@

π + 2πn ∀n ∈ Z; 2

x= −

C

π + 2πn ∀n ∈ Z. 2 >K3C HhNHh+ ymax = − 1.

R P @ E(C?>>@ K @ " * K\R F P?C?) F '?*@b '?F Q

+2πn, 1 ∀n ∈ Z F J?>@L π − +2πn, −1 ∀n ∈ Z 2

_?P HhF Q S(* y = 1; R F P?C?) F '?*@b '?F Q _?P HhF Q %@UC?y'?= * " P?−S@1.b@L R P @ E(C?> P @ K3_?F b F" '¾'?" (2πn; '?C"U_?P HhF Q ' @¾C?π'?+* " 2πn) P?S@b@L ∀n ∈ Z R P @ E(C?>UP @ K3_?F b yF=" 'M1;'?"(S +T"(_?P HhF Q (π + 2πn; 2π(n + 1)) ∀n ∈ Z y = − 1.

! $

£

6 75 4¾8742 7 5 s+T_?NB>b@U' @C?'?* " P?S@b@L

12◦ .

(2πn; π + 2πn) ∀n ∈ Z;

S F R '?NB*@M' @C?'?* " P?S@b@L

?7 ?036 0[¡0+4 3Yc'?" * 13 . % : 534 7 s" P?* C?>@b '?+T"%@ K3C Hh_?* F * + 14 x = πn ∀n ∈ Z. 2%" ◦S .FK3* F P?F '? '?C?" _?P?"WA"3b +%_?P?C cosec x → − ∞ x → 2πn − 0 ∀n ∈ Z; _?P?C cosec x → + ∞ x → (π + 2πn) − 0 ∀n ∈ Z. (P @ S FK3* F P?F '?'?C?"_?P?"WA"3b +%_?P?C cosec x → + ∞ x → 2πn + 0 ∀n ∈ Z; _?P?C cosec x → − ∞ x → (π + 2πn) + 0 ∀n ∈ Z. %@ >b F '?'?+L`C&R F P?C?) F '?*@b '?+LM@ K3C Hh_?* F *,'?" * (π + 2πn; 2π(n + 1))

◦

∀n ∈ Z.

y 2π

−π

−

3π 2

π − 1 2 O

3π 2 π −1 2

x

5π 2 π

3π

−2π

S.TEU V

[?Y2 4?032 2 75

(" F R P @ '?C?J?" '?' @M'?C&K3'?C?) N =?'?C1K3S " PLN

15◦ . ]Y4 7 4 4¾2Y4 ?$?C&S CBAD@

!

y(x) = A sin(ωx + ϕ)

∀x ∈ R,

R\A" *@ >?C?",_?FK3* F?'?'?+T" =¡J?* F A > 0, ω > 0, 0 6 ϕ < 2π, ' @ ) +TA, S@ " ω, * K\ ϕuA> *.=I13/&D&(13M P?C K W A3= L (FK3* F?'?'?+T" =S LFA ?C?"S,E(F P HhNb N =BC Hh"+&c*,' @ ) S@ '?COA (+*,u1 A3[ \]AX25471347A V /&AX\]A Z 6/&A3[ A+A Z D&47,CEA V ω

ϕ

y A A sin ϕ −ϕ ω

− π+ϕ ω

− 2π+ϕ ω

O

− π+2ϕ 2ω

− 3π+2ϕ 2ω

π−ϕ ω π−2ϕ 2ω

2π−ϕ ω

x

3π−2ϕ 2ω

−A

S.TEU V 5W

@ P HhF '?C?>@?S b??" * K\%_?" P?C?FAC?J?"K3>BF Q<E(NB'?>?$?C?" QKO_?" P?C?FAFH

:(C KWb F

ν=

@¾?S b??" * K\P?" " '?C?"HAC E%E(" P?" '?$?C @b '?F R F¥NBP @ S '?" '?C R3@ P HhF '?C?J?"K3>BF R F>BF b " @ '?C y 00 = − ω 2 y.

!

d

d 3 O d

$

'

]"

&

w

!

!

~O

?Nb??P?'?+LUR3@ P HhF '?C?J?"K3>?CLM>BF b " @ '?C?QO

n ∈ Z,

1

y

1

y

x −1

O

V

S.TEU V

1

−1

x = cos 3t, y = sin 2t

x −1

O

5V

S.TEU V

1

−1

x = sin 5t, y = sin 3t

w

1

~O

y

1

y

x −1

O

uV

S.TEU V

x −1

1

−1

O

x = cos 5t, y = cos t

1

V

S.TEU V

y

1

−1

x = sin 5t, y = cos 2t

1

y

x −1

O

uV

S.TEU V

x −1

1

−1

O

x = cos 6t, y = cos 5t

1

S.TEU V

1

−1

V

y

x = cos 4t, y = sin 3t

1

y

x −1

O

S.TEU V

1

x −1

−1

V

x = cos t, y = sin 3t

O

1

−1

S.TEU V V

x = sin 4t, y = sin 5t

d d #

d

$

! $

!! $

]" ' (P?F C?) S FA' @

(" P?S F F P @ ) ' @

sin x

cos x

− cos x + C

cos x

− sin x 1 cos2 x 1 − sin2 x

sin x + C

ctg x

tg x sec x

cosec x

− ctg x cosec x

$

arcsin x

9¦$ !! $ ]"

C?)%F * P?" ) >@

x h

−

{*(47AX/

C?* " P?S@b@

+

C?)%F * P?" ) >@

∀x ∈ [ − 1; 1], ∀x ∈ [ − 1; 1].

C?)%F * ?P " ) >@ ' @ ) +TS@ " * K\M*@ >BF " J?* F " R F,>B[F− K3C?1; '?N1]KP @ S " '

x [0; π],

x.

∀x ∈ [ − 1; 1],

cos arccos x = x

∀x ∈ [ − 1; 1].

π π − ; , 2 2

x.

0 6 arccos x 6 π

; /&251>=

; , 2 2

sin arcsin x = x

{*((1325D&/&,251>= J?C KWb@

C?)ZF * P?" ) >@ ' @ ) +TS@ " * K\*@ >BF " [ − 1; 1] i π π J?* F " R FK3C?'?NKP @ S " '

π π 6 arcsin x 6 2 2

−

arccos x

ln | sin x| + C x π ln tg + +C 2 4 x ln tg + C 2

_?P?F C?) S F b ' @M_?FK3* F?'?' @W

{*(25D&/&,251>= J?C KWb@

− ln | cos x| + C

sec x

(C

J?C KWb F

!

=

(0; π),

π π < arctg x < ∀x ∈ R, 2 2 tg arctg x = x ∀x ∈ R.

arcctg x

x.

0 < arcctg x < π

∀x ∈ R,

ctg arcctg x = x

∀x ∈ R.

arcsin( − x) = − arcsin x

∀x ∈ [ − 1; 1].

arccos( − x) = π − arccos x ∀x ∈ [ − 1; 1]. arctg( − x) = − arctg x

∀x ∈ R.

arcctg( − x) = π − arcctg x arcsin x + arccos x =

π 2

arctg x + arcctg x = arcsin sin x = x

∀x ∈ R.

∀x ∈ [ − 1; 1]. π 2

∀x ∈ R.

h π πi ∀x ∈ − ; . 2 2

arccos cos x = x ∀x ∈ [0; π]. π π arctg tg x = x ∀x ∈ − ; . 2 2

arcctg ctg x = x ∀x ∈ (0; π). √ cos arcsin x = 1 − x2 ∀x ∈ [ − 1; 1]. tg arcsin x = √

x 1 − x2

∀x ∈ ( − 1; 1).

√ 1 − x2 ctg arcsin x = ∀x ∈ [ − 1; 0) ∪ (0; 1]. x √ sin arccos x = 1 − x2 ∀x ∈ [ − 1; 1].

/
0, y > 0

"KWb C

x < 0, y < 0

C

x2 + y 2 > 1. p √ arcsin x + arcsin y = − π − arcsin x 1 − y 2 + y 1 − x2 ,

C

x2 + y 2 > 1. p √ arcsin x − arcsin y = arcsin x 1 − y 2 − y 1 − x2 ,

"KWb C

Cb C

x2 + y 2 6 1. p √ arcsin x − arcsin y = π − arcsin x 1 − y 2 − y 1 − x2 ,

"KWb C

xy > 0

C

x2 + y 2 > 1. p √ arcsin x − arcsin y = − π − arcsin x 1 − y 2 − y 1 − x2 ,

"KWb C

x > 0, y < 0

C

x < 0, y > 0 x2 + y 2 > 1. p arccos x + arccos y = arccos xy − (1 − x2 )(1 − y 2 ) ,

"KWb C

x + y > 0.

!

& d Ow

d

p

arccos x + arccos y = 2π − arccos xy −

"KWb C

(1 − x2 )(1 − y 2 ) ,

x + y < 0. p arccos x−arccos y = −arccos xy+ (1 − x2 )(1 − y 2 ) , arccos x − arccos y = arccos xy +

p

"KWb C

x+y , 1 − xy

arctg x + arctg y = − π + arctg

x+y , 1 − xy

arctg x − arctg y = arctg

x−y , 1 + xy

arctg x − arctg y = π + arctg

2 arcsin x = − π − arcsin 2x 1 −

x2

,

2 arccos x = arccos(2x2 − 1),

2 arccos x = 2π − arccos(2x2 − 1),

2 arctg x = arctg

2x , 1 − x2

"KWb C

"KWb C "KWb C "KWb C

C

|x| 6

"KWb C

xy > 1.

C

xy > 1.

xy > − 1.

x 0.

"KWb C

(1 − x2 )(1 − y 2 ) ,

arctg x + arctg y = arctg arctg x + arctg y = π + arctg

"KWb C

xy < − 1.

C

xy < − 1.

√

2 . 2

√

2 < x 6 1. 2

√ 2 −16x < − . 2 0 6 x 6 1. − 1 6 x < 0. |x| < 1.

£d

2 arctg x = π + arctg

2x , 1 − x2

2 arctg x = − π + arctg

2x , 1 − x2

"

$

{*(25D&/&,2

!

"KWb C "KWb C

x > 1. x < − 1.

B 0

! $ ('

!! $ ]" ' "$

¡ uü ¡epn op³ y(x) = arcsin x E(NB'?>?$?C 3 . arcsin( − x) = − arcsin x ∀x ∈ [ − 1; 1]. K3C HHh" * P?C?J?" '1F * '?FK3C?* "3b '?F,' @J @b@>BF F P AC?' @ *U P?C K DW ]00[D4 7 4?0 536Y+ 4 .] x = 0. 5 . ^T[D7< y 0 6 4 2Y6 7 75 7 25 8 Y (F 6b F.C?* "3b ' @%' @c_?F b NBC?'?* " P?S@b% F * P?C?$ @ * "3b ' @' @c_?F b NBC?'?* " P?S@b% (0; 1]; ^T[D7< 0 6 4 % 72 7 72 2 75 [−1; 4 0). 7 . sF ) P @ K3*@ " *' @F * P?" ) >B" −1 1 x /&4*) )y0[¡032 9 4[ 0 < [Y+−s1;(1]. P?C?* C?J?"K3>?CL 8 . * F J?" >U'?" * 6 75 4 874 2 75 ,s+G − _?NB9>b.@' @cF * P?" ) >B" S F R '?NB*@' @cF *G P?" ) >B" O F J?[0; >@,1]; _?" P?" R C?@ [ − 1; 0]. S.TEU VuW ?7 ?036 65 [¡0 < Yc'?" * (0, 0). 10 . :%534 7 '?" * 11 . 1◦ .

DY 5

◦

◦

◦ ◦

◦

π 2

◦

O

◦

◦

π 2

◦

12◦ .

]Y4 7

0 0 2Y ?032 40

◦

max (arcsin x) =

[−1;1]

π 2

AFK3* C?R3@ " * K\%S* F JG

>B" ' @ C Hh" '? " "c) ' @J?" '?C?" π AFK3* C?R3@ " * K\`S x = 1; min (arcsin x) = − [−1;1] 2 * F J?>B" x = − 1. ¡u®±¡u ¡epn op³ y(x) = arccos x {*((1325D&/&,2 E(NB'?>?$?C(F P @ * ' @(>%K3N" '?C!& >BFK3C?'?NK @s' @ [0; π]. 7 [¡0 0 D032 4 ◦ DY 5 1 .

D(arccos x) = [ − 1; 1].

$

2◦ .

£d

X12 7 0 5 87 2Y ?032 4

y π

8

O

S.TEU V

]Y4 7 12◦ . max (arccos x) = π

'?C?"

{*(47AX/

1◦ . 2◦ .

X12 7

AFK3* C?R3@ " * K\US,* F J?>B"

2Y6 7 75 7 25 ' @#_?F b NBC?'?* " P?S@b "

0 5 87 2Y ?032 4

−1

π/2

x = 1.

D(arctg x) = R. π π E(arctg x) = − ; . 2 2 ] 0 ? 0 2 Y 3◦ .

+

arctg(−x) = −arctg x ∀x ∈ R.

π 4 O

− π4

1

−π/2

S.TEU V

x

OP @ E(C?> K3C HHh" * P?C?J?" 'F * '?FK3CG * "3b '?F,' @J @b@>BF F P AC?' @ * ]0 0[D4 7 4 ?0 536+Y 4◦ . ] x = 0. ◦ 5 . ^T[D7 < 0 6 4 2Y6 7 75 ◦ J?C 7 KW6b 2F 5.S FHU8 Yb ?NB(J?F " b FC?* "3b ' F @*MP?' C@ G

(0; +∞); $ @ * "3b ' @U' @J?C KWb F S FH b NBJ?" ( − ∞; 0). 7 ZK3N" '?C!&*@ ?' R " ' K @,' @C?'G

π π − ; . DY 25 2 7 [¡0 0 D032 4

2 420

x = − 1;

min (arccos x) = 0

* " P?S@b

2 7

6 4 %72 7 72 2 75 4 ?P " ) >B" [ − 1; 1]. ◦ /&4*) )y0[¡032 9 4[ 0 U'?" * ?7 ?036 65 [¡0 @_?" P?" R C?@ π [0; 1]. 0, . 2 4 7 '?" * ◦ :%53 . 0 0(4&2Y4 B" ' @ C Hh" '? " "c) ' @J?"3G

[−1;1]

[−1;1]

]00[D4 7 4?0 536Y+

4◦ . ] 5◦ . ^T[D7 &'?" * 8◦ . R. ?7 ? 3 0 6 6 5 ¡ [ 0 < c Y ? '

" * 9◦ . 6 7 5 48742 75 Ts+T_?NB>b@ ' @J?C KWb F S FHb NBJ?" 10◦ . S F R '?NB*@(' @J?C KWb F S FHMb NBJ?" OF J?>@_?" P?" R C?@ (0, 0). (−∞; 0]; [0; +∞). 7[D4 72 Y 2 f0BF *@ '?R " ' K @(' @ C?'?* " P?S@b (0; π). DY 5 7 [¡0 0 D032 4 ◦ 1 . D(arcctg x) = R. X12 7 0 5 87 2Y ?032 4 2◦ . E(arcctg x) = (0; π). 5 ,2 4 ?0 2 7 ◦ ]0 D8 D0 3 . 2 4Z20 ?0 2 7 P?C K BW

y π π/2 π 4

/ - ]00[D4 7 4?0 536Y+ 4 . ] D0 '?" *OP @ E(C?> '?" _?" P?5"K3." >@ " *FK3C&@ K3$?C K K ^T[D7< 0 6 4 2Y6 7 7 5 67 .25 8 Y (F b FC?* "3b ' @ ' @SK3" QUJ?C KWb F S F QM_?P HhF Q ◦

1

O

S.TEU V

◦

x

◦

+ 4

6 4 %72 7 72 2 75 4 3c +TS@ " *' @SK3" Q(J?C KWb F S F Q_?R.PG ◦ ^T[D7 b@ ' @J?C KWb F S FHb NBJ?" 10◦ . S F R '?NB*@%' @J?C KWb F S FHMb NBJ?" OF J?>@_?" P?" R C?@ π [0; +∞); (−∞; 0]. 0, . 2 7[D4 72 Y 2 f0?$?C P?C K W y(x) = arcsin sin x ∀x ∈ R y

π 2

−3π −2π − 5π − 3π 2 2

−π − π2

π O − π2

S.TEU V

π 2

2π 3π 2

3π 5π 2

x

£d

3B ?$?C

π

O

2π

3π

x

π π y(x) = arctg tg x ∀x ∈ − + πk; + πk , ∀k ∈ Z 2 R P @ E(C?>&_?FK3* P?F " 'U' @P?C K 2 W

,3

y π 2

−3π

−π −2π 3π − π2 − 5π − 2 2

π O

2π 3π 2

− π2

S.TEU V

3B ?$?C

(P?F C?) S FA' @

(" P?S F F P @ ) ' @

arcsin x

√

1 ∀ x ∈ (− 1 ; 1) 1 − x2

x arcsin x +

arccos x

√

−1 ∀ x ∈ (− 1 ; 1) 1 − x2

x arccos x −

1 1 + x2

arctg x

arcctg x

− (C

$#

"$ !

1 1 + x2

√

1 − x2 + C

√

1 − x2 + C

x arctg x −

1 ln(x2 + 1) + C 2

x arcctg x +

1 ln(x2 + 1) + C 2

_?P?F C?) S F b ' @M_?FK3* F?'?' @W $

! ! $

]"

¹Ouü

'

cos x = cos y

cos x = cos y ⇐⇒ x = ± y + 2πn ∀x, y ∈ R, ∀n ∈ Z.

¹Ou®±¡u

sin x = sin y ⇐⇒

∀x ∈

sin x = sin y

x = y + 2πn, x = π − y + 2πn

¹OuwOu

∀x, y ∈ R, ∀n ∈ Z.

tg x = tg y

tg x = tg y ⇐⇒ x = y + πn π π π π − + πk; + πk , ∀y ∈ − + πl; + πl , ∀k, l, n ∈ Z. 2 2 2 2

¹Ou ©u

ctg x = ctg y

ctg x = ctg y ⇐⇒ x = y + πn ∀x ∈ (πk; π(k + 1)), ∀y ∈ (πl; π(l + 1)), ∀k, l, n ∈ Z.

$

¹Ou®¦¡u

cos x = − cos y

cos x = − cos y ⇐⇒ x = π ± y + 2πn ∀x, y ∈ R, ∀n ∈ Z.

¹OuwªOu

sin x = − sin y ⇐⇒

sin x = − sin y

x = − y + 2πn, x = π + y + 2πn

¹Ou¡u

∀x ∈

∀x, y ∈ R, ∀n ∈ Z.

tg x = − tg y

tg x = − tg y ⇐⇒ x = − y + πn π π π π − + πk; + πk , ∀y ∈ − + πl; + πl , ∀k, l, n ∈ Z. 2 2 2 2

¹Ouw»Ou

ctg x = − ctg y

ctg x = − ctg y ⇐⇒ x = − y + πn ∀x ∈ (πk; π(k + 1)), ∀y ∈ (πl; π(l + 1)), ∀k, l, n ∈ Z.

¹Ouw¹Ou

sin x = cos y π sin x = cos y ⇐⇒ x = ± y + 2πn ∀x, y ∈ R, ∀n ∈ Z. 2

¹Ou¨¬Ou

tg x = ctg y π tg x = ctg y ⇐⇒ x = − y + πn 2 π π ∀x ∈ − + πk; + πk , ∀y ∈ (πl; π(l + 1)), ∀k, l, n ∈ Z. 2 2

¹Ou¨ü

sin x = − cos y π sin x = − cos y ⇐⇒ x = − ± y + 2πn ∀x, y ∈ R, ∀n ∈ Z. 2

¹Ou¨?±¡u

$ +

tg x = − ctg y π tg x = − ctg y ⇐⇒ x = + y + πn 2 π π ∀x ∈ − + πk; + πk) , ∀y ∈ (πl; π(l + 1)), ∀k, l, n ∈ Z. 2 2

#

yKWb C yKWb C

!#"$ % '¥$ ! ! $ ]"

&% |a| > 1, |a| 6 1,

'

©D¬Ouüp¢ lehgepohg sin t = a * FNBP @ S '?" '?C?" '?"%C Hh" " *,P?" " '?C?QO sin t = a * F`Hh'?F"K3* S FH P?" " '?C?Q& N A" * P?C K C&P?C K ? W

{( − 1)n arcsin a + πn} ∀n ∈ Z

J @ K3* '?FK3* CO=

sin t = 0 ⇐⇒ t = πn ∀n ∈ Z, π sin t = − 1 ⇐⇒ t = − + 2πn ∀n ∈ Z, 2 π sin t = 1 ⇐⇒ t = + 2πn ∀n ∈ Z. 2 sin

y

sin (a > 1)

a 1 a

Pπ−arcsin a Pπ −1

Pπ 2

Parcsin a P0 1 x

O

Pπ −1 P−π−arcsin a

a>0

S.TEU V

yKWb C yKWb C

|a| > 1, |a| 6 1,

1 P0 1 x

O

Parcsin a a −1 (a < − 1) a

−1 n> } p

y

n> } p

a60

S.TEU V

©D¬Ou®±¡uh¢ lehgepohg cos t = a * FNBP @ S '?" '?C?" '?"%C Hh" " *,P?" " '?C?QO cos t = a * F`Hh'?F"K3* S FH P?" " '?C?Q& N A" * P?C K ?(CUP?C K W

{ ± arccos a + 2πn} ∀n ∈ Z

J @ K3* '?FK3* CO=

π + πn ∀n ∈ Z, 2 cos t = − 1 ⇐⇒ t = π + 2πn ∀n ∈ Z, cos t = 0 ⇐⇒ t =

cos t = 1 ⇐⇒ t = 2πn ∀n ∈ Z.

$

} p

−1 n> } p

−16a60 S.TEU VW

06a61 S.TEU V

D© ¬OuwOup¢ lehgepohg tg t = a (P?CMb&c FH A" Q K3* S C?* "3b '?FH Hh'?F"K3* S FHP?" " '?C?Q& N A" * a P?C K C&P?C K BW {arctg a + πn} ∀n ∈ Z y

Pπ+arctg a 1 Parctg a

Pπ −1

S.TEU V

Pπ −1

tg

P0 1 0

O

a>0

n> } p

S.TEU V

x

Parctg a

−1

x a

−1

n> } p

$5¨

P0 1 0

O

Pπ+arctg a

y 1

tg

a

a60

&

#

D© ¬Ou ©up¢ lehgepohg ctg t = a (P?CMb&c FH A" Q K3* S C?* "3b '?FH Hh'?F"K3* S FHP?" " '?C?Q& N A" * a P?C K C&P?C K W {arcctg a + πn} ∀n ∈ Z. y Parcctg a

a

Pπ −1

y ctg

0 1

P0 1

O

0 1

x

−1 Pπ+arcctg a n> } p

!#"$

%

O

S.TEU V $

P0 1

ctg

a

x

Pπ+arcctg a −1 n> } p

a60

S.TEU V

Pπ −1

Parcctg a

a>0

!! $ ]"

"$

©üü1glDgehm lDk sin t < a yKWb C a 6 − 1, * F,'?" P @ S " ' K3* S F,P?" " '?C?QU'?"%C Hh" " * yKWb C a > 1, * FP?" " '?C?"H N A" *BF * F P?+L,K3C?'?NK3F CBAD@b "3C?* t, '?C"(R F P?C?) F '?*@b '?F QM_?P HhF Q y = a. y0 032 40% 5 7 %7 0 4 2 4 2 7 76W[ 2 75 4D; %7 [?Y 5 5 032 4 [D4 P?C K C 3W −1?sin P?N '?FK3* C¾S +sA"3b??"H ANBR N =>@AD@Z* F J?>@ >BF * F P?F Q_?P?F " $?C?P?N " * K\ ' @¥_?F b NBC?'?* " P?S@b FK3C F P AC?' @ * .yP* F t [ − 1; a) ANBR3@ ^ " )(>BF '?$?F S R\A"

π − arcsin a + 2πn < t < 2π + arcsin a + 2πn,

Pπ−arcsin a P2π+arcsin a

$ !

&

#

?Dx NBJ?" * FH _?" P?C?FAD@,)@ _?C K3+TS@ "HSK3"BF * F P?+T" N AF S b " * S F P−!&c1 *6LaF *?$?C?C,K3C?'?NKy' @LFAC H1Hh'?F"K3* S F x NBJ?" * FH ) ' @J?" '?C?Q'?" )@ S 2π C K3C HhF Q,_?" P?"Hh" '?'?F Q _?P?C`>BF * F P?+L,K3C?'?NK3F CBAD@b "3C?* t, S +T"(R F P?C?) F '?*@b '?F Q`_?P HhF Q arcsin a + 2πn < t < π − arcsin a + 2πn,

R\A"

y = a. y

(a > 1)

a 1 a −3π

−2π

π

O

−π

−1 −2π + arcsin a

2π

3π

t

π − arcsin a

−π − arcsin a

3π − arcsin a arcsin a 2π + arcsin a

n> } p

a>0

S.TEU VuW

−3π − arcsin a

−2π + arcsin a −π − arcsin a −3π

−2π

y

arcsin a

π − arcsin a

1 O

−π

a −1 a n> } p

π

2π + arcsin a 2π

3π

t

(a < − 1)

a60

S.TEU V

$$

#

76W[ 2 75 4

&

y0 032 40%5 7%7 0 4 2 42 7 sin y a

sin y 1

(a > 1)

1 Pt

Pt Pπ−arcsin a

Parcsin a

a

Pπ −1

P0 1 x

O

Pt

Pt Pπ −1

O

Pπ−arcsin a

Parcsin a a −1 a (a < − 1)

−1 n> } p

a>0

S.TEU V

P0 1 x

n> } p

a60

S.TEU V

7 [?Y 5 5 032 4 [D4 % P?C K 3C 3 BW −16a?sin P?N '?Fa.K3* C¾S +sA"3b??"H ANBR N =>@AD@Z* F J?>@ >BF * F P?F Q_?P?F " $?C?P?N " * K\`' @?CL¡=fJ?* F* F J?>?C b "3,@ * ' @ S +sA"3b " '?'?F Q ANBR " .y* C H Hh'?F"K3t* S FH ?S b??" * K\UF O"WAC?'?P" t'?C?"C?'?* " P?S@b F S (arcsin a + 2πn; π − arcsin a + 2πn) ∀n ∈ Z.

©üwOu1glDgehm lDk cos t < a yKWb C a 6 − 1, * F,'?" P @ S " ' K3* S F,P?" " '?C?QU'?"%C Hh" " * yKWb C a > 1, * FP?" " '?C?"H N A" *@ >BF * F P?F Q_?P?F " $?C?P?N " * K\#' @ _?F b NBC?'?* " P?S@b FK3C @ K3$?C K K .yP* F t [ − 1; a) ANBR3@ ^ " )(>BF '?$?F S P P ?Dx NBarccos J?" * FaH 2π−arccos _?" P?C?FAD@,a)@ _?C K3+TS@ "HSK3"BF * F P?+T" N AF S b " * S F P−!&c1 *6LaF *BF * F P?+L>BFK3C?'?NK3F CBAD@ K3* S F() ' @J?" '?C?Q,'?2π " )@ S C K3C HhF Q,_?" P?"Hh" '?'?F Q t, b "3C?*,S +T"(R F P?C?) F '?*@b '?F QM_?P HhF Q y = a. y0 032 40% 5 7 %7 0 4 2 4 2 7 76W[ 2 75 4 %7 [?Y 5 5 032 4 [D4 P?C K ?(C1 W −16ab@A+TS@ "H_?F b NBC?'?* " P?S@b J?* F¾K3F F *G (a; 1], S " * K3* S N " *'?" P @ S " ' K3* S N t > a. D%@`"WAC?'?C?J?'?F QZF >?cos P?N '?FK3* C¾S +sA"3b??"H ANBR N =>@AD@Z* F J?>@ >BF * F P?F QM_?P?F " $?C?P?N " * K\U' @_?F b NBC?'?* " P?S@b FK3C&@ K3$?C K K .y* FANBPR3@ t (a; 1] " )(>BF '?$?F S ^P − arccos a Parccos a ?Dx NBJ?" * FH _?" P?C?FAD@,)@ _?C K3+TS@ "HSK3" 1)

a − π2

− 5π 2

π 2

1

5π 2

a −3π

−π

−2π

−2π − arccos a

π

O −1

− 3π 2

arccos a

− arccos a

3π t

2π

2π + arccos a

3π 2

−2π + arccos a

2π − arccos a n> } p

a>0

S.TEU V

y −2π + arccos a − arccos a

−2π − arccos a −3π

2π − arccos a

−π

−2π

− π2

− 5π − 3π 2 2

a O −1 a

n> } p

S.TEU V

Pπ 2

Pπ −1

arccos a

1

2π + arccos a π 3π 2

Pπ 2

a>0

S.TEU V W

y 1

Pt

Parccos a Pπ a −1

P0 cos 1 x

O

Pt

P− arccos a −1

−1 n> } p

(a < − 1)

y

O

5π 2

a60

1 Parccos a Pt P0 cos a 1 x Pt P− arccos a

3π t

2π

π 2

n> } p

a60

S.TEU V

&

#

© ü®¦¡u1glDgehm lDk tg t < a (P?C b&c FH A" Q K3* S C?* "3b '?FH Hh'?F"K3* S FHP?" " '?C?Q ?S ?b ?" * K\ a F O"WAC?'?" '?C?"C?'?* " P?S@b F S π + πn; arctg a + πn ∀n ∈ Z. 2 )f4?0 536 70s[¡0 032 40AD@ '?F,' @P?C K ?

−

< [?Y xNBJ?" * FH G_?" P?C?FAC?J?'?FK3* CE(NB'?>?$?C?C*@ '?R " ' KC* F R F = J?* F@AD@&* F J?>@ >BF * F P?+LU*@ >@O= J?* FM_?P Hh+T" _?" P?"K3" >@ &c*FK3`*@ '?R " ' K3F S' @,b NBPJ?" t OPt .y* FANBR C ^P π Parctg a C ^P π Pπ+arctg a " )(>BF '?$?F S ( − ∞; a). − ?Dx NBJ?" * FH _?" P?C?FAD@,)2@ _?C K3+TS@ "HSK3"?$?C?C*@ '?R " ' KC* F R F = J?* F@AD@&* F J?>@ >BF * F P?+LU*@ >@O= J?* FM_?P Hh+T" _?" P?"K3" >@ &c*FK3`*@ '?R " ' K3F S' @,b NBPJ?" t OPt .y* FANBR C ^ C ^ " )(>BF '?$?F S π

(a; + ∞).

Parctg a P

Pπ+arctg a P 3π

? Dx NBJ?" * FH _?" P?C?FAD@,)@ _?C K3+TS@ "HSK3"?$?C?C¾>BF *@ '?R " ' K%C¾* F R F = J?* FUF ' @ F _?P?"WA"3b " ' @BF P'?arctg $?F S a Pπ Pπ+arcctg a −1 ?x NBJ?" * FH _?" P?C?FAD@%)@ _?C K3+TS@G "H¥SK3"Hh'?F"K3* S F() ' @J?" '?C?Q *@G S.TEU V >?CL¡= J?* F* F J?>?C b "3,@ *' @,t S +G A"3b " '?'?+LÀNBR3@L¡ Pt t

t

0

t

t

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#

&

©üw»Ou1glDgehm lDk ctg t > a P?C b&c FH A " Q K3* S C?* "3b '?FH ( Hh'?F"K3* S FHP?" " '?C?Q ?S b??" * K\ a F O"WAC?'?" '?C?"C?'?* " ?P S@b F S (πn; arcctg a + πn) ∀n ∈ Z.

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1 ctg Pt P0 1 x

t

t

t

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!

d

$

9

$

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©¡±¡uüp¢ lehgepohg

yKWb C yKWb C

π |a| > , 2 π |a| 6 , 2

arcsin x = a

* FNBP @ S '?" '?C?"P?" " '?C?Q&'?"%C Hh" " * * FP?" " '?C?"H N A" *

©¡±¡u®±¡up¢ lehgepohg

x = sin a.

arcsin f (x) = g(x)

arcsin f (x) = g(x) ⇐⇒   |f (x)| 6 1,    |g(x)| 6 π ,  π 2 |g(x)| 6 , ⇐⇒ ⇐⇒ 2    f (x) = sin g(x).  f (x) = sin g(x)

yKWb C yKWb C

¡© ±¡uwOup¢ lehgepohg arccos x = a Cb C * FNBP @ S '?" '?C?"P?" " '?C?Q&'?"%C Hh" " * a > π, * FP?" " '?C?"H N A" *

a

arctg x = a

* FNBP @ S '?" '?C?"P?" " '?C?Q&'?"%C Hh" " * * FP?" " '?C?"H N A" *

x = tg a.

& #d #

©¡±¡uwªOup¢ lehgepohg

arctg f (x) = g(x)   |g(x)| < π , 2 arctg f (x) = g(x) ⇐⇒  f (x) = tg g(x).

yKWb C yKWb C

¡© ±¡u¡up¢ lehgepohg arcctg x = a Cb C * F,NBP @ S '?" '?C?"(P?" " '?C?Q&'?"%C Hh" " * a >π, * FP?" " '?C?"H N A" *

a60 0 < a < π,

x = ctg a.

©¡±¡uw»Oup¢ lehgepohg

arcctg f (x) = g(x)

arcctg f (x) = g(x) ⇐⇒

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0 < g(x) < π, f (x) = ctg g(x).

9¦$ ! ! $

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yKWb C

y

(

a6 −

* F

π , 2

a6 −

a>

( − ∞; tg a).

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yKWb C

arctg x > a

π , 2 R.

y

* F&Hh'?F"K3* S FH

π 2

(tg a; + ∞).

©DOu¡u1glDgehm lDk y π π/2 ctg a

P?C K

* FP?" " '?C?Q&'?" *

P?" " '?C?QU N A" * yKWb C |a| < π , * F Hh'?F"3G K3* S FH¾P?" " '?C?Q, N A2" *%J?C KWb F S F Qb NBJ

a

P?C K

O

S.TEU V

x

arcctg x < a

tg a x

O

a − π2

S.TEU V

P?C K 3

yKWb C * F&Hh'?F"K3* S FHP?"3G " '?C?Q& N aA> " * π, yKWb C a 6 0,R.* FP?" " '?C?Q&'?" * yKWb C 0 < a < π, * FHh'?F"3G K3* S FH¾P?" " '?C?Q N A" *%J?C KWb F S F Qb NBJ (ctg a; + ∞).

& d & ua ud

©DOuw»Ou1glDgehm lDk

yKWb C

arcctg x > a

P?C K

* F,P?" " '?C?Q&'?" *

y π

yKWb C aa >6 π,0, * F1Hh'?F"K3* S FH P?"3G " '?C?QU N A" * yKWb C 0 < R.a < π, * FHh'?F"3G

π/2 a

K3* S FH¾P?" " '?C?Q, N A" *%J?C KWb F S F Qb NBJ

S.TEU V

p

A±

!

√ B=

!

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v q u u A + A2 − B t 2

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v q u u A − A2 − B t 2

,

A > 0, B > 0, A2 − B > 0. p √ √ √ a + b + 2 ab = a + b (a > 0, b > 0) .

p √ √ √ a + b − 2 ab = | a − b | (a > 0, b > 0).

" ! "$

) 2 6 9 40 57 DY 5 7[¡0 0 D 032 4

p45 787

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y = f (x)

f.

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yKWb CE(NB'?>?$?C

_?" P?C?FAC?J?"K3>@CC Hh" " *%_?" P?C?FA * F%E(NB'?>?$?C f T, \R A" C _?FK3* F?'?'?+%= @J?C KWb F *@ >" g(x) = Af (kx + b), A, k b k 6= 0, _?" P?C?FAC?J?' @?=_?P?C?J?"H " "%_?" P?C?FA1P @ S " ' T .

87 \[?Y 5 Y 0 ' @J?C KWb F S FH _?P?FHh"3NB* >B" "KWb CÀ b? C?) * @ >?CL¡=DJ?* F S +T_?F b '?" * X,K\Z'?" P @ S " 'G

NB'?>?$?C < C b&c +L x1 K3* S F

|k|

f x2 X, f (x2 ) > f (x1 ) :

x2 > x 1 ,

∀x1 , x2 ∈ X : x2 > x1 =⇒ f (x2 ) > f (x1 ). f8 Y 0 ' @J?C KWb F S FHZ_?P?FHh"3NB* >B" "KWb C?CL¡= J?* F S +T_?F b '?" X, * K\&'?" P @ S " ' K3* x1 x2 X, x2 > x 1 , f (x2 ) < f (x1 ) :

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∃U(x0 ) : f (x) 6 f (x0 ) ∀x ∈ U(x0 ).

f

fmax .

O F J?>?C&HhC?'?C HhNH@CM* F J?>?C&H@ >K3C HhNH@?NB$?'?C?>?CO$? C?CO 1C?'?C HhNHh+ C,H@ >K3C HhNHh+' @ ) +TS@&c* K\ 65 [¡0 O F J?>?CGH >K3* P?"HhNH@ME(NB'?>?$?C?C&?S b?!&c* K\ZS '?NB* P?" '?'?C HhCZ* F J?>@ HhC1" "

F b@ K3* CMF _?P?"WA"3b " '?CO (P?C _?" P?"3LFA"`J?" P?" )U* F J?>?NHhC?'?C HhNH@¥NB +TS@ '?C?"MK Hh" '?" * K\S F )3G P @ K3*@ '?C?"H=?@,_?P?C&_?" P?"3LFA"(J?" P?" )%* F J?>?NUH@ >K3C HhNH@S F ) P @ K3*@ '?C?"(K Hh"3G '?" * K\UNB +TS@ '?C?"H @ >?C H F P @ ) FH=O_?P?C _?" P?"3LFA"J?" P?" )`* F J?>? C H >K3* P?"HhNH@1C?)Hh" 'G " * K\MLB@ P @ >?* " P&HhF '?F * F '?'?FK3* C1E(NB'?>?$?C?CO yKWb C¥S '?NB* P?" '?'¥* F J?>@ x = x C?)F b@ K3* C¾F _?P?"WA"3b " '?C¥E(NB'?>G $?C?C ?S b??" * K\* F J?>BF Q H >K3* P?"HhNH0 @E(NB'?>?$?C?C * F#_?P?F C?) S FA' @ f f, C b & C , S *

F ? J B > " ? _ ? P

F ? C

)

S F A ' @ ? ' % "

F _?P?"WA"3b " ' @? 0 f 0 (x0 ) = 0 s'?NB* P?" '?'U* F J?>@C?x)(=F xb0@ K3* CMF _?P?"WA"3b " '?CfUE(NB'?>?$?C?C ' @ ) +TS@G "KWb C1 S H * F QZ* F J?>B"?$?C?C ?$?C F P @ * C H@#* F R\AD@ C * F b >BF * F R\AD@?=f>BF R\AD@¥>@AF "ZK3S F " ) ' @J?" '?C?"(F ' @_?P?C?'?C H@ " ** F b >BFFAC?'&P @ ) 2&c@U_?P H@O=?_ @ P @b?b "3b ' @UFK3C1@ K3$?C K K =?"KWb C&_?" P?"K3" >@ " *,R P @G E(C?>UF P @ * C HhF Q&E(NB'?>?$?C?CO=?* F,FAC?'UP @ ) yKWb C E(NB'?>?$?C HhF '?F * F '?' @U' @UJ?C KWb F S FH _?P?FHh"3NB* >B" =O* F&F ' @MF G P @ * C H@' { @ H * FH _?P?FHh"3NB* >B" (NK3* ¥F P @ * C H@E(NB'?>?$?C F _?P?"WA"3b " ' @¾' @¥J?C KWb F S FH_?P?FHh"3G NB* >B" C¾C Hh" " *`SM>@J?"K3* S "K3S F f" R F`Hh'?F"K3* S@) ' @J?" '?C?Q1_?P?FHh"3NG X(FK3*@ S C H S K3F F * S " * K3* S C?"M>@AFHhN * F > C?) * F¥"WAC?' K3* S " 'G Y. '?F ",) ' @J?" '?C?" _?P?C >BF * F P?FH OF R\yAD@&_?F Yb NBJ?C H E(NB'?>?$?!C &= A b? >BF * F P?F Q F x, b@ K3* 1F _?P?"WA"3b " '?fC(x) "K3=* y. @ZHh'?F"K3* S FH ) ' @J?" '?C?Q .y*@E(NB'?>?$?C#' @ ) +TS@ " * K\ Y,7 [?Y 2 7 > E(NB'?>?$?C?C f C ?S b??" * K\ X. F F ) ' @J @ " * K\ −1 f

.

/<

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f (x) > m ∀x ∈ D(f ) x1 ∈ D(f ), f (x1 ) = m. (C KWb F ' @ ) +TS@ " * K\M2Y4 7 4 2Y ?032 40 E(NB'?>?$?C?C "KWb C f, ) ' @J?" '?C M C¾K3NB"K3* S N " **@ >BF " J?* F f (x) 6 M ∀x ∈ D(f ) x2 ∈ D(f ), f (x2 ) = M. yKWb C f (x) > m A b?SK3"3L x C?) Hh'?F"K3* S@ G C K3NB"K3* S N " * *@ >BF " J?* F * FJ?C KWb F ' @ ) +TS@ " * K\`2Y4 ?$?C?C

A b? SK3"3L

C?)¥Hh'?F"K3* S@

G

f.

CK3NB"K3* S N " *

f (x) 6 M *@ >BF " J?* F * FxJ?C KWb F ' @ ) +TS@ " G* K\2Y4 7 4 x2 ∈ G, f (x2 ) = M, M 2Y ?032 40 ) 2 6 9 4 4 2Y %2 7 0 5 8 0 C&F F ) ' @J @ " * K\ f G max f.

(P?CKH * FH

max f

D(f )

¨

' @ C? F b " "c) ' @J?" '?C?"%E(NB'?>?$?C?C

G

f.

!

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cos x = cos y VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdV uW sin x = sin y V~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV uW tg x = tg y V~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdV uW ctg x = ctg y V~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV cos x = − cos y VdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdV sin x = − sin y VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdV tg x = − tg y VdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~V ctg x = − ctg y V~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdV sin x = cos y V~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdV tg x = ctg y VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV sin x = − cos y VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~V tg x = − ctg y

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sin t = a dV V~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdV cos t = a VddV VdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdV tg t = a dV VdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~V ctg t = a

b b b

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b

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b b b

b

b

b

b

arcsin x = a VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV arcsin f (x) = g(x) V~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV arccos x = a VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdV arccos f (x) = g(x) VdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdV arctg x = a VdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdV arctg f (x) = g(x) VdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdV arcctg x = a VdV~VdVdVdV~VdVdVdVdV~VdVdVdVdV~VdVdVdV~VdVdVdVdV~VdVdVdV~VdV arcctg f (x) = g(x)

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