">;?@!AAA:B6#$8*:$01C=,0+0'C8&#D,+')36#1%!
79
!"#$%&''%()%*%+*),-).%
!"#$%$&'(")*+",'&-+-./"0*+$123)*+-*4"5)26"
%
%
%
%
%
%
%
%
%
tan A =
%
/-0*11%.2*.%.2-%34,2.%.34*),1-%5-'4)4.4()%('%.2-%.*),-).%3*.4(%467%
length of side opposite angle A
%
length of side adjacent to angle A
!
#" /-846-%.246%5-'4)4.4()%.(%'4)5%.2-%.*),-).%('%*)9%*),1-%('%3(.*.4():%,48-)%4)%-4.2-3%3*54*)6%(3%
5-,3--6"%%;<=1*4)%>29%9(?3%5-'4)4.4()%46%3-*6()*@1-"%
A" /-846-%.246%5-'4)4.4()%.(%'4)5%.2-%.*),-).%('%*)9%*),1-%('%3(.*.4()%53*>)%4)%6.*)5*35%=(64.4()%
()%.2-%?)4.%04301-"%%;<=1*4)%>29%9(?3%5-'4)4.4()%46%3-*6()*@1-"%
%
%
B-%2*8-%(@6-38-5%.2*.%()%.2-%?)4.%04301-%.2-%8*1?-%('%64)-%*)5%0(64)-%0*)%@-%3-=3-6-).-5%>4.2%.2-%
1-),.2%('%*%14)-%6-,C-)."%
%
$" D)540*.-%()%.2-%'(11(>4),%54*,3*C%>2402%6-,C-).E6%1-),.2%3-=3-6-).6%.2-%8*1?-%('%2-*F!G%*)5%
>2402%3-=3-6-).6%.2-%8*1?-%('%7'2F!G%'(3%.2-%,48-)%*),1-%!"%
%
%
%
%
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%&'#$()*!+(*(,-!./,0&)$!1!"
+ .!!! !
"#$%&'%(!)&(%*!+,%!-*%.+#/%!-0110&'!2++*#3)+#0&450&-011%*$#.647,.*%26#8%!9:;!<&=0*+%(!6#$%&'%!
Trigonometric Functions 79
!
!
80
!"#$#%&'%()'*%(%)&+#%'#,-#+.%."(.%/(+%0#%1#2&+#1%*+%."#%3+&.%/&$/)#%'*%."(.%&.'%)#+,."%$#4$#'#+.'%."#%
5()3#%*2%!"#6!78%%9*+'&1#$%."#%)#+,."%*2% DE %&+%."#%3+&.%/&$/)#%1&(,$(-%0#)*:8%%;*.#%."(.%"$%&'(+1%
"$()'($#%$&,".%.$&(+,)#'8%%<$&.#%(%/*+5&+/&+,%($,3-#+.%#=4)(&+&+,%:">%."#%)#+,."%*2%'#,-#+.%%&'&'%
#?3&5()#+.%.*%."#%5()3#%*2%!"#6!7%2*$%."#%,&5#+%(+,)#%!8%
!
%
%
%
@+%."#%/**$1&+(.#%(=#'%0#)*:%'A#./"%."#%,$(4"%*2% y = tan(" ) %0>%/*+'&1#$&+,%."#%)#+,."%*2%'#,-#+.%
%&'('%!%$*.(.#'%."$*3,"%(+,)#'%2$*-%B%$(1&(+'%.*%C#%$(1&(+'8%%D=4)(&+%(+>%&+.#$#'.&+,%2#(.3$#'%>*3%
+*.&/#%&+%>*3$%,$(4"8%
%
!
!
!
!
!
!
!
!
!
!
%
%
%
%
!
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%&'#$()*!+(*(,-!./,0&)$!1!"
+ .!!! !
"#$%&'%(!)&(%*!+,%!-*%.+#/%!-0110&'!2++*#3)+#0&450&-011%*$#.647,.*%26#8%!9:;!<&=0*+%(!6#$%&'%!
Trigonometric Functions 80
!
!
81
!"#$%&'()*+',+-./')0' y = tan(" ) '1('2)%34&$+4%,'#/$'5$%,#/')0'3$,6$%#'!"'-3'!'+)#-#$3'#/+)*,/'
%$,-#47$'-%,5$3'0+)6'8'+-&4-%3'#)'9:"'+-&4-%3;'
<34%,'()*+'*%4#'24+25$'&4-,+-63'0+)6'#/$'#-3='#$%&'(#)&&*+($,-(%)&(.,/%(0/'1*&>',47$'$"-2#'7-5*$3'0)+'
#/$'0)55)?4%,'#+4,)%)6$#+42'$".+$334)%3@'
!
'
'
#"%
=' '
$ 6&
'
#"%
=' '
$ 4&
'
#"%
=' '
$ 2&
'
A;'' tan
'
!
'
C;'' tan
!
'
F;'' tan
!
!
# 5" %
=' '
$ 6&
'
# 3" %
=' '
$ 4&
'
:;'' tan
D;'' tan
G;'' tan(" ) = ''
'
!
!
'
'
'
!
!
!5(7(623!58(!
I''J'0*%2#4)%'2345(43'25-33404$&'-3'-%'!""#$%&'()!&#40'236
# 7" %
='
$ 6&
B;'' tan
# 11" %
='
$ 6 &
E;'' tan
# 7" %
='
$ 3&
H;'' tan
I''J'0*%2#4)%'2345(43'25-33404$&'-3'-%'*+*&#$%&'()!&#40'236!5(7(23!58(
(
K-3$&')%'#/$3$'&$04%4#4)%3'-%&'()*+'?)+='4%'#/43'6)&*5$>'&$#$+64%$'/)?'#)'25-3340('$-2/')0'#/$'
0)55)?4%,'#+4,)%)6$#+42'0*%2#4)%3;'
•
L/$'0*%2#4)%'9(7(+/,345(?)*5&'1$'25-33404$&'-3'-%'M)&&'0*%2#4)%>'$7$%'0*%2#4)%>'%$4#/$+'-%')&&'
)+'$7$%'0*%2#4)%N;''O47$'$74&$%2$'0)+'()*+'+$3.)%3$;'
•
L/$'0*%2#4)%'9(7(1:+345(?)*5&'1$'25-33404$&'-3'-%'M)&&'0*%2#4)%>'$7$%'0*%2#4)%>'%$4#/$+'-%')&&'
)+'$7$%'0*%2#4)%N;''O47$'$74&$%2$'0)+'()*+'+$3.)%3$;'
•
L/$'0*%2#4)%'9(7(%$,345(?)*5&'1$'25-33404$&'-3'-%'M)&&'0*%2#4)%>'$7$%'0*%2#4)%>'%$4#/$+'-%'
)&&')+'$7$%'0*%2#4)%N;''O47$'$74&$%2$'0)+'()*+'+$3.)%3$;'
'
'
'
'
'
'
'
(
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%&'#$()*!+(*(,-!./,0&)$!1!"
+ .!!! !
"#$%&'%(!)&(%*!+,%!-*%.+#/%!-0110&'!2++*#3)+#0&450&-011%*$#.647,.*%26#8%!9:;!<&=0*+%(!6#$%&'%!
Trigonometric Functions 81
!
!
!"#$%&''%()%*%+*),-).%/%+-*01-2%3(.-4%
!"#$%$&'(")*+",'&-+-./"0*+$123)*+-*4"5)26"
!"#$%&'5%%+1-%6726(4-%('%.184%.*49%84%.(%-:.-);%.1-%;-'8)8.8()%('%.1-%.*),-).%'2(<%.1-%28,1.%.28*),=-%
.28,()(<-.280%2*.8(%;-'8)8.8()> tan(A) =
;-'8)8.8()5% tan(" ) =
length of side opposite angle A
%>.(%*)%*),=-%('%2(.*.8()%
length of side adjacent angle A
y
"%%+1-%,2*61%('%.1-%.*),-).%'7)0.8()%84%(?.*8)-;%?@%2-62-4-).8),%.1-%.*),-).%('%
x
*)%*),=-%('%2(.*.8()%?@%.1-%=-),.1%('%*%=8)-%4-,<-).%2-=*.-;%.(%.1-%7)8.%0820=-A%*);%.2*098),%.1-%=-),.1%
!
('%.1-%=8)-%4-,<-).%*4%.1-%*),=-%('%2(.*.8()%8)02-*4-4%*2(7);%.1-%7)8.%0820=-"%%+1-%.28,()(<-.280%
!
tan(" ) =
8;-).8.@%
sin(" )
%84%*=4(%-:6=(2-;%8)%.-2<4%('%.1-%7)8.%0820=-"%
cos(" )
(%#')*+,-.,#.&)/%0"&5%
!/12/13%B:6=*8)%1(C%.1-%7)8.%0820=-%8)%.1-%0((2;8)*.-%6=*)-%-)*?=-4%.1-%-:.-)48()%('%.28,()(<-.280%
'7)0.8()4%.(%*==%2-*=%)7<?-24A%8).-262-.-;%*4%2*;8*)%<-*472-4%('%*),=-4%.2*D-24-;%0(7).-20=(09C84-%
*2(7);%.1-%7)8.%0820=-"%
/12/14%EFG%H4-%46-08*=%.28*),=-4%.(%;-.-2<8)-%,-(<-.280*==@%.1-%D*=7-4%('%48)-A%0(48)-A%.*),-).%'(2%
5I$A%5IJ%*);%5I!A%*);%74-%.1-%7)8.%0820=-%.(%-:62-44%.1-%D*=7-4%('%48)-A%0(48)-A%*);%.*),-).%'(2%K/:A%
KF:A%*);%LK/:%8)%.-2<4%('%.1-82%D*=7-4%'(2%:A%C1-2-%:%84%*)@%2-*=%)7<?-2"%
/12/16%EFG%H4-%.1-%7)8.%0820=-%.(%-:6=*8)%4@<<-.2@%E(;;%*);%-D-)G%*);%6-28(;808.@%('%.28,()(<-.280%
'7)0.8()4"%
7'8,+'.)*+,-.,#.&9))/1:/16;)/1:/1<;)/1:/1=;)/1:/1>)
?,"-0@)AB@%8')(8,&&C9%
M-<8);%4.7;-).4%.1*.%C-%1*D-%2-;-'8)-;%48)-%*);%0(48)-%'(2%*),=-4%('%2(.*.8()%;2*C)%8)%4.*);*2;%
6(48.8()%?@%748),%.1-%D*=7-4%('%7A%/%*);%1"%%N49%1(C%.1-@%<8,1.%2-;-'8)-%.*),-).%748),%.1-4-%4*<-%
D*=7-4"%%O.7;-).4%41(7=;%)(.-%.1*.%.1-%;-'8)8.8()% tan(" ) =
y
%84%8);-6-);-).%('%.1-%D*=7-%('%18"
x
B:*<8)-%.1-%.C(%7)8.%0820=-%;2*C8),4%8)%P7-4.8()%$%.(,-.1-2%*4%*%0=*44"%%Q)%.1-%;2*C8),4%=*?-=%
4-,<-).%NR%*4%7"9":'2;!<A%4-,<-).%SR%*4%/"9"2-*;!<"*);%4-,<-).%NS%*4%1"9%#"%%Q)%.1-%4-0();%;2*C8),%
!
)(.-%.1*.%!NSR%84%48<8=*2%.(%!NTB%*);%.1*.%.1-%<-*472-%('%4-,<-).%NB%84%#"%%H48),%.184%8)'(2<*.8()%
*49%4.7;-).4%.(%0()48;-2%C1*.%.184%8<6=8-4%*?(7.%.1-%<-*472-%('%4-,<-).%TB"%%U8D-%4.7;-).4%*%
0(76=-%('%<8)7.-4%.(%47,,-4.%.1*.%48)0-%.1-%.28*),=-4%*2-%48<8=*2%.1-@%0*)%C28.-%.1-%62(6(2.8()%
!
!
DE BC
DE y
%(2%
=
= "%%+1-@%41(7=;%2-0(,)8V-%.1*.%.1-%=-),.1%('%4-,<-).%TB%84%;-'8)-;%8)%.1-%
x
AE AC
1
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"#$%&'#$()*!+(*(,-!./,0&)$!1!"
+ .!!! !
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!
!
!"#$%&"'%()"(%&$%)"*$%+$,-.$+%!"#$!%/%%0)"(%-!1%()$%2$.3()%4,%!$3#$.(%56%7$87$!$.(!%()$%*"29$%4,%
!"#$!%%-.%()$%!"#$%&"'%()"(%()$%2$.3()%4,%!$3#$.(%:;%7$87$!$.(!%()$%*"29$%4,%&'($!%%".+%()$%2$.3()%
4,%!$3#$.(%<;%7$87$!$.(!%()$%*"29$%4,%()#$!%/%%=49%#"'%"2!4%&".(%(4%84-.(%49(%()"(%()$%(7-34.4#$(7->%
-+$.(-('% tan(" ) =
sin(" )
%-!%87$!$.(%-.%()-!%+-"37"#/%
cos(" )
?4&%()"(%&$%)"*$%"%&"'%4,%*-!9"22'%7$87$!$.(-.3%()$%#"3.-(9+$%4,%()$%*"29$%4,%!"#$!%*+"!!-3.%
!(9+$.(!%(4%&47@%4.%+$($7#-.-.3%&)"(%()-!%-#82-$!%"A49(%()$%!)"8$%".+%,$"(97$!%4,%()$%37"8)%4,%%
!,+-+!"#$!%/%%:2!4%)"*$%()$#%&47@%4.%()$%7$!(%4,%()$%("!@%A'%9!-.3%()$-7%9.-(%>-7>2$%+-"37"#!/%
%
!"#$%&'()*+,$$(-&%.#/0((
B,%!(9+$.(!%"7$%)"*-.3%"%)"7+%(-#$%!@$(>)-.3%()$%37"8)1%,4>9!%()$-7%"(($.(-4.%4.%!#"22%-.($7*"2!%4,% !/%%
C47%$D"#82$1%&)"(%)"88$.!%(4%()$%2$.3()%4,%!$3#$.(%56%"!%!%-.>7$"!$!%,74#%E%7"+-".!%(4% "2 %7"+-".!F%%
G)"(%)"88$.!%&)$.% " = #2 F%%G)"(%)"88$.!%&)$.%!%-.>7$"!$!%,74#% "2 %(4%!F%%H4&%&492+%'49%+7"&%
":56%4.%()-!%-.($7*"2F%%G)"(%"A49(%.$3"(-*$%".32$!%4,%74("(-4.F%
!
G"(>)%"!%!(9+$.(!%>4#89($%*"29$!%4,%!"#$!%%9!-.3%-.,47#"(-4.%7$>47+$+%4.%()$-7%9.-(%>-7>2$%
!
!
+-"37"#!/%%I(9+$.(!%#"'%.$$+%)$28%!-#82-,'-.3%()$%7"(-4!%,47#$+%A'%
y
/%%:224&%!(9+$.(!%(4%2$"*$%
x
()$!$%7"(-4!%9.!-#82-,-$+%9.(-2%()$%&)42$%>2"!!%+-!>9!!-4.%&)$.%'49%>".%+-!>9!!%!4#$%4,%()$%
"7-()#$(->%-.*42*$+1%)48$,922'%A'%9!-.3%&47@%,74#%!(9+$.(!%&)4%"7$%!9>>$!!,92%"(%!-#82-,'-.3%()$!$%
7"(-4!/%%J44@%,47%!9>)%!(9+$.(!/%
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(".3$.(%,9.>(-4.!/%%G)"(%7$87$!$.("(-4.!%+4%()$'%+7"&%984.%(4%#"@$%()$!$%+$>-!-4.!K%()$%!'##$(7'%
4,%84-.(!%"749.+%()$%9.-(%>-7>2$1%"%37"8)%4,%()$%,9.>(-4.1%47%!4#$%4()$7%&"'!%4,%7$"!4.-.3F%
%
1234.33()56%$'(7$,33/0%
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"7-()#$(->%4,%!-#82-,'-.3%()$!$%7"(-4!/%
0)$%>2"!!-,->"(-4.%4,%!-.$1%>4!-.$%".+%(".3$.(%"!%$*$.%47%4++%,9.>(-4.!%".+%()$%$*-+$.>$%9!$+%
!
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!
!
82
!
&'()*+%,'-+%./0%
&'()*%
>0=#$?!!@#A#(!.&(!&0&4*#A#(!+*.&'B0*1.+#0&'!0B!B)&$+#0&
!"#$#%&'()*+$*,$'$-'.#+($,&+/()*+$)0$1)2#+3$$4.)(#$'$
+#5$#%&'()*+$5)("$("#$1)2#+$(.'+0,*.6'()*+03$$!"#+$
07#(/"$("#$+#5$,&+/()*+$*+$("#$0'6#$1.'-"$'0$("#$
-'.#+($,&+/()*+3$$CB!+,%!B)&$+#0&!,.'!.'D1=+0+%'E!'8%+$,!
+,%1!#&:!
%
!!!!!!!!! ! ! ! ! !
!O;QP!GGG:B6#$8*:$01R=,0+0'R8&#A,+')36#1%!
!"#$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&'()*+*#$,'(-%./+-,(*+0% !"#$%
F%*+#$.6!',#B+?!!(0G&!H!
,0*#I0&+.6!',#B+?!6%B+!J!
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Trigonometric Functions 82
!
!
83
!"#$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&'()*+*#$,'(-%./+-,(*+0% !"#$%
!
9:!!!! ! !!
>%*+#$.6!',#?+@!!&0&%!
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(#6.+#0&@!!9!!
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F.&G%@!
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!
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0 3!!! !
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Trigonometric Functions 83
!
!
84
!"#$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&'()*+*#$,'(-%./+-,(*+0% !"#$%
!
>:!!!?*#.&@6%!A7?!#'!.&!%B)#6.+%*.6!+*#.&@6%:!!!
A7!C!D!!!!!!!!!!!!!!!"!#'!.&!.6+#+)(%!
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6%&@+,!0E!+,%!'#(%'F!A2F!.&(!+,%!!"#$%!6%&@+,!
0E!!"!!:!
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0E!+,%!=0#&+'!.&(!.&@6%'!0E!*0+.+#0&!#&!+,%!)&#+!$#*$6%'!3%60I:!
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Trigonometric Functions 84
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85
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Trigonometric Functions 85
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86
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Trigonometric Functions 86
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