!"#$!%& "$(!)*+) (+%*,-#.*#& /!% %*,+)&% #$)"+ /! .*0!%&#*1. *.,!."*0+ 2!3)!)+ 4567 ,$,+)8& 9:; 5< 9: =>?@A 9: 4567 2?>BC>D 9: ;>D 6EF55 G 6HF55 ,)*I+.+J!,)8& K(&),! 4L "#
$%&'()* +%, ,- (./,0' 1, 2'0*3 1, -%4 (*5%0*- &*0* %( 60,* &*0578%-*0 3, &%,1, /'1,-74*0 8'( -* ,9&0,37:( ! " # $ % & ()*
+ ,
"-.
; 1'(1, ! ,3 ,-
(./,0' 1, 2'0*3 1, -%4 (*5%0*- < " ,3 ,- 1=* 1,- *>'; 8'(371,0*(1' 8'(371,0*(1' " # / 8'/' ,- %(' 1, ,(,0'? @,5,0/7(, ,- (./,0' 1, 2'0*3 1, -%4 (*5%0*-A *# ,- B 1, ,(,0' C# ,- DE 1, ,(,0' 8# ,- F 1, G,C0,0' F#
$%&'()* +%, -* ,8%*87:( 1, -* -'()75%1 1, -* '(1* +%, 3, &0'&*)* &'0 %(* 8%,01* ,( %( 7(35*(5, 1, 57,/&' " ; ,( 3,)%(1'3; H7,(, 1*1* &'0 -* G%(87:(A
0 " # 1 ()* 2" % 3 $7 ,- &,0='1' 1, -* G%(87:( ,3 45 ; ,- H*-'0 /=(7/' 1, -* -'()75%1 1, -* '(1* ,3 1, 4 &%-)*1*3 < ,- H*-'0 /697/' /697/' ,3 /. &%-)*1*3; 1,5,0/7(,A *# -* 0,)-* 1, 8'00,3&'(1,(87* 1, 0 ? C# ,- H*-'0 (%/I078' 1, 1 % 2 % 3 ? D#
@*1* -* G%(87:( 06 7 - .*8 * 9 : ; 7 1,G7(71* &'0 0 < # =>?
+ ,
<-/
;
71,(57G7+%, -* &0'&'3787:( JKL$KA *# 0 ,356 *8'5*1* &'0 -* 0,85* @ # -/ ? C# M- &,0='1' G%(1*/,(5*- 1, 0 ,3 A # . ? 8# 0
B ,
# -/
1# L* G%(87:( C < # -0 < (' ,3 *8'5*1*? ,# 0 ,3 ,350785*/,(5, ,350785*/,(5, 80,87,(5, 80,87,(5, ,( ,- 7(5,0H*-' 7(5,0H*-' /8 . ? N#
$7
"D* E # -.
&0'&'3787:(A
D6 ()F E #
& &
J6 F(F F(F E # -
& .
<
FG( E H I ; 1,5,0/7(, ,- H*-'0 1, H,01*1 1, 8*1*
B# $7 K # DLF ()* .<
+
<
M K M 5 ; 1,5,0/7(, ,- H*-'0 (%/I078' 1,A
,
FG( , K N "D* K O#
$,*( -'3 6()%-'3 0 < tan
*#
P#
$7
(
<
1 7
2!
< " <
( );
cot !
=
1
C# !
2
,
()
arctan arctan 3 < 0 <
<
2
,
=
()
arcta arctan n 2 ? M- H*-'0 1,
) ,3 7)%*- *A
!
3!
2
sec
<
(! ) <
!
5
=
csc
5
8#
arccos
1#
7
(3 ) ;
,# !
1
1,5,0/7(, -'3 H*-'0,3 1,
x
5 6
sen
(! ) ;
( ) ;
tan !
(! ) ?
Q#
$7( %3*0 8*-8%-*1'0*; 1,5,0/7(, ,- H*-'0 1,A *# FG( .&&O C# ()* /&O
R#
S'(371,0*(1' -'3 H*-'0,3 &*0* -'3 8%*-,3 ,356 1,G7(71* -* ,9&0,37:(; 71,(57G7+%, -' +%, 1,C, 8'-'8*03, ,( ! &*0* +%, 8*1* ,9&0,37:( 3,* %(* 71,(571*1 507)'('/I5078*A
( )
*#
! csc
C#
cos 2 x
8#
1#
x
( )
sen
=
= !
"
(2 ) x
1 csc
2
1
( )
+
sec
x
2
( ) x
( ) ( ) " csc ( x ) sen ( x ) 1 cos ( x ) ! tan ( x ) 1 $ cos ( x ) # & ' sen x c o s x ( ) % " ( ) tan x ! sen x
=
4
+
+
=
+
(2 ) ! cos ( ) () ( ) ! cos ( )
sen
,#
1 ! sen
+ sen
x
2
2
x
(2 ) ! " cos ( ) ( ) sec ( )
sen
G#
x
x
sen x
1
x
x
=
x
=
x
sec
( ) x
"E# @,/%,350,; 1, 3,0 &'37C-,; -* 37)%7,(5, 71,(571*1 507)'('/I5078*A
( ) cos ( )
( ) cos ( 2 )
sen ! + sen 2!
1+
!
+
( )
= tan !
!
""# @,5,0/7(, ,- H*-'0 (%/I078' 1, D # / - PQP,
+ R
?
"F# K- 8'(371,0*0 -'3 6()%-'3 ,( ,- &07/,0 8%*10*(5,; 8*-8%-, ,- H*-'0 (%/I078' 1, 8*1* ,9&0,37:( 507)'('/I5078*A
' ! 1 $ + ) # 3& " % ( ' ! 1 $ * cot ) 2 arctan # 2 & " % ,+ (
*#
co s
C#
arcsen
"D# S'(371,0, +%, 37
( )
*#
sen z =
C#
cos
( )
z =
arcsen
u
=
tan
! 2 $ * # 3 & " % ,+
! $ 3, 8%/&-, +%,A # 2 & " % z
2u 1+ u
2
1! u
2
1+ u
2
"N# @*1' ,- 8'(T%(5' S) # I8 .5 < ,- &0,178*1' T E 6 FG( .E .E - ()* E # I; -* 3%/* 1, -'3 ,-,/,(5'3 1,- 8'(T%(5' 1, H,01*1 1T E ,3 7)%*- *A *# C# 8# 1# ,#
U+ V W+ X X+ , U+ , U+ X
"B# $,* ,- 0,G,0,(87*- Re
( )
p ! : 1 "
"O# $,* Re
=
( )
2 cos x
=
=
"#0 ,4 ! ) ; 1,5,0/7(, ,- 8'(T%(5' 1, H,01*1 1,- &0,178*1'
0
"#0 ,2 ! %$ < ,- &0,178*1'
( )
( )
( )?
p x : sen 2 x = 1 + cos 2 x
@,5,0/7(, ,-
8'(T%(5' 1, H,01*1 Ap ( x ) ? "P# $,*
()
p x :
,2 co cos
8'(T%(5' 2
() x
=
1 ! sen
0,G,0,(87*-
Re
=
( )? x
@,5,0/7(, -* 3%/* 1, -'3 ,-,/,(5'3 1, Ap ( x ) ?
!"0,2 0, 2! #$
<
,-
&0,178*1'A
"Q# $,* ,- 8'(T%(5' 0,G,0,(87*- Re
()
()
q x :
cos x
<
=
! 3! $ # 0, & < -'3 &0,178*1'3 " 2%
( ()
=
( )
sen 2 x
>
0 <
0?
@,5,0/7(, ,- 8'(T%(5' 1, H,01*1A A p x
"R# $7 Re
()
p x :
!
( )) ?
q x
#$ !" , " &% ; 1,5,0/7(, ,- 8'(T%(5' 1, H,01*1 1,- &0,178*1'A
( )
( ( ) )
p ! : µ sen 2! " 1 < 0
FE# @*1'
,-
T < 6 *# C# 8# 1# ,#
8'(T%(5'
-. FG( FG(
Y
,
8
X+ ,
[
]+ ,
8
BB+ ,
I8 5 [ &58 45 X+ , X+ ,
8
U+ ,
[
<
,-
&0,178*1'
1,
%(*
# I ? M(5'(8,3; ,- 8'(T%(5' 1, H,01*1 1T < ,3A
X
.58 .58 Z5 [ Z58 Z58 \5 +
S) # I8 45
^+ ,
8
8 .5 [ \58
]+ , ]+ ,
H*07*C-,
