!"#$%&'&()"*+',-./0+'&+1"20.3"
,+-(./0$123$(.4+$53&67$.8$+6.52-.7+$0.8*$9.076+$.7$7(+$
5+.9($2861$72$(.4+$&7$'+7$0:+/7$.:.1$51$7(+$&8;92<&8'$
7&*+"$
=/-&8'$5-+.>$&0$8+?7$:++>$.8*$123$.-+$/6.88&8'$.827(+-$
7-&/$72$7(+$5+.9("$$)(&0$7&<+$123$*+9&*+$72$/.1$.77+87&28$72$7(+$7&*+0$02$7(.7$123$9.8$>++/$7-.9>$2@$
(2:$<39($7&<+$123$(.4+$72$53&6*$.8*$.*<&-+$123-$0.8*$9.076+"$$$
A23$(.4+$.$@-&+8*$:(2$&0$&8$9.693630$:(2$:&66$5+$'2&8'$28$0/-&8'$5-+.>$:&7($123"$$A23$'&4+$123-$
@-&+8*$02<+$*.7.$@-2<$7(+$.6<.8.9$.5237$(&'($7&*+0$.628'$7(+$29+.8B$.0$:+66$.0$.$928723-$<./$2@$7(+$
5+.9($123$&87+8*$72$4&0&7B$.8*$.0>$(+-$72$92<+$3/$:&7($.8$+C3.7&28$@2-$7(+$:.7+-$6+4+6$28$7(+$5+.9($
28$7(+$*.1$2@$123-$7-&/"$$D992-*&8'$72$123-$@-&+8*E0$.8.610&0B$7(+$:.7+-$6+4+6$28$7(+$5+.9($:&66$@&7$7(&0$
+C3.7&28F$$$$
$
$
$
$
$
#" %
f (t) = 20sin t $$
$6 &
G8$7(&0$+C3.7&28B$(H/I$-+/-+0+870$(2:[email protected]$7(+$:.7+-6&8+$&0$.524+$2-$5+62:$&70$.4+-.'+$/20&7&28"$$)(+$
*&07.89+$&0$<+.03-+*$&8$@++7B$.8*$/$-+/-+0+870$7(+$+6./0+*$7&<+$H&8$(23-0I$0&89+$<&*8&'(7"$
!
#"$ J(.7$&0$7(+$(&'(+07$3/$7(+$5+.9($H92</.-+*$72$&70$.4+-.'+$/20&7&28I$7(.7$7(+$:.7+-6&8+$:&66$5+$
*3-&8'$7(+$*.1K$$H)(&0$&0$9.66+*"4&14"/&',"I$$J(.7$&0$7(+$62:+07$7(.7$7(+$:.7+-6&8+$:&66$5+$*3-&8'$
7(+$*.1K$$H)(&0$&0$9.66+*$%$5"/&',"I$
$
L"$ =3//20+$123$/6.8$72$53&6*$123-$9.076+$-&'(7$28$7(+$.4+-.'+$:.7+-6&8+$M307$.0$7(+$:.7+-$(.0$
<24+*$5+62:$7(.7$6&8+"$$%2:$<39($7&<+$:&66$123$(.4+$72$53&6*$123-$9.076+$5+@2-+$7(+$&892<&8'$
7&*+$*+07-210$123-$:2->K$
$
N"$ =3//20+$123$:.87$72$53&6*$123-$9.076+$#O$@++7$5+62:$7(+$.4+-.'+$:.7+-6&8+$72$7.>+$.*4.87.'+$2@$
7(+$*.</$0.8*"$$J(.7$&0$7(+$<.?&<3<$.<2387$2@$7&<+$123$:&66$(.4+$72$<.>+$123-$9.076+K$$%2:$
9.8$123$9284&89+$123-$@-&+8*$7(.7$123-$.80:+-$&0$92--+97K$
$
$
$
$
$
$
$
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%&'#$()*!+(*(,-!./,0&)$!1!" .!!!
"#$%&'%(!)&(%*!+,%!-*%.+#/%!-0110&'!2++*#3)+#0&450&-011%*$#.647,.*%26#8%!9:;!<&=0*+%(!6#$%&'%!
!
!
!
Trigonometric Functions 68
!
+
$
!"##$%&'($)&*+$
!>;?@!AAA:B6#$8*:$01C=,0+0'CD0%('#6/.!
68
69
!
"#!!!!$%&&'()!*'%!+,-.!.'!/%012!*'%3!4,(.1)!56!7)).!,/'8)!.9)!,8)3,:)!+,.)310-)!.'!:08)!*'%!;'3)!
.0;)!.'!,2;03)!*'%3!+'3<#!!=9,.!0(!.9)!;,>0;%;!,;'%-.!'7!.0;)!*'%!+011!9,8)!.'!;,<)!*'%3!
4,(.1)?!!@'+!4,-!*'%!4'-80-4)!*'%3!730)-2!.9,.!*'%3!,-(+)3!0(!4'33)4.?!
!
!
!
!
!
!
!
!
6# A'%!;,*!9,8)!,-(+)3)2!.9)!&3)80'%(!B%)(.0'-(!%(0-:!,!:3,&9!'7!.9)!.02)!7%-4.0'-#!!C(!.9)3)!,!
+,*!*'%!4'%12!%()!,1:)/3,!,-2!.9)!0-8)3()!(0-)!7%-4.0'-!.'!,-(+)3!.9)()!B%)(.0'-(#!!C7!('D!(9'+!
*'%3!+'3<#!
!
E1:)/3,04!+'3<!7'3!B%)(.0'-!FG!
!
!
!
!
!
!
!
!
!
E1:)/3,04!+'3<!7'3!B%)(.0'-!"G!
!
!
!
!
!
!
!
!
!
!
H# $%&&'()!*'%!2)402)!*'%!'-1*!-))2!.+'!9'%3(!.'!/%012!,-2!,2;03)!*'%3!4,(.1)#!!=9,.!0(!.9)!
1'+)(.!&'0-.!'-!.9)!/),49!+9)3)!*'%!4,-!/%012!0.?!@'+!4,-!*'%!4'-80-4)!*'%3!730)-2!.9,.!*'%3!
,-(+)3!0(!4'33)4.?!
!
!
!
!
!
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%&'#$()*!+(*(,-!./,0&)$!1!"
+ .!!! !
"#$%&'%(!)&(%*!+,%!-*%.+#/%!-0110&'!2++*#3)+#0&450&-011%*$#.647,.*%26#8%!9:;!<&=0*+%(!6#$%&'%!
Trigonometric Functions 69
!
!
!
"#$$!%&'(!)&*+!,!)+-.(+/!012+3!
!"#$%&'&()"*+',-./0+'&+1"20.3"
!"#$%&'4!!56!2(&3!2-37!328*+623!319&*&:;!2(+&/!86*+/32-6*&6'!1:!83&6'!2/&'161<+2/&.!:86.2&163!21!
<1*+9!=+/&1*&.!>+(-?&1/!>;!-==9;&6'!2/&'161<+2/;!21!-!.162+@2!2(-2!&3!=+/&1*&.!A(&'(!-6*!91B!2&*+3CD!
>82!612!.&/.89-/!<12&16#!!)(+;!9+-/6!(1B!21!&62+/=/+2!-<=9&28*+!-6*!=+/&1*!&6!2+/<3!1:!2(&3!6+B!
.162+@2#!!)(+;!-931!.163&*+/!83&6'!&6?+/3+!2/&'161<+2/&.!:86.2&163!21!-63B+/!E8+32&163!->182!2(+!
2&<+!-2!B(&.(!2(+!2&*+!/+-.(+3!?-/&183!(+&'(23F2(+!&6?+/3+!E8+32&16!:/1<!B(-2!&3!/+=/+3+62+*!>;!
2(+!:86.2&16 B(&.(!<1*+93!2(+!(+&'(2!1:!2(+!2&*+!-2!?-/&183!2&<+3#!!G28*+623!-63B+/!2(+3+!&6?+/3+!
E8+32&163!>12(!'/-=(&.-99;!-6*!-9'+>/-&.-99;#!
(%#')*+,-.,#.&)/%0"&4!
/12/13!H(113+!2/&'161<+2/&.!:86.2&163!21!<1*+9!=+/&1*&.!=(+61<+6-!B&2(!3=+.&:&+*!-<=9&28*+D!
:/+E8+6.;D!-6*!<&*9&6+#!!
/14/15!I&6*!&6?+/3+!:86.2&163#!
-#!!G19?+!-6!+E8-2&16!1:!2(+!:1/<!:A@C!J!.!:1/!-!3&<=9+!:86.2&16!:!2(-2!(-3!-6!&6?+/3+!-6*!B/&2+!-6!
+@=/+33&16!:1/!2(+!&6?+/3+#!!
.#!!AKC!L+-*!?-98+3!1:!-6!&6?+/3+!:86.2&16!:/1<!-!'/-=(!1/!-!2->9+D!'&?+6!2(-2!2(+!:86.2&16!(-3!-6!
&6?+/3+#!
6'7,+'.)*+,-.,#.&8)))
9,"-0:);<:%7')(7,&&=8!
562/1*8.+!2(+!.162+@2!1:!2&*+3!21!328*+623!-6*!(-?+!2(+<!-63B+/!E8+32&163!$!-6*!M!&6*&?&*8-99;D!-6*!
2(+6!*&3.833!2(+&/!-63B+/3!-3!-!.9-33#!!N+/&:;!2(-2!328*+623!-/+!->9+!21!&62+/=/+2!2(+!=-/-<+2+/3!1:!
2(&3!+E8-2&16!&6!2+/<3!1:!2(&3!6+B!.162+@2#!!)(+;!3(189*!>+!->9+!21!+@=9-&6!2(-2!2(+!MO!&6!2(+!
+E8-2&16!2+993!2(+<!2(-2!2(+!2&*+!B&99!/-6'+!MO!:++2!->1?+!1/!>+91B!-?+/-'+!2&*+D!-6*!2(-2!2(+! " 6 !2+993!
2(+<!2(-2!2(+!=+/&1*!1:!2(+!2&*+!:86.2&16!A16+!.1<=9+2+!.;.9+!1:!(&'(!-6*!91B!2&*+3C!&3!$M!(18/3D!
B(&.(!.-6!>+!:186*!>;!.-9.89-2&6'!
2"
"
6
#!!5:!328*+623!-/+!(-?&6'!*&::&.892;!/+9-2&6'!-!2/&'161<+2/&.!
!
:86.2&16!21!-!616P.&/.89-/!.162+@2D!-37!2(+6!(1B!2(+;!B189*!-63B+/!2(+3+!E8+32&163!&:!2(+!+E8-2&16!
B-3!/+:+//&6'!21!-!I+//&3!B(++9#!!Q1&62!182!2(-2!+?+6!2(18'(!2(+!(+&'(2!1:!2(+!2&*+3!(-3!612(&6'!21!*1!
B&2(!-6'9+3!1:!/12-2&16D!B+!.-6!R>1//1BS!2(+!=+/&1*&.!>+(-?&1/!1:!.&/.89-/!2/&'161<+2/&.!:86.2&163!
!
21!*+3./&>+!12(+/!=+/&1*&.!.162+@23!38.(!-3!2&*+3D!?&>/-2&6'!32/&6'3D!1/!2(+!.;.9+!1:!-?+/-'+!
2+<=+/-28/+3!1?+/!2(+!.18/3+!1:!-!;+-/#!!
!
I1991B&6'!2(&3!&62/1*8.2&16D!(-?+!328*+623!B1/7!16!E8+32&163!TPUD!-6*!E8+32&16!"!&:!2(+;!(-?+!2&<+#!
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%&'#$()*!+(*(,-!./,0&)$!1!"
+ .!!! !
"#$%&'%(!)&(%*!+,%!-*%.+#/%!-0110&'!2++*#3)+#0&450&-011%*$#.647,.*%26#8%!9:;!<&=0*+%(!6#$%&'%!
!
!
!
!
"#$%&'(!)*+,%%!-'&.$/0!
"#!$%!&'#$($)&#*+!#,&#!%#-+*'#%!.$//!&'%.*0!1-*%#$2'%!3!&'+!4!56!0*7*00$'8!#2!#,*!80&),!27!#,*!
%$#-&#$2'9!)*0,&)%!56!)/2##$'8!&!,20$:2'#&/!/$'*!&#!!!;!<=>!?720!1-*%#$2'!3@!20!!!;!A!?720!1-*%#$2'!4@!
&'+!#,*'!-%$'8!#,*!7*&#-0*%!27!#,*!(&/(-/&#20!#2!+*#*0B$'*!#,*!)2$'#%!27!$'#*0%*(#$2'!27!#,*%*!/$'*%!
.$#,!#,*!#0$82'2B*#0$(!80&),C!!D,$%!%#0&#*86!$%!&'!$B)20#&'#!2'*!720!&//!%#-+*'#%!#2!(2'%$+*0!&%!$#!
+*E*/2)%!(2'(*)#-&/!-'+*0%#&'+$'8!720!#,*!)025/*B9!%2!*'(2-0&8*!&'+!%-))20#!$#C!!F*/*(#!%#-+*'#%!
#2!)0*%*'#!+-0$'8!#,*!+$%(-%%!%#&8*!27!#,*!/*%%2'!.,2!(&'!+*%(0$5*!#,*$0!.20G!-%$'8!#,$%!%#0&#*86C!!
H/%2!.&#(,!720!%#-+*'#%!.,2!-%*!#&5/*%!#2!&'%.*0!1-*%#$2'%!3!&'+!4!&'+!%*/*(#!%#-+*'#%!#2!)0*%*'#!
#,$%!%#0&#*86!&/%29!$7!$#!$%!)0*%*'#!$'!#,*!.20G!27!#,*!%#-+*'#%C!!I2#*!#,&#!%$'(*!#,*!#$+*!82*%!#,02-8,!
#.2!(2B)/*#*!(6(/*%!+-0$'8!&!J4<,2-0!+&69!#,*0*!&0*!&(#-&//6!#.2!$'#*0E&/%!27!#$B*!.,*'!#,*!%&'+!
(&%#/*!5-$/+$'8!(0$#*0$&!(&'!5*!%&#$%7$*+!720!1-*%#$2'%!3!&'+!4C!!K&G*!%-0*!%#-+*'#%!&0*!&'%.*0$'8!#,*!
1-*%#$2'%!$'!#*0B%!27!#,*!#$B*!27!#,*!+&6LB20'$'89!&7#*0'22'!20!*E*'$'8L.,*'!#,*!%&'+!(&%#/*!
(2-/+!5*!5-$/#C!
!
F#-+*'#%!%,2-/+!5*!7&B$/$&0!.$#,!-%$'8!$'E*0%*!7-'(#$2'%9!&'+!-%$'8!$'E*0%*!#0$82'2B*#0$(!7-'(#$2'%!
#2!7$'+!B$%%$'8!&'8/*%!$'!0$8,#!#0$&'8/*%C!!D,*6!)025&5/6!.$//!'2#!,&E*!(2'%$+*0*+!#,*!$%%-*%!#,&#!
&0$%*!.,*'!$'E*0%*!#0$82'2B*#0$(!7-'(#$2'%!&0*!&))/$*+!#2!)*0$2+$(!7-'(#$2'%C!!H//2.!#,*%*!$%%-*%!#2!
%-07&(*!'&#-0&//6!&%!%#-+*'#%!.20G!2'!#,*$0!&/8*50&$(!&))02&(,*%!#2!&'%.*0$'8!1-*%#$2'%!3!&'+!4!
?%**!1-*%#$2'!A@C!!M$%#*'!720!,2.!%#-+*'#%!+*&/!.$#,!#,*%*!$%%-*%!&'+!%*/*(#!%#-+*'#%!#2!+$%(-%%!#,*%*!
$%%-*%!&'+!,2.!#,*6!0*%2/E*+!#,*B!+-0$'8!#,*!.,2/*!(/&%%!+$%(-%%$2'C!!N*7*00$'8!#2!#,*!80&),%!27!
#,*!%$#-&#$2'%!.$//!,*/)!B&G*!#,*%*!$%%-*%!B20*!(/*&0C!!O20!*P&B)/*9!%#-+*'#%!B$8,#!)*0720B!#,*!
72//2.$'8!&/8*50&!720!1-*%#$2'!3Q!
"10 = 20sin( #6 t )
"10
20
!
!
!
!
= sin( #6 t )
" 12 = sin( #6 t )
sin "1 (" 12 ) = sin "1 (sin( #6 t ))
!
sin "1 (" 12 ) = #6 t
H#!#,$%!)2$'#!#,*!1-*%#$2'!B$8,#!&0$%*9!RS,&#!$%!#,*!$'E*0%*!%$'*!27!<=TJUV!!D,&#!$%9!RS,&#!$%!#,*!&'8/*!
.,2%*!%$'*!$%!<=TJUV!!"7!.*!*P&B$'*!#,*!-'$#!($0(/*!.*!25%*0E*!#,&#!#,*!%$'*!7-'(#$2'!,&%!&!E&/-*!27!!
7"
11"
!
<=TJ!&#!#,*!)0$'($)&/!E&/-*%!27!
6 !&'+! 6 C!!W%$'8!#,*%*!E&/-*%!#2!%2/E*!720!"#.*!7$'+!.*!(&'!5-$/+!2-0!
%&'+!(&%#/*!5*#.**'!XQ>>!&'+!==Q>>!&CBC!$'!#,*!B20'$'8C!!Y2.*E*09!#,*!80&),!%-88*%#%!&'2#,*0!
$'#*0E&/!27!#$B*!5*#.**'!XQ>>!&'+!==Q>>!)CBC!?20!=ZQ>><J3Q>>!B$/$#&06!#$B*@!&%!&!)2%%$5/*!5-$/+$'8!
#$B*!$7!.*!.&'#!#2!%#&6!2'!#,*!5*&(,!/&#*!$'!#,*!*E*'$'8C!!D,$%!$'#*0E&/!.20G%!5*(&-%*!#,*!%$'*!*1-&/%!
!
!
<=TJ!&#! 196" !&'+! 236" !&/%2C!!"'!7&(#9!#,*0*!&0*!&'!$'7$'$#*!'-B5*0!27!&'8/*%!720!.,$(,!#,*!%$'*!27!#,*!
&'8/*!$%!<=TJ9!5-#!2'/6!#,*%*!72-0!&'8/*%!B&G*!%*'%*!$'!2-0!(2'#*P#!27!%)*'+$'8!&!+&6!&#!#,*!5*&(,C!!"7!
!
!
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%&'#$()*!+(*(,-!./,0&)$!1!"
+ .!!! !
"#$%&'%(!)&(%*!+,%!-*%.+#/%!-0110&'!2++*#3)+#0&450&-011%*$#.647,.*%26#8%!9:;!<&=0*+%(!6#$%&'%!
!
!
!"#$%&'#$(#$)"#*+,*%,+$(&#!"#-"$# sin
"1
(" 12 ) = " 0.5236 or " # 6 .#!)/*)#&"+,,0#1("2'3$#4/$#(%$#*('$"5$#+$#
+,,.#2/'*"#/$#!(%,1#6"+'#!#7#89.#(&#('"#)(%&#:"4(&"#6/1'/-)$#(4#$)"#1+0#!"#;,+'#$(#</2/$#$)"#:"+*)=#
>,,(!#2$%1"'$2#$(#!&"2$,"#!/$)#$)"2"#/1"+2#%'$/,#$)"0#+&&/<"#+$#2/6/,+&#"5;,+'+$/('2#/'#$"&62#(4#$)"#
*('$"5$.#+2#-/<"'#+:(<"=##?@+/$#%'$/,#$)"#!)(,"#*,+22#1/2*%22/('#$(#;(/'$#(%$#$)"#1/,"66+#$)+$#$)"#
!
/'<"&2"#2/'"#4%'*$/('#*+'#(',0#)+<"#('"#(%$;%$.#+'1#/$#/2#:0#*('<"'$/('#$)+$#6+$)"6+$/*/+'2#)+<"#
&"2$&/*$"1#(%$;%$#<+,%"2#4(&#$)"#/'<"&2"#2/'"#4%'*$/('#$(#$)"#/'$"&<+,# " #2 $ sin "1 ( x ) $
#
2
=##A($"#$)+$#
$)"#*+,*%,+$(&#(%$;%$#!+2#/'#$)/2#/'$"&<+,=B#
#
C)"#+,-":&+#4(&#D%"2$/('#E#4(,,(!2#$)"#2+6"#&"+2('/'-#+2#+:(<".#)(!"<"&#$)"&"#+&"#'(#"+2/,0#
!
/1"'$/4/+:,"#+'-,"2#('#$)"#%'/$#*/&*,"#4(&#$)"#D%"2$/('.#F@)+$#+'-,"#)+2#+#2/'"#<+,%"#(4# 3 4 GH##
I('2"D%"'$,0.#!"#+&"#,"4$#2(,</'-#$)/2#"D%+$/('#%2/'-#+#*+,*%,+$"1#<+,%"#4(&# sin
( ) = 0.848 =##
"1 3
4
J(,</'-#$)"#"D%+$/('#4(&#!#%2/'-#$)/2#<+,%"#0/",12#!#7#9=KL.#:%$#!"#*+'3$#2$+&$#:%/,1/'-#(%&#2+'1#*+2$,"#
9=KL#)(%&2#+4$"&#6/1'/-)$#M+;;&(5/6+$",0#9NOP#+=6=Q.#2/'*"#$)"#,"<",#(4#$)"#$/1"#/2#&/2/'-#+$#$)+$#$/6".#
!
'($#&"*"1/'-=##J$%1"'$2#!/,,#)+<"#$(#&"+2('#$)&(%-)#)(!#$)"0#6/-)$#%2"#$)/2#<+,%"#$(#1"$"&6/'"#$)"#
!
/'$"&<+,#(4#$/6"#!)"'#$)"6#6/-)$#:%/,1#$)"/&#*+2$,"=##R5+6/'/'-#+#%'/$#*/&*,"#1/+-&+6#!(%,1#2%--"2$#
$)+$#$)"#2/'"#4%'*$/('#+,2(#)+2#+#<+,%"#(4# 3 4 #+$#+'#+'-,"#"D%+,#$(#!#S#T=UEU#(&#+'#+'-,"#6"+2%&/'-#
L=LVOK#&+1/+'2=##J(,</'-#$)"#"D%+$/('#4(&#!#%2/'-#$)/2#<+,%"#0/",12#!#7#E=OU#)(%&2#+4$"&#6/1'/-)$.#(&#
+;;&(5/6+$",0#ENLO#+=6=#+2#$)"#$/6"#!)"'#!"#*+'#2$+&$#:%/,1/'-#(%&#2+'1#*+2$,"=##C(#4/'1#$)"#$/6"#
!)"'#$)"#&/2/'-#$/1"#!/,,#!+2)#+!+0#(%&#2+'1#*+2$,"#!"#!/,,#'""1#$(#4/'1 $)"#'"5$#+'-,"#!)"&" $)"#
!
2/'"#4%'*$/('#"D%+,2# 3 4 =##C)/2#(**%&2#+$#L!#WT=UEU#(&#+'#+'-,"#6"+2%&/'-#P=9O9L#&+1/+'2=##J(,</'-#
$)"#"D%+$/('#4(&#!#%2/'-#$)/2#<+,%"#0/",12#!#7#9O=KL#)(%&2#+4$"&#6/1'/-)$#(&#9NOP#;=6=#+2#$)"#"'1#(4#
$)"#/'$"&<+,#/'#!)/*)#!"#*+'#+16/&"#(%&#!(&X=##M@"#*(%,1#+,2(#:%/,1#(%&#2+'1#*+2$,"#:"$!""'#ENLO#
;=6=#+'1#9NOP#+=6=#$)"#'"5$#6(&'/'-=Q#
!
!
"#$%&$$!'()*+,!-+.$$/0!
C)"#1/2*%22/('#'""12#$(#4(*%2#('#$)"#/1"+2#+'1#2$&+$"-/"2#4(&#+'2!"&/'-#/'<"&2"#$&/-('(6"$&/*#
D%"2$/('2.#+2#)/-),/-)$"1#:0#$)"#/22%"2#+11&"22"1#+:(<"=##Y/2*%22#$)"#*('<"'$/('#4(&#&"2$&/*$/'-#$)"#
(%$;%$#(4#$)"#/'<"&2"#2/'"#4%'*$/('#$(#$)"#/'$"&<+,# " #2 $ sin "1 ( x ) $
#
2
=##C)/2#+,,(!2#$)"#1(6+/'#(4#$)"#
/'<"&2"#2/'"#4%'*$/('#$(#*(<"&#+,,#;(22/:,"#<+,%"2#(4#"#$ "1 # x # 1#+'1#0"$#0/",1#+#%'/D%"#(%$;%$=##
I('2"D%"'$,0.#$)"#+'2!"&#(:$+/'"1#%2/'-#+#*+,*%,+$(&#6+0#'($#:"#$)"#+'2!"&#$)+$#4/$2#$)"#*('$"5$=##
C)"#%'/$#*/&*,"#+'1#$)"#-&+;)#(4#$)"#2/$%+$/('#2"&<"#+2#+/12#4(&#4/-%&/'-#(%$#)(!#$(#/'$"&;&"$#
!
*+,*%,+$"1#<+,%"2#4(&#$)"#/'<"&2"#2/'"#2(#$)+$#&"2%,$2#4/$#$)"#2*"'+&/(=##Z2"#2$%1"'$#!(&X#+'1#2$%1"'$#
!
$)/'X/'-#+:(%$#D%"2$/('2#O#+'1#E#$(#)",;#6+X"#$)"2"#;(/'$2=##J$+&$#!/$)#-&+;)2#+'1[(&#$+:,"2#$(#4/&2$#
4/'1#$)"#+'2!"&2#$(#D%"2$/('2#O#+'1#E.#+'1#$)"'#;%&2%"#$)"#+,-":&+/*#/1"+2#(%$,/'"1#+:(<"=#
#
\4#$)"&"#/2#$/6".#1/2*%22#D%"2$/('#]#%2/'-#$+:,"2#+'1#-&+;)2#$(#6+X"#2"'2"#(4#$)"#2*"'+&/(=##C)/2#/2#
'($#+'#/'<"&2"#$&/-('(6"$&/*#D%"2$/('.#2/'*"#!"#'""1#(',0#"<+,%+$"#$)"#-/<"'#4%'*$/('#+$#;+&$/*%,+&#
$/6"2#$(#4/'1#$)"#,(!"2$#;(/'$#('#$)"#:"+*)#!)"&"#!"#*+'#:%/,1#$)"#*+2$,"=##^(!"<"&.#/$#6+0#:"#
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%&'#$()*!+(*(,-!./,0&)$!1!"
+ .!!! !
"#$%&'%(!)&(%*!+,%!-*%.+#/%!-0110&'!2++*#3)+#0&450&-011%*$#.647,.*%26#8%!9:;!<&=0*+%(!6#$%&'%!
!
!
!"##"$%&'(')(!*'*+,"-*('.*('",*/(0'(1."$.(1*(/.)%&!(*20&%0'*('.*(#%-$'")-(1"'.)%'('.*(%/*()#(0('03&*(
)+(0(4+05.6(
(
!"#$%&'()&*'+,(-&.,(/01((23#$0%04&.3#5(67%5.#0%8(9:;;(
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%&'#$()*!+(*(,-!./,0&)$!1!"
+ .!!! !
"#$%&'%(!)&(%*!+,%!-*%.+#/%!-0110&'!2++*#3)+#0&450&-011%*$#.647,.*%26#8%!9:;!<&=0*+%(!6#$%&'%!
!
!
70
!
%&'()*$+&,*$-./$$
%&'()$
>0=#$?!>.&@%&+'!#&!*#@,+!.&@6%!
+*#@0&01%+*A!
tan A =
!
!
D:!!!E#&(!!"# !!!"# !"# ! !!
!
!
!
%
B%$.66!+,.+!+,%!*#@,+!+*#.&@6%!(%C#&#+#0&!0C!
+,%!+.&@%&+!*.+#0!#'?!
!F;DG!HHH:C6#$8*:$01I=,0+0'IJ0%('#6/.!
!"#$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&'()*+*#$,'(-%./+-,(*+0% !"##$
length of side opposite angle A
!
length of side adjacent to angle A
!
!
F:!!!E#&(!!"# !!!"# !"# ! !!
!
!
!
!
9:!!!E#&(!!"# ! !"# !"# ! !
!
!
!
!
G: E#&(!!"# ! !"# !"# ! !
!
!
!
!
!
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%&'#$()*!+(*(,-!./,0&)$!1!"
+ .!!! !
"#$%&'%(!)&(%*!+,%!-*%.+#/%!-0110&'!2++*#3)+#0&450&-011%*$#.647,.*%26#8%!9:;!<&=0*+%(!6#$%&'%!
Trigonometric Functions 70
!
!
71
!"#$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&'()*+*#$,'(-%./+-,(*+0% !"##$
!
%&'$
>0=#$?!@.+,%1.+#$.6!10(%6#&A!)'#&A!'#&%!.&(!$0'#&%!B)&$+#0&'!
@.&C!*%.646#B%!'#+).+#0&'!')$,!.'!'0)&(!D./%'E!D%.+,%*!=.++%*&'E!.&(!%6%$+*#$.6!$)**%&+'!$.&!3%!
10(%6%(!3C!'#&%!.&(!$0'#&%!B)&$+#0&':!!>,%!+.36%!3%60D!',0D'!+,%!(%=+,!0B!D.+%*!F#&!B%%+G!.+!+,%!
%&(!0B!.!D,.*B!.'!#+!/.*#%'!D#+,!+,%!+#(%'!.+!/.*#0)'!+#1%'!()*#&A!+,%!10*&#&A:!
!"F+#1%G"
#"F(%=+,G!
!
1#(&#A,+!
K:LJ!
H!2:@:!
LH:LJ!
I!2:@:!
J!2:@:!
LI:;K!
LH:LJ!
K!2:@:!
K:LJ!
L;!2:@:!
M:NJ!
&00&!
N:HJ!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
+!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
O%!$.&!)'%!.!+*#A0&01%+*#$!B)&$+#0&!+0!10(%6!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
LH!B+!
P!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
+,%!(.+.:!!7)==0'%!C0)!$,00'%!$0'#&%:!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! ! !"#$ !" ! ! ! !! !!!"!!!!!"!!"#$!!!"!!"#!!"#$!
>,%"$%&'(!)#*"D#66!3%!+,%!(#'+.&$%!B*01!+,%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
./%*.A%!0B!+,%!,#A,%'+!.&(!60D%'+!/.6)%':!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
>,#'!D#66!3%!+,%!./%*.A%!(%=+,!F#+:!!
!!H!B+!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
&00&!
1#(&#A,+!
!!J!2:@:!
M:!!!78%+$,!+,%!6#&%!+,.+!',0D'!+,%!./%*.A%!(%=+,:!
J:!!!Q#&(!+,%!.1=6#+)(%:!!! ! !
!
=%*#0(!B0*!0)*!10(%6!D#66!3%!
!!
!
!!"! ! !"# ! !
N:!!!Q#&(!+,%!=%*#0(:!!!! ! ! !"#!!"#$ ! !!"!!!"!" :!!7#&$%!.!&0*1.6!=%*#0(!B0*!'#&%!#'!HR:!!>,%!&%D!
(%$#1.6:G!
!
!!"!! !
!!
!
!!!F<'%!+,%!&"C0)!$.6$)6.+%(E!(#/#(%!.&(!+)*&!#+!#&+0!.!
!
K:!!!S#A,!+#(%!0$$)**%(!I!,0)*'!.B+%*!1#(&#A,+:!!>,%!B0*1)6.!B0*!+,%!(#'=6.$%1%&+!#'!! ! !!!<'%!,!.&(!
!
'06/%!B0*!-."
T:!!!50D!+,.+!C0)!,./%!C0)*!/.6)%'!B0*!/0",0"-E!.&(!#E!C0)!$.&!=)+!+,%1!#&+0!C0)*!%U).+#0&:!!!!!!!!!!!!!!!!!
!!!!!!!! ! !"#$ !" ! ! ! !!
L;:!!!<'%!C0)*!10(%6!+0!$.6$)6.+%!+,%!(%=+,!.+!T!2:@:!.&(!9!V:@:!
LL:!!!2!30.+!&%%('!.+!6%.'+!L;!B%%+!0B!D.+%*!+0!(0$8!.+!+,%!D,.*B:!!W)*#&A!D,.+!#&+%*/.6!0B!+#1%!#&!+,%!
.B+%*&00&!$.&!#+!'.B%6C!(0$8X!
!
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%&'#$()*!+(*(,-!./,0&)$!1!"
+ .!!! !
"#$%&'%(!)&(%*!+,%!-*%.+#/%!-0110&'!2++*#3)+#0&450&-011%*$#.647,.*%26#8%!9:;!<&=0*+%(!6#$%&'%!
Trigonometric Functions 71
!
!
72
!"#$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&'()*+*#$,'(-%./+-,(*+0% !"##$
!
%&$
>0=#$?!<'#&@!+,%!$.6$)6.+0*!+0!A#&(!.&@6%'!0A!*0+.+#0&!
!"#$%&'($)*+)'+*,&($*-.$/0*,$%&'$1-&/$*2&',$/0#(#$"3-#$*-.$)&"3-#$*(#$4&"3,35#$*-.$
-#6*,35#$3-$,0#$'-3,$)3()+#$,&$73-.$,0#$,/&$*-6+#"$,0*,$*(#$"&+',3&-"$,&$,0#$#8'*,3&-9$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
:*1#$"'(#$!!!"!!"#!$%$!! ! ! ! !"9$$$$$$$$$$$$$!"#$%&'()'$)(*#%&+,#$)-&./&+/*&01&%(-0(1+23&
$"#$&40))&1#*0'/&*,(*&5#$%&'()'$)(*#%&40))&+#6/*06/+&708/&5#$&(&1/7(*08/&(17)/2&&9,(*&0+&./'($+/&*,/&
'()'$)(*#%&0+&:%#7%(66/-&*#&%/+*%0'*&*,/&(17)/&#;&%#*(*0#1&+#&*,(*&*,/&018/%+/&#;&*,/&;$1'*0#1&0+&()+#&(&
;$1'*0#12&&<01'/&*,/&%/=$/+*/-&(1+4/%+&,(8/&.//1&%/+*%0'*/-&*#&:#+0*08/&%#*(*0#1+>&0;&*,/&'()'$)(*#%&708/+&
5#$&(&1/7(*08/&(17)/&#;&%#*(*0#1>&5#$&40))&1//-&*#&;07$%/&#$*&*,/&:#+0*08/&'#*/%601()&(17)/&;#%&*,/&(17)/&
*,(*&5#$%&'()'$)(*#%&,(+&708/1&5#$2&&$
$
!
BC:!!!!"# ! ! ! ! !
!
!
!
BD:!!!!!"# ! ! ! !
!
BH:!!!!"# ! ! ! ! !!!!!!!!! !
B9:!!!!"# ! ! ! ! !
!
!
!"
!
!
BE:!!!!"# ! ! ! !
!"
!"
!
!
!
BF:!!!!"# ! ! !
!
!!
!"
!
BG:!!!!"# ! ! ! !!!!!!
!
!
!
BI:!!!!"# ! ! ! ! !
!
!
!
!
!
?#*/@&& A,/1&5#$&(+B&5#$%&'()'$)(*#%&;#%&*,/&(17)/>&5#$&(%/&&C$1-#017D&*,/&*%07&;$1'*0#12&&E01-017&*,/&
(17)/&0+&;01-017&*,/&018/%+/&*%07&;$1'*0#12&&A,/1&5#$&+//&J;3-.!!!"#!!
!
!
!&&>&5#$&(%/&./017&(+B/-&*#&;01-&
*,/&(17)/&*,(*&6(B/+&0*&*%$/2&&9,/&(1+4/%&4#$)-&./&*,/&+(6/&(+&*,/&(1+4/%&5#$%&'()'$)(*#%&7(8/&5#$&01&
FGH2&&I1#*,/%&1#*(*0#1&*,(*&6/(1+&*,/&018/%+/&+01/&;$1'*0#1&0+&&!"#$%&
!!
!
!
!
!
!
!
!&
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%&'#$()*!+(*(,-!./,0&)$!1!"
+ .!!! !
"#$%&'%(!)&(%*!+,%!-*%.+#/%!-0110&'!2++*#3)+#0&450&-011%*$#.647,.*%26#8%!9:;!<&=0*+%(!6#$%&'%!
Trigonometric Functions 72
!
!