27101574 Mth 133 Trigonometry

5/20/2018 27101574 Mth 133 Trigonometry

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MTH 133 TRIGONOMETRY

NOUN 2

MTH 133: TRIGONOMETRY

COURSE GUIDE

NATIONAL OPEN UNIVERSITY OF NIGERIA


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TABLE OF CONTENT

INTRODUCTION

WHAT YOU WILL LEARN IN THIS COURSE

COURSE AIMS

COURSE OBJECTIVES

WORKING THROUGH THIS COURSE

- COURSE MATERIALS - STUDY UNITS

- SET TEXTBOOKS

- ASSIGNMENT FILE

- PRESENTATION SCHEDULE

ASSESSMENT

- TUTOR-MARKED ASSIGNMENTS AND MARKING SCHEMES

- FINAL EXAMINATION AND GRADING

- COURSE MARKING SCHEME

- COURSE OVERVIEW

SUMMARY

TRIGONOMETRY

COURSE GUIDE

TABLE OF CONTENTS

(2) COURSE CONTENT

(3) INTRODUCTION

NOUN 3


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() WHAT YOU WILL LEARN

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(") ASSESSMENT

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3.1 COURSE MATERIALS

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NOUN 18

MTH 133: TRIGONOMETRY

COURSE DEVELOPMENT

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UNIT 1

TRIGONOMETRIC RATIOS I

TABLE OF CONTENTS

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35151

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INTRODUCTION

OBJECTIVES

TRIGONOMETRIC RATIOS

TRIGONOMETRIC RATIOS OF ACUTE ANGLES

RELATIONSHIPS BETWEEN TRIGONOMETRIC RATIOS

TRIGONOMETRIC RATIOS OF ANY ANGLE5

CONCLUSION

SUMMARY

TUTOR -MARKED ASSIGNMENT 1 AND MARKING SCHEME5

REFERENCES AND OTHER MATERIALS

1.0 INTRODUCTION

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UNIT

TRIGONOMETRIC RATIOS II

TABLE OF CONTENTS

158 INTRODUCTION

258 OBJECTIVES351 TRIGONOMETRIC RATIOS - RECIPROCALS

35151 SECANT/ COSECANT AND COTANGENT OF ACUTE ANGLES

35152 RELATIONSHIP BETWEEN THE TRIGONOMETRIC RATIOS AND

THEIR RECIPROCALS

NOUN 2


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352 USE OF TABLES

35251 USE OF NATURAL TRIGONOMETRIC TABLES

35252 USE OF LOGARITHM OF TRIGONOMETRIC TABLES

58 CONCLUSION!58 SUMMARY

"58 TUTOR-MARKED ASSIGNMENT AND MARKING SCHEME

#58 FURTHER READING AND OTHER RESOURCES

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T:%. +:* *. +:+ ,'.*, %. +:* $*,%7$', ' .% 0 .% %. +:*

$*,%7$', ' ,'.*, 5 F$' +:* ;'4* $+%'. .'/ '6 , .** E?*$,%.* >

(%) %0 +:* '+:*$ $*,%7$',.5 N'9 +$ +:%.5 +:* ;'4* *?7* .*$4*. . &6%0*5

(%%) 4*$% +:+ ,'. .% ,'+ '$ +$%&*5 I. +:%. .6$7$%.%&5T:%. %. +:* ;*6+ ' +:* +$%&''*+$%, $+%'.

N'+* $' +:* .6 ' &*. ' +$%&* &%4%& 18/ +:* ''9%& $*+%'.

, ;* 7$'4*05

F%& 2522

Y'6 .:'60 $*, +:+ % 6%+ 1/

S% (8'- ) ,'. 0

C'. (8 - ) .% '9 *+ 6./ .** +:* +&*+5

T (8 -) BCAC % %&> 252 %5*5

, ,'+

A.' .*, (88 ) ,'.*, 5 T:%. ;$%&. 6. +' +:* ,',6.%' +:+ +:* +&*+ ' ,6+* &* %. *@6 +' +:* ,'+&*+ ' %+. ,'7**+5 I5*5 ,'+ 38 + "8 0 +

18 ,'+ 8 .' .*, 18 ,'.*, 8

NOUN 31


32/131


N'9 &' +:$'6&: +:*.* *?7*.

15 F%0 +:* 46* ' 8 % +:* ''9%&

() .*, ,'.*, 38 (;) .% !8

(,) ,'+ 28 + (0) .*, 8 ,'.*,

S'6+%'>

() ,'.*, 38 .*, (8 - ),'.*, 38 .*, (8 - 38) .*, "8

(;) 1.% !8 ,'.*, !8

(,) ,'+ 28 + (8 - )+ (8 - 28) + #8

(0) .*, 8 ,'.*, (8 - ) ,'.*,(8 - 8) ,'.*, !8

25 I +:* 0%&$/ ' +:* $%&:+ %0 +:* ''%9%&>

() .*,

(;) ,'.*,

(,) ,'+

1!

() .*, :7'+:*6. 1 1#

0,*+ ,'. 1!

(;) ,'.*, 1 ::'+:*6. 1#

.%

'77'.%+* 1!(C) ,'+ 1 0,*+ 1!

+ '77'.%+*

N'9 '4* +' +:* *?+ .+*7/ +:* $*+%'.:%7 ;*+9** +$%&''*+$%,

$+%'. ' '+:*$ &*.5

I+ :. ;** *.+;%.:*0 +:+>

NOUN 32


33/131


(1) +:* .*,+ ' &* %. +:* $*,%7$', ' +:* ,'.%* ' +:* &* %5*5 .*,

1,'.

(2) ,'.*, 1.% 0

(3) ,'+ 1+

I+ +:* *. +:+ 9:+*4*$ 77%*. +' +:* +$%&''*+$%, $+%'. +:*%$

$*,%7$',./ .' +:* ''9%& $* +$6* % +:* %$.+ @60$+ %5*5 O 8(,6+*) +:* $*,%7$',. +$%&''*+$%, $+%'. $* 7'.%+%4*5

S*, -= ,'.*, 0 ,'+

I +:* .*,'0 @60$+ 8 18 (';+6.*) .%,* ' +:* .%* %. 7'.%+%4*' %+. $*,%7$', +:* ,'.*,+ 9% .' ;* 7'.%+%4* % +:* +:%$0 0 '6$+:

@60$+. $*.7*,+%4* ' +:* +&*+ 0 ,'+&*+ '$ 18 2#8 $*7'.%+%4* 0 ,'.%* 0 .*,+ % 2#8 8 3"8 $* 7'.%+%4* $*.7*,+%4*5

S' +:* ''9%& $*+%'.:%7. $* *.+;%.:*015 .*, ,'.*, (8 - )

,'.*, .*, (8 - )

+ ,'+ (8 - )

,'+ + (8 - )

25 .*, (18 - ) %. *&+%4*/ %*. ;*+9** 8 0 18

,'.*, (18 - ) %. 7'.%+%4*

,'+ (18 -) %. *&+%4*5

35 .*, (- 18) %. *&+%4*/ %*. ;*+9** 18 0 2#8,'.*, (- 18) %. *&+%4*

,'+ (- 18) %. 7'.%+%4*

5 .*, (3"8 - ) %. 7'.%+%4*/ %*. ;*+9** 2#8 0 3"8

,'.*, (3"8 -) %. *&+%4*

,'+ (3"8 - ) %. *&+%4*

:4%& .** +:* $*+%'.:%7. ;*+9** +:* +$%&''*+$%, $+%'. 0 +:*%$

$*,%7$',./ *+ 6. '4* ' +' %0 &*. 6.%& +:* +$%&''*+$%, +;*.5

3..1 USE OF TRIGONOMETRIC TABLES

I +:* +$%&''*+$%, +;*. '$ .%*/ ,'.%* 0 +&*+ ' &*. , ;* 6.*0

+' %0 +:* 46*. ' +:*%$ $*,%7$',.5 I +:* '6$ %&6$* +;*. 4%;* '

+:* +;*. '$ .%*/ ,'.%* 0 +&*+ $* 4%;* .' 9:+*4*$ ';+%. %

+:*%$ ,.* .' 77%*. +' +:*%$ $*,%7$',.5

NOUN 33


34/131


T:* *?,+ 46*. ' +:* +$%&''*+$%, $+%'. ';+%*0 6.%& +:* 6%+ ,%$,*

'+ ;* ,,6$+* 06* +' *.6$**+ *$$'$.5 S' +' ';+% +:* *?,+ 46*.

' +:* +$%&''*+$%, $+%'./ '6 6.* +:* '6$ %&6$* +;*. '$ ,,6+'$.5

T:* +;*. +' ;* 6.*0 :*$* $* *?+$,+ ' +:* N+6$ .%* 0 ,'.%* '

.**,+*0 &*. ;*+9** 1880 8+ +:* %+*$4 ' "1'$ 8518 5 T:* 6+$%&''*+$%, +;*. 9% ;* .677%*0 + +:* *0 ($* +;*. '$ '& .%*/ '&

,'. 0 '& +)

N'+* +:+ +:* 0%*$*,* ,'6 9. + +:* *?+$** $%&:+ - :0 ,'$*$ ' +:* +;*

%. '%++*0

E?+$,+. $' +6$ ,'.%* '$ ,'. ? ( WAEC/ '6$ %&6$* +;*)

8 " 12 1 2 38 3" 2 !

X 858 851 852 -"-3-8 85! 85" 85# 85 85

18 85 ! 2 3 3" 33 2 2" 23 28

28 8553# 31 3! 3# 3#3 3"# 3"1 3! 3 32

38 85""8 "!2 "3 "3 "3! "1" !8# ! !8 !1

8 85#""8 #"" #"3 #"2# #"1! #"8 #!3 #!1 #!#8 #!!

NOUN 3

8 " 12 1/ 2 38 3" 2 !

X 858 851 852 853 85 85! 85" 85# 85 85

28 85328 8533# 853!3 853" 853" 853!82 853!1 853!3! 853!!1 3!"#

38 85!888 85!81! !838 !8! !8"8 !8#! !88 !18! !128 !13!

8 85"2 "1 "!! "" "1 " "!8 "!21 "!3 "!#

!8 85#""8 #"#2 #"3 #" #8! ##1" ##2# ##3 ## ##"8

"8 85""8 "" "# "" "! #8 #12 #21 #2 #3

#8 853# 83 8 1! 21 2" 32 32

8 85 !1 ! !# "8 "3 "" " #1 #

85 85 85 85 85 158888 158888 158888 158888 15888


35/131


!8 85"2 "1 "81 "3 "3# "3"1 "3# "33 "328 "38#

"8 85!888 ! #8 !! 3 2 8 # "3

#8 85328 38 33# 33#1 33!! 333 3322 338! 32 32#2

8 851#3" 1#1 1#82 1"! 1"" 1"!8 1"33 1"1" 1! 1!3

8581#! 81!# 818 81222 818! 88# 88#8 88!2 883! 881#

A&% +:* 0%*$*,* ,'6 %. '%++*05

E?7*>

F%0 +:* 46* ' +:* ''9%& &*.>

(%) .% 28 5 "8 (%%) ,'. 388121 (%%%) .% #881 (%4) ,'. 85 #8

S'6+%'.>

(I) F$' +:* .%* +;* +' %0 .% 28 5 "8''< + +:* *+ :0 ,'6 $


36/131


S% (2881!1) 853"1

Y'6 .** +:+ *%+:*$ 9 +:* 46* ' .% (28 1!1) %. 85 3"1

N'+* +:+ +:* 46*. ' +:* (28 1!1)%. +:* .* % +:* +9' *+:'0. ;'4* ;6+ %

'.+ ,.*./ +:* 46*. $* '+ +:*$* $* .%&:+ 0%*$*,*. + +%*.5

(%%) F$' +:* ,'.%* +;*

,'. (!8 1#1) %. *$*$ ,'. !8 1

,'. (!8 1) 85 "3

76. +:* 0%5 F'$+* 2 85"38

,'. (!88 1#1) 85"38

OR

,'. (!8121) 85 "81

%6. +:* 0%5 F'$ !

1

-12,'. (!8 1#1) 85 "3

O;.*$4* +:+ +:* 0%*$*,* 9. 00*0 +' +:* %$.+ *+:'0 %. ,'. !8 110

.6;+$,+*0 $' +:* .*,'0 *+:'0 %5*5 ,'. !8 1215 T:%. %. ;*,6.* +:* &*

%,$*.*./ +:* 46* $*06,*. % ,'.%*5 Y'6 , :4* ,$%+%, ''< + +:*

+;*. '$ ,'.%*5 I+ %. &''0 +' '+* +:+ +:* 46*. ' .%* %,$*.*. '$ O +'

1 9:%* +:* 46*. ' ,'.%* 0*,$*.*. $' 1 +' O5

T:* .* *+:'0. . 6.*0 % %0%& +:* +&*+ ' &*. $' +:*%$

+&*+. +;*.5

F'$ &*. &$*+*$ +: 8/ +:* .* +;*. $* 6.*0 % %0%& +:*%$

+$%&''*+$%, $+%'. ;6+ %$.+/ '6 0*+*$%* +:* @60$+ 0 .%& ' +:*

&* 0 +$*+ ,,'$0%&5

E?7*.>

F%0 (1) .% 128 (2) .% ( - 38)

(3) ,'. ( - 18 ) () ,'. 2"8

S'6+%'.>

(1) S% 128 %. % +:* .*,'0 @60$+ 0 .%* %. 7'.%+%4* .% (18 - 1288 .%

"8 0 .%,* .% "8 %. 7'.%+%4*/ $' +:* .%* +;*5

S% 128 .% "8 85""8

(2) S% (-38) %. % +:* '6$+: @60$+/ 9:*$* +:* .%* %. *&+%4*5

NOUN 3"


37/131


( - 38) - .%* (3"8 - 38) - .% 338 - .% 38 $' +:* .%* +;* .%38 85!888

- .% 388 - 85!888

(3) C'. (- 18) %*. % +:* '6$+: @60$+/ 9:*$* ,'.%* %. 7'.%+%4*5

,'. ( -18) ,'. (3"8 - 18) ,'. 3!8 ,'. 18

,'. ( -18) +:* ,'.%* +;* %. 85

,'. (- 18) 85() C'. (2"8) %. % +:* 3$0 @60$+ 9:*$* ,'.%* %. *&+%4* ,'. (2"8 - 18)

,'. 85 F$' +:* ,'.%* +;* ,'. 8 851#3" 0 .%,* ,'.%* %. *&+%4*

% +:* 3$0 @60$+ ,'. 2"88 - ,'. 88 - 851#3"

3.3. USE OF LOGARITHMS OF TRIGONOMETRIC FUNCTIONS

A +%*. '6 %&:+ ;* ,*0 9%+: 7$';*. 9:%,: $*@6%$* 6+%7%,+%' 0

0%$*,+%' % .'4%& +$%&*.5 H*$* +:* 6.* ' +;*. ' +$%&''*+$%,6,+%'. ;*,'*. +%* ,'.6%& 0 **$& .77%&5 I+ %. ;*.+ + +:%. .+&*

+' 6.* +:* +;*. ' +:* '&$%+:. ' +$%&''*+$%, 6,+%'. 0%$*,+5

E?7*.>

F%0 (1) '& ,'. 28 "15

S'6+%'>

T:* 6.* ' +:* +;*. ' ,'.%* 9% '9 '6 +' (1) %0 ,'. 28 " 1$' +:*+;* 5

(%%) %0 +:%. 46* $' +:* ,'' '&$%+: +;*

%5*5 ,'. 5 (28 "15) 8531 ($' N+6$ ,'.%*)

T:* '& 8531 5#2# ($' ,'' '&$%+:)

B6+ 6.%& +:* '&$%+: +;* ' ,'.%* &' .+$%&:+ 0 %0 '& 28 "15

'& ,'. (28 "1) 5 #2#

H*$* '6 , .** +:+ 77%& +:* '& ,'. +;* %. *.%*$ 0 .+*$5

(%%) L'& .% (2 13

1

)L'& .% (2 131) 5 "12#/

P6. 0%*$*,* '$ 11 858882 (,'+ $' +:* 0%*$*,* +;* + +:* $%&:+

:0 *?+$** ,'6)

'& .% 2 131 5 "82

(%%%) '& + 8 1#1

$' +:* '& +&*+ +;*

NOUN 3#


38/131


'& + 8 1#1 5 2"

76. +:* 0%5 F'$ !1

'& + 8 1#1 5 2##

A+*$+%4*/ '6 , ''< '$ +:* '&$%+:>

'& + 8 11 5 2

%6. +:* 0*5 '$ 11 - 2'& + 8 11 5 22

T:* +9' $*.6+. % +:%. ,.* $* '+ +:* .*5 T:* .*,'0 $*.6+ %. 7$**$;*

;*,6.* +:* .*$ +:* 0%*$*,* +:* '$* ,,6$+* +:* 46* ' +:* &*

;*%& .'$+ '$5

I +:* &*. $* % $0%6. ,'4*$+ +' 0*&$**.5

E?*$,%.* 252

U.%& +:* '6$ %&6$* +;* '$ ,,6+'$

15 %0 +:* 46* ' *,: ' +:* ''9%&

5 .% 32 1# .5 85!31

;5 .% 12"85388 .5 85332

,5 .% 38 11 .5 - 85332

05 ,'. 3!'5 #' . 85121

*5 ,'. 13#'1"1 . -85#3"

5 ,'. (-8') . 85#""8

&5 .% (-8') . -85"2

:5 + 128'

. -15#321%5 + 2"!' . 115381

5 + 12'"1 . 8522""

25 F%0 +:* @60$+ ' +:* ''9%& &*. 0 0*+*$%* 9:*+:*$ +:*

+$%&''*+$%, $+%'. ($*,%7$',.) $* 7'.%+%4* '$ *&+%4*5

() 188' (;) 118' (,) 123'

(0) 2' (*) 28' () 231'

(&) 2"' (:) 312' (%) !1' () 1'5

S'6+%'.>

5 20/ ' .%* 0 ,'.%* 7'.%+%4*

;5 20/ ' .% 0 ,'.*, 4*

,5 20/ ' .% 0 ,'.*, 4*

05 1.+ +$%& $+%'. 7'.%+%4*

*5 1.+/ +$%&5 R+%'. 7'.%+%4*

NOUN 3


39/131


5 3$0/ ' + 0 ,'+ 7'.%+%4*

&5 +:/ ,'. 0 .*, 7'.%+%4*5

:5 3$0 ' + 0 ,'+ 7'.%+%4*

%5 3$0/ ' + 0 ,'+&*+ 7'.%+%4*5

$.0 CONCLUSION

I 6%+ 1 0 2/ '6 :4* *$+/ +:* 0*%%+%' ' +:* +$%&''*+$%, $+%'. 0

+:*%$ $*,%7$',./ 0 :'9 +' %0 +:* +$%&''*+$%, $+%'. ' &%4* &*

0 +:* 6.* ' +$%&''*+$%, +;*. % %0%& &*.5 Y'6 .:'60 :%4* .'

*$+ +:+ +:* 46* ' &* 0*7*0. ' +:* ;.%, &* 0 %+. .%&

0*7*0. ' +:* @60$+ % 9:%,: %+ %. '605 H'9*4*$/ '6 **0 ;* 9$*

+:+ +:* '.+ ,'' 6.*0 +$%&''*+$%, $+%'. $* +:* .%* ,'.%* 0

+&*+ 0 +:* ;.%, &* 8 %*. ;*+9** O 0 3"8 %5*5 8 8 3"85

".0 SUMMARY

I +:*.* +9' 6%+. '6 :4* .** +:+ +:* +$%&''*+$%, $+%'. 0 +:*%$

$*,%7$',. 9%+: $*.7*,+ +' $%&:+ &*0 +$%&* %.

.% '77'.%+* :7'+:*6.

,'. 0,*+:7'+:*6.

+ '77'.%+*0,*+

T:* ,$' SOH CAH TOA *%&

S .%*/ O '77'.%+* '4*$/ H :7'+*6.*

C ,'.%*/ A 0,*+ '4*$/ H :7'+*6.*

T +&*+/ ' '77'.%+* '4*$/ A 0,*+

C ;* 6.*0 +' $**;*$ +:* +$%&''*+$%, $+%'. +:*%$ $*,%7$',. $*

';+%*0 $' +:*.*5

Y'6 :4* .' *$+ +:+>

(%) +:* .%* '$ ,'.%* '$ +&*+ ' ,6+* &* *@6. +:* ,'.%* '$ .%* '$

,'+&*+ ' %+. ,'7**+$ &*5

S% ,'. (8 - ) .% (8 ) ,'.

C'. .% (8 -8 ,'. (8 ) -.%

T ,'+ (8 - ) + (8 ) - ,'+

T:%. *. +:+ '6 , 6.* +:* .%* +;* %0 +:* ,'.%*

NOUN 3


40/131


' &*. $' 8 +' 8 + +:* .* %+*$4 ' "1 '$ 8 51

(%%) +:* +;*. ' +$%&''*+$%, 6,+%'. , .' ;* 6.*0 % %0%& +:*

$+%'. ' &%4* &*. ; ;*$%& % %0 +:* ''9%& 9:*$*8 %.

,6+* '$ ';+6.*5

(%%%) .% (18 - ) - .% .% (- 18) - .%

,'. (18 - ) -,'. ,'. (- 18) -.%

+ (18 - ) -+ + (- 18) +

(%4) .% (18 ) - .%

,'. (18 ) - ,'.

+ (18 ) +

(4) .% (3"8 - ) - .%

,'. (3"8 - ) ,'.

+ (3"8 - ) -+

I 6.%& +:* +;* .'*+%*. &*. ;* *?7$*..*0 % $0%./ %$.+ ,'4*$+

+:* &*. % $0%. +' 0*&$**. ;*'$* %0%& +:* +$%&''*+$%, $+%'. ' +:*

&%4* &*. '$ ,'4*$+ $' 0*&$**. +' $0%. ;*'$* %0%& +:*

+$%&''*+$%, $+%'./ % %+ %. % $0%.

8.0 TUTOR MARKED ASSIGNMENT

I +:* 0%&$ ;*'9/ %0 +:* +$%&''*+$%, $+%'. %0%,+*05

D

() .% / ,'. / .*, / ,'+ (;) ,'. / + / ,'.*, / .%

(,) + / ,'. / .*, ,'.*, .%

(0) .% ,'. + / ,'+ .*, .*, ,'.*,

NOUN 8

A


41/131


25 E?7$*.. +:* ''9%& %+*$. ' +:* +$%&''*+$%, $+%'. '

() %5 C'. (8 ) %%5 S% (8 )

(;) %5 C'.*, (8 ) %%5 S*, (8 )

(,) %5 C'. (8 ) %%5 S*, (18 )

(0) %5 S% (3"8 ) %%5 T (3"8 )

35 F%0 +:* ;.%, &*. ' +:* ''9%& 0 +:*%$ $*.7*,+%4* @60$+.5

() 128' (;) -38' (,) -228'

(0) 1' (*) 12!' () 21'

5 6.* +$%&''*+$%, +;*. +' %0 +:* 46* ' +:* ''9%&>

() .% 11# (;) ,'. 1151 (,) + 2#!

(0) .% 28 5 #( *) ,'. 121

!5 U.* +:* '&$%+: +;* '$ +$%&5 F6,+%'. +' %0 +:* 46* ' +:* ''9%&5

() '& ,'. 3 1#1 (;) '& .% 23 2!1 (,) '& + 11 "1

UNIT 3

INVERSE TRIGONOMETRIC FUNCTIONS OR

CIRCULAR FUNCTIONS

TABLE OF CONTENT

158 INTRONDUCTION

258 OBJECTIVES

351 INVERSE TRIGONOMETRIC FUNCTIONS

3515 DEFINITION AND NOTATION OF INVERSE TRIGONOMETRIC

FUNCTIONS OR CIRCULAR FUNCTIONS

351525 INVERSE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

351535 PRINCIPAL VALUES OF INVERSE TRIGONOMETRIC FUNCTIONS5

3525 TRIGONOMETRIC RATIOS OF COMMON ANGLES

NOUN 1


42/131


352515 TRIGONOMETRIC RATIOS OF 38 AND "8

352525 TRIGONOMETRIC RATIOS OF !

352535 TRIGONOMETRIC RATIOS OF 8/ 8 AND MULTIPLES OF 8

58 CONCLUSION !58 SUMMARY

"58 TUTOR MARKED ASSIGMENTS

#58 FUTHER READINGS AND OTHER RESOURCES5

1.0 INTRODUCTION

V*$ '+* '6 .** $*+%'. %


43/131


O$ + 5

T:* ;'4* $* *?7* 9:* +:* 46*. ' %.


44/131


+:%. @6*.+%' , .' ;* .++*0 +:6.> F%0 %

() .% 851#8 (;) ,'. 852! (C) + 151###

S'6+%'.>

1() F$' +:* .%* +;* (N+6$ .%* +;*) +:$'6&: +:* ;'0 +:* 46*851#8 ('$ 46* ,'.* +' %+) %. ',+*05 T:* 46* %. 851##1 '60 60*$ 18

125 T:* 0%*$*,* ;*+9** 851#8 0 851##11 %. 5 T:%. %. '60 60*$ 3 %

+:* 0%*$*,* ,'65 S' % +;6$ '$

55 851##1 18 121

76. 0%*$*,* '& - 31

851#8 188 1!

+:* &* 9:'.* .%* %. 851#8 %. 18 1!1

A+*$+%4* 851# , ;* ',+*0 60*$ 18 1 0 0%*$*,* ;*+9**

851# 0 851#8 %. ;6+ ,'+ ;* '60 % +:* 0%*$*,* ,'6 .'

,:''.* +:* 6;*$ *$*.+ +' %5*5 '60 60*$ 31

851# 18 11

76. 0%*$*,* '$ 31

851## 18 1!1

55 +:* &* 9:'.* .%* %. 77$'?%+* *@6 +' 851#8 (851##) %5* 18 1!1

18 1!1

;) C'. 852!

$' +:* ,'.%* +;*/ +:* 46* 852! %. '60 60*$ #! 81

+:* &* 9:'.* ,'.%* %. 852! %. #! 55 #!

,) + 15 1###/$' +:* +6$ +&*+ +;* +:* 46* ,'.*.+ +' 151### %. 151#!8 '60

60*$ 3"15 T:* 0%*$*,* ;*+9** +:* +9' 46*. %. 2# 9:%,: %. '60

60*$ 1

I +:* ;'4* *?7*. &*. $* ,6+* &*.5 T:* %4*$.* +$%&''*+$%,

6,+%'. , ;* *?+*0*0 +' &*5

3.1. INVERSE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE.

NOUN

55 851#!8 3"1

76. +:* 0%*$*,* '$ 2# 1

851### 81


45/131


T:* %4*$.* +$%&''*+$%, 6,+%'. :*$* $* *?+*0*0 +' %,60* 46*. '

&%4* &*. ;*+9** 8 0 3"8 0 ;*'0

E?7* 2>

F%0 +:* 46* ' 8 ;*+9** 8 0 3"8 % +:* ''9%&>() .% 85" (;) ,'. - 85"#2 (,) + 852"

S'6+%'.>

() .% 85" >= .%-(85")

55 "3 11 S%,* .% %. 7'.%+%4* +:* +:* &* 6.+ *%+:*$ ;* %+:* 1.+ '$ 20 @60$+ +:6.

F%& 3518

I +:* %$.+ @60$+ "3'110 % +:* 20@60$+

18 "3

'

1

1

11"

'

1

1

5

(;) C'. 85"#2

F$' +:* ,'.%* +;*. ,'. -1 85"#2 #'181;6+ ,'.%* %. *&+%4*/

+:*$*'$* %*. *%+:*$ % +:* 20'$ 3$0@60$+5

NOUN !


46/131


F%& > 3511

I +:* 20 Z@60$+

18 - #8

18 132 !8QI +:* 3$0 @60$+

18 # 181 22#8181

(,) + - 852"/ :*$* 1"8"Z ;6+ .%,* + %. *&+%4*/ %*. *%+:*$ % +:*20 '$ +: @60$+.5

F%& > 3512

I +:* 20 @60$+

18 - 1" "1 1"38!1

% +:* +: @60$+ 3"8 - 1" "1

33 !1

N'+* $' +:* 7$*4%'6. 6%+. +:*$* $* .*4*$ 46*. ' 9%+: +:* .* 46*;6+ % 0%*$*+ @60$+.5 F'$ *?7* .% 38 .% 1!8 .% 38 .%

#!8 *+,/ :*,* +:* %4*$.* +$%&''*+$%, 6,+%'. $* 46*0

NOUN "


47/131


*?7$*..%'.5 T:%. * +:+ '* 46* ' %. $*+*0 +' %%%+* 6;*$ '

46*. ' +:* 6,+%'5 H*,* +' ';+% 7'..%;* &*. ' &%4*+$%&''*+$%, $+%' *%+:*$ 00 '$ .6;+$,+ (3"8 K ) 9:*$* K %. %+*&*$-

7'.%+%4*/ *&+%4* '$ *$'

E?7* 3

F%0 7'..%;* &*. ' 8 % *?7* 2()/ (;) 0 (,) ;'4*5

() &%4* .% 85" 0 "3 110 11" 11

T:*.* +9' VALUES OF ARE THE BASIC ANGLES

ALL POSSIBLE ANGLES OF ARE"3811[ (3 "8


48/131


(%%) I ,'. / +:* ,'.-1 / %. +:* %4*$.* ,'.%* ' 5 0 +:*

7$%,%7 46* ' +:* %4*$.* ' ,'.%* %. +:* 46* ' % +:* $&*

8 +' (18)5 T:%. %. +:* .* '$ $, ,'+ / 0 $, .*,

E?7*/ % ,'.-112 !/ +:* $, .*, +:* 7$%,%7 46*

C'.-1(- 12) -(-!) +:* 7$%,%7 46* %. ,'.-12) -(13!)

(%%%) T:* 7$%,%7 46* ' +:* %4*$.* ' +&*+ %. +:* 46* ' % +:*

$&* -2 (- 8) +' 2 (- 8)5 T:%. %. +:* .* '$ $, ,'.*, 5

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T:* 7$%,%7 46* '

() T-1(-1) - (!)5 (;) C'+-13 " (38)

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T:* $*+%'.:%7 ;*+9** +:* 46*. ' %4*$.* 6,+%' 0 %+. 7$%,%746* %. &%4* ; +:* '$6* ;*'9 (V&'0. 3"")5

(%) A$, .% ? < (-1)

() A$, .%] < (-1)


49/131


(1) 9$%+* 0'9 +:* 46*. '

() .%-(- 12) (;) ,'.-(-1) (,) +-(-1)

A.> ()-211118/ 338 (;) 18 (,) 13!/ 31!

(2) U.* +;*. +' *46+*>

() +- 2 (;) ,'.-1 (1) (,) .%- (3!)

A.> () "3 2" (;) 2" (,) 3" 2"(3) F%0 +:* 46* ' +:* ''9%& &*.>

() .%-1(85#!8) (;) ,'.-1(8521) (,) +- (25!#)

A.> () 8 (;) 212 (,) " "

5 F%0 +:* &*. ;*+9** 8'0 3"8'5

() .% - 85!128 (;) + '5!!" (,) ,'. - 8"8A.> () 218 / 32 12 (;) 3 2/ 223 2 (,) 12# 38/ 238

!3

!5 F%0 7'..%;* &*. % @6*.+%' ()

A.> () 218 [ (3"8 A$, ,'+3 < $, ,'+3 9:*$* < %. %+*&*$ < "

'$ < 8/ A$, ,'+3 " 38 (&*. % $0%.)

< 1/ A$, ,'+3 " " 218

< -/ A$, ,'+3 " -!" -1!8 *+,5

3. TRIGONOMETRIC RATIOS OF COMMON ANGLES

T:* &*. 8/ 38/ !/ "8/ 8 $* ,*0 ,'' &*. ;*,6.* +:* $*

$*@6*+ 6.*0 % +:*+%,. 0 *,:%,. % 7:.%,.5

A+:'6&: +:* +$%&''*+$%, $+%'. ' ,'' &*. 8/ 38/ !/ "8/ 8/

(0 6+%7*. ' 8 67 +' 3"8 ) , ;* '60 $' +:* +$%&''*+$%, +;*./

+:* , ;* *.% 0*+*$%*0 0 $* 9%0* 6.*0 % +$%&''*+$%, 7$';*.5

3... THE ANGLE OF 309 AND 809

C'.%0*$ *@6%+*$ +$%&* ABC ' .%0*. 2,5 A +%+60* AD (.** %&352) A

NOUN


50/131


F%&> 352

A +%+60* AD (.** %& 352)

B%.*,+. BAC .' +:+ BAD CAD 38

ABC ACB "8

; P+:&'$. +:*'$* AD 3 6%+.5

H*,*/ +:* 46* ' +:* +$%&''*+$%, $+%'. ' "8 0 38 $*

S% "8 32 0 .% 38 12

C'. "8 12 0 ,'. 38 32

T "8 3 0 + 38 I3

C'+ "8 13 0 ,'+ 38 3

S*, "8 2 0 .** 38'-23

C'.*, "8 230 ,'.*, 38 2

3... THE ANGLE $"9

C'.%0*$ $%&:+-&*0 %.'.,**. +$%&* ABC 9%+: AB BC 1 6%+/

B8 0AC!

AC 2 6%+. (P+:&'$. +:*'$*)

NOUN !8


51/131


F%& 3521

H*,* +:* +$%&''*+$%, $+%'. ' ! $* .% ! 12

C'. ! 12T ! 1

C'+ ! 1

S*, !2

C'.*, ! 2

3..3. ANGLES 09 AND 09

I+ %. 0%%,6+ % 7$,+%, 7$';*. +' %0 &*. 8 0 8 % $%&:+ -

&*0 +$%&* . ,6+* &*. ;6+ 9%+: *?+*0*0 +$%&''*+$%, 6,+%'./

+:*.* &*. $* ,'.%0*$*05

(.** %& 3522 ;*'9)

U.%& 6%+ ,%$,* *+ P1(?/ ) ;* 7'%+ ' +:* ,%$,*5 I P/ %. $'++*0

;'6+ 8 % +:* +% ,',


52/131


I PIA

S% P1A ;6+ P18 1 6%+ (6%+ $0%6.)P18

S% P1A ,''$0%+* ' P1 7$'*,+%' ' OP/ ' +:* Y-?%.

,'. OA OA OA ;6+ OA BP1OP1 1

55 ,'. BP1 ? ,''$0%+* ' P1 7$'*,+%' ' 8 P1' +:* X-?%.

T:6. % P %. 7'%+ ' ,%$,* 9%+: ,*+*$ 8 0 6%+ $0%6. 0 OP

S% ,''$0%+* '$ +:* 7$'*,+%' ' OP ' +:* -?%. 0

,'. / ? ,''$0%+* '$ +:* 7$'*,+%' ' OP ' +:* ?-?%.T:6. '$ &*. 8 0 8

S% 8 ,''$0%+* 1

,'. 8 ? ,''$0%+*

8 (8 :. ' 7$'*,+%' ' +:* ? ?%.)

+ 8 .% 88 1 (%%%+)

,'.88 8

F%&> 35253

S%%$ '$ 8

S% 8 ,''$0%+*

8(8 :. ' 7$'*,+%' ' -%?.)

,'. 8 ? ,''$0%+*

NOUN !2


53/131


1 (8 %*. ' +:* ?-?%.

+ 8 .% 88 8

,'.8 1 8

A+*$+%4*

I $%&:+-&*0 +$%&* ABC/ 9%+: A 8 0 C 9:%,: %. 4*$ .

F%&> 35255

T:* $+%'. ' $*>

S% ABBC

C'. ACBC

+ ABAC

.'

.% AC

BC

+ AC

AB

W:* &*+. .*$ 0 .*$/ R ;*,'*. $&*$ 0 $&*$/ +:*.* $**?7$*..*0 +:6. .

+*0. +' O %5*5 ---- 8 +*0. +' 8 %5*5 ---- 8

B--=A 0 BC--=AC . AB--= 8

S% 8AB --= 8 8 8

BC AC AC

C'. 8 AC -= AC 1

BC ACT 8 AB -= 8 8

AC AC

S% 8 AB --= AC 1

AC AC

C'. 8 AB 8 8

BC AC

T 8 AC -= AC

NOUN !3


54/131


AB 8

O$ .%,* 8 0 8 $* ,'7**+$ &*. +:*

S% ,'. (8 - 8) ,'. 8 8C'. 8 .% (8 -8) .% 8 1

T 88 .% 8 1

,'. 8 8

H*$* %. +:* .6$ ' +:* ,'' +$%&''*+$%, $+%'.5 T:* +$%&''*+$%,

$+%'. ' +:*.* .7*,% &*. 0 +:+ ' 6+%7*. ' 8 $* 7$*.*+*0 % +:*

+;* 1 ;*'95

T;* 1> T$%&''*+$%, R+%'. ' .7*,% &*.

A&* S% A C'. A T A C'+ A S*, A C'.*,

A' A

8

'

8 1 8 1 38' ] 32 13 3 23 2

!' 13 12 1 1 2 2

"8' 32 ] 3 13 2 23

8' 1 8 8 1

18' 8 -1 8 -1

2#8' -1 8 8 -1

3"8' 8 1 8 1

E?7*>

W%+:'6+ 6.%& +;*.,,6+'$ %0 +:* 46* ' +:* ''9%&>

(1) (%) CO. 8' 1 (%%) .% "8 (%%%) 2 - 3 1

,'."8 .%38 +2"8

(2) % 388 *46+* .%2 +2 ? ,'.

1-+ ? ,'.2S'6+%'.>

15 (%) ,'. 8 1 $' ;'4* +;*/ ,'. 8 8

55 C'.8 1 8 1 1

55 ,'. 8 1 1

(%%) S .% "88 + "8 0 + "8 3

,'. "8

OR

S% "8 32 0 ,'. "8 12

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NOUN !


55/131


.% "8' 3 ,'. "8'

(%%%) 2 3 1 S6;.+%+6+%& +:* 46*./

.%38 +2"8

2 - 3 1 2 ? 2 - -3 1

] (3)2 3 1 1

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1-+2? ,'.

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,'.38

? ,'. 38

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38'

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%+ +:* ;*,'*.

A+*$+%4* .6;.+%+6+*0 '$ + 38 1/13

1 - ^

32 312

3

32 3 ? 12 3

32 3

NOUN !!


56/131


E?*$,%.* 352

S%7% +:* ''9%& 9%+:'6+ 6.%& +;*. '$ ,,6+'$.

15 () .%3 338 ? +2 28

C'.38A. -23

(;) 3 .%2 "8 +2"8'

2 ,'.+ "8 A.

25 I "8/ ,,6+*/ 9%+:'6+ +;* '$ ,,6+'$

() .% ,'. A. 2(3 1

1,'.2 !

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35 () (.% 13! ,'. 31!)2 A. 2

(;) + 288 + 31! A. 25

1 +238 1 +2"8

5 I .% A 3! 0 .% B !135 9:*$* A 0 B $* ,6+*/ %0 9%+:'6+

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(;) ,'. A ,'. B .% A .% B A. 33"!

(,) + A - + B A. 1"331 (+ A)(+B)

!5 I A %. % +:* '6$+: @60$+ 0 ,'. A !13 %0 +:* 46* '

13 .% A ! .*, A 9%+:'6+ 6.%& +;*.

!+A ",'.*,A A. -23#

$.0 CONCLUSION

I +:%. 6%+/ '6 :4* *$+ +:* %4*$.* +$%&''*+$%, 6,+%'. '$ ,%$,6$

6,+%'./ +:*%$ 0*%%+%'. '$ *%&. 0 '++%'./ '6 :4* .' *$+ +'

%0 +:* %4*$.* +$%&''*+$%, 6,+%'. $' +$%&''*+$%, +;*./ +:*

7$%,%7 46* ' %4*$.* +$%&''*+$%, &*./ +:* $*+%' ;*+9** %4*$.*

+$%&''*+$%, 6,+%'. 0 +:*%$ 7$%,%7 46*. 0 .' +:* +$%&''*+$%,

$+%'. ' ,'' &*. - :'9 +:* $* 0*$%4*0 0 :'9 +' %0 +:*%$ $+%'.

9%+: '6+ 6.%& +;*.5

".0 SUMMARY

NOUN !"


57/131


I +:%. 6%+/ '6 :4* *$+ +:+ +:* %4*$.* ' +$%&''*+$%, 5 $+%'. %. +:*

&* 9:'.* +$%&''*+$%, $+%'./ %. &%4*5 A0 +:*.* 46*. , ;* '60 %

+:* ;'0 ' +:* +$%&''*+$%, $+%' +;* $' 9:*$* +:* &*. $* $*0 '5

Y'6 :4* .' *$+ +:+ +' %0 7'..%;* &*. ' &%4* 7$';* %$.+

%0 +:* ;.%, &*. +:* 00 '$ .6;+$,+ (3"8


58/131


8.0 TUTOR-MARKED ASSIGNMENTS

15 9$%+* &* % +:* %$.+ @60$+ 9:'.* +&*+ %.

() 851" (;) 15" (,) 2582"!

25 F%0 +:* 46*. ' 8 %& ;*+9** 8 0 3"8 9:*

() .% ](;) ,'. .% 2!

(,) + -1

35 %0 +:* &*. ;*+9** 8 0 #28 9:'.* +&*+ %. 135 .%7% 9%+:'6+ +;*. '$ ,,6+'$ +:* ''9%&>

() .% 1!8 - !,'. 388 #+22!

+ 13! 3.% 218

(;) .% "8 ,'.38 .% 38 ,'. "8

!5 % + #2 0 %. $**?/ %0 9%+:'6+ +;*. '$ ,,6+'$+:* 46* '/

() .*, (;) .%

7.0 FURTHER READING AND OTHER RESOURCES

7.1 REFERENCES

A%&'/ J5 C5 (*0) 11> I+$'06,+'$ U%4*$.%+ M+:*+%,. 1> A&*;$/

T$%&''*+$ 0 C'7*? N6;*$.5 O%+.: A$%,/ F*7

P6;%.:*$. L+05

B60/ B5 D/ 0 M6H'0 A5 (18)> P6$* +:*+%,. '$ A04,*0L*4* L'0'> B6+:*$9'$+: 0 C'5 (P6;%.:*$.) L+05

V&'0. M+:*+%, H0;'' ***+$ M+:*+%,.5

M.,'> MR P6;%.:*$.5

OTHER MATERIALS>

A +:*+%,. +*?+;''< +:+/ '6 , '6$ :0. '/ 9:%,: ,'+%

+:*.* +'7%,.5

NOUN !


59/131


UNIT $

GRAPHS OF TRIGONOMETRIC FUNCTION AND

THEIR RECIPROCALS

TABLE OF CONTENTS

158 INTRODUCTION

258 OBJECTIVES

351 GRAPHS OF TRIGONOMETRIC RATIOS

35151 GRAPH OF .%

35152 GRAPH OF ,'.

35153 GRAPH OF +

352 GRAPH OF RECIPROCALS AND INVERSE TRIGONOMETRIC

FUNCTIONS

35251 GRAPH OF ,'+ 8

35252 GRAPH OF INVERSE TRIGONOMETRIC FUNCTIONS

35253 PROBLEMS INVOLVING COMPOSITE FUNCTIONS

58 CONCLUSION

!58 SUMMARY

"58 TUTOR-MARKED ASSIGNMENTS


NOUN !


60/131


1.0 INTRODUCTION

G$7:. $' ***+$ +:*+%,. :*7 +' *.+;%.: +:* $*+%' ;*+9**

%0*7*0*+ 4$%;* 0 '+:*$ 4$%;* 0*7*0*+ 4$%;*5 H'7* '6

D$9 +:* &$7:. ' +$%&''*+$%, 6,+%'. 0 +:*%$ $*,%7$',.

,,6$+*5

R*0 46*. ' &%4* &* '$ +:* &$7:. ,'$$*,+5

D*+*$%* +:* 7*$%'0%,%+ (7*$%'0) 0 7%+60* ' &%4*

+$%&''*+$%, $+%'.5

F%&> 51> G$7: ' .%

NOUN "8

#


61/131


F%&> 515 G$7: ' .% (; 7$'*,+%')

3515 GRAPHS OF TRIGONOMETRIC FUNCTIONS

I 0$9%& +:* &$7:. ' +$%&''*+$%, $+%'./ ;.,%.. (?-?%.) %. +

() ..%& 0%*$*+ 46*. +' 8 + %+*$4. ' 38 +' 3"8 %5*5 8 8/

38/ "8/ 8 555/3"8

(;) %0 +:* ,'$$*.7'0%& 46*. ' .% 8 / +:%. %. 6.*0 +' $' +:* +;*

' 46*.

(,) ,:''.* .6%+;* .,* +:* 7'+ +:* 46*.

(0) '% +:* 7'++*0 7'%+. 9%+: *%+:*$ +:* $** :0 '$ ;$'' .+%,< +' &*+

.''+: ,6$4*5 T:%. ,6$4* %. +:* +:* &$7: ' .% 5

T:6. ''9%& +:* .+*7./ +:* +;* ' 46*. 77$'?%+*0 +' 2 0*,%

7,*. %.5

NOUN "1


62/131


T;* > T;* ' V6* '$ .% '$ 8 3"8

8' 38' "8' 8' 128' 1!8' 18' 218' 288 2#8' 388' 338' 3"

88 88 88 88

S% 8 85! 85# 1 85# 85! 8 -85! -85# -1 -85# -85!

S;*:

L*+ 1, $*7$*.*+ 1 6%+ ' +:* ?%. %5 *5 +:* :'$%'+ '$ ?-?%.5

L*+ , $*7$*.*+ 1 6%+ ' +:* .% ?%. %5*5 +:* 4*$+%, '$ -?%.5

T:* 7'%+. $+* +:* 7'++*0 ' &$7: .:**+ 0 %. '%*0 ; ;$'' .+%,*'&* +, '< +, 6 ';%

15 T:* +&*+ ,6$4* %. 0%.,'+%6*. ;*,6.* + %. '+ 0*%*0 + 80 2#8 $*.7*,+%4* %5*5 + 8' + 2#8'

25 T:* &$7: ' + :. 3 7$+. * 8 -= 8/ 8 2#88/3"88

35 T:* +&*+ ,6$4* %0*%%+* 77$',:*. +:* 4*$+%, %*. + 8 0

2#8 ;6+ *4*$ +'6,:*. +:*5 S6,: %*. (+ 8 0 2#8) $* ,*0

.7+'+*.(:*$* +:* ,6$4* 77$',:*. .+$%&:+ %* 7$* +' +:* -

?%. 0 0%.+,* $' %+ ; [ 8/ [2#8/ [!8 *+,5 ;6+ *4*$

$*,:*. +:*.* .+$%&:+ %*.5 P6+ % '+:*$ '$/ +:* %*. + 8 0

2#8 $* .%0 +' ;* .7+'+%, ,6$4*.5

3.. GRAPHS OF RECIPROCAL OF TRIGONOMETRIC FUNCTIONS

3..1. GRAPH OF 6 +'

T:%. %. +:* $*,%7$', ' +:* &$7: ' + 0 %. .:'9 ;*'95

NOUN "!


66/131


T;* $ T;* 3: T;* +, 2;4* ,+ 6 +' 00@ 0 @ 3800

8' 38' "8' 8' 128' 1!8 18' 218' 28' 2#8' 388' 338' 3"8

C'+ 15#3 85! 8 -85! -15#3 15#3 85! 8 -85! -15#3

NOUN ""


67/131


B5 A$,+

C 8 A$,,'.*,

D A$,.*,

E 5 A$,'+

NOUN "#

-18 -! 8 ! 18

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2

2

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2

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68/131


F5 8 $,.% 5 R&*> U$*.+$%,+*05

3..3. APPLICATIONS OF GRAPHS OF TRIGONOMETRIC

F6,+%'.> ,'7'.%+* 6,+%'.5

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(%) () D$9 +:* &$7: ' .% 28 '$ 46*. ' ;*+9** 0 3"8

(;) 6.* '6$ &$7: +' %0 +:* 46* ' +:* ''9%& 9:*

(%%) 2!' (%%%) 3!' (%4) !8

S'6+%'>

(%)


69/131


+'6,:*. +:* &$7:/ 0$9 :'$%'+ %* +' +:* -?%. (.% 2)?%. +:* $*0 ' +:* 46*. '$ %+. 77$'?%+%'.

(;) 2! *. +:+ .% 2 .% 2 ?2! .% !85 T:* !8 %*.;*+9** 38 0 "8 .' %+. 46* 9% ;* ;*+9** +:* 46*. '

38(85!) 0 "8 (85#)5 T:%. 46* %. 77$'?%+* 85! (.**&$7: ;*'9)

(,) 3! *. +:+ .% 28 .% 2? 3! .% #8 #8 %*.;*+9** "8 0 85 S' %+. 46* 9% ;* ;*+9** +:* 46*.

' "8 0 8 %5*5 (85# 0 8)5 F$' +:* &$7: %+ %.

77$'?%+* 85

(0) !8 -----= 2 .% 2?!8 18855 .% 188 %. 77$'?%+* -852! $' +:* &$7:

F&?: $.8

25 D$9 +:* &$7: ' 3 - 2!.% ? '$ 46*. ' ? ;*+9** 8 0 3"8

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3 - 2 .%?

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?8 88 388 "88 88 1288 1!88 188 2188 288 2#88 3888 3388 3"

NOUN "


70/131


S%? 8 85! 85# 158 85# 85! 8 -85! -85!# -1 -85# -85! 8

3 3 3 3 3 3 3 3 3 3 3 3 3 3

2.%? 8 158 15# 258 15# 158 8 -158 -15# -258 -15# -158 8

Y 3-2.%? 3 258 152" 158 152" 258 3 58 5# !58 -5# -58 3

O;.*$4* +:+ 9* %$.+ '60 +:* 46*. ' 2.% ? ('$ +:* &%4* 46*. ' ? )

;*'$* .6;+$,+%& +:* $' 3 . .** % +:* .+ " $'9 ' +:* +;* '

46*. ;'4*5

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(1) () C'.+$6,+ +;* '$ ,'. ? - 3 .%? '$ 46*. ' ? $' 8

+' 18 +

3 %+*$4

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+' 0$9 '6$ &$7:5

(2) D$9 +:* &$7: ' .%? ,'.? '$ +:* %+*$4 8 ? 3"8

6.* '6$ &$7:

+' %0

() +:* ?%6 46*. ' .%? ,'. ?

NOUN #8


71/131


(;) +:* %%6 46*. ' .%? ,'. ?

E*&* $.3

15 D$9 +:* &$7: ' +:* ''9%& '$ 46*. ' 8 $' 8 +' 3"8

%,6.%4*5

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25 9%+:'6+ 7'++%& +:* &$7:/ %0 +:*

(%) 7%+60* (%%) 7*$%'0%,%+ ' +:* ''9%&

6,+%'.5

() ! .% # (;) ! .% ( 3"8)(,) ,'.! (0) -2 ,'. 2?

S+4'&+%:

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(;) %5 ! %%5 3"8'

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(0) % 2 %%5 18

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T;*: 6 + &% ,+ 09 @ @ 380 &% '


72/131


2 Y .%2 1 3"82188

Y 2.% 2 18

Y !.% ! 18

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N'+* +:+ '$ &$7: ' A .% / +:* 7%+60* %. A %5*5 9:*$* A %.

,'.++ (,'*%,%*+ ' .% ) 0 7*$%'0%,%+ ' 3"8/ 9:%* % +:*&$7: %. +:+ ' A .% 8 +:* 7%+60* %. .+% A/ 7$'4%0*0 A %.

,'.++ 0 %+. 7*$%'0%,%+ %. 3"8 9:*$* %. ,'.++5 T:%. .' 77%*.

+' ,'.%*/ .*,+ 0 ,'.*,+5A7%+60* %. 9. 7'.%+%4* 6;*$5

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H4%& +$*+*0 +:* &$7: ' +$%&''*+$%, 6,+%'. 0 +:*%$ $*,%7$',. 0

.' % +:%. 6%+/ '6 :4* .** +:+ +:* +$*+*+ ' &$7:. :*$* $* +:* .*

9%+: +:* +$*+*+ ' &$7:. ' &*;$%, 6,+%'./ +:* ' 0%*$*,* %. %

+:* 46*. ..%&*0 +' +:*

%0*7*0*+ 4$%;* (?) 9:%,: % +:%. ,.* $* &*.5 T:* 7$',*..*. $* +:*

.* +:6.

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I +:%. 6%+/ 9* :4* ++*7+*0 +' 0$9 +:* &$7:. ' +$%&''*+$%, 6,+%'./+:*%$ $*,%7$',. 0 %4*$.* 6,+%'.5 T:* 7$'7*$+%*. ' +:*.* &$7:. '

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0 ,'+&*+ $* 0%.,'+%6'6. 6,+%'.5

NOUN #2


73/131


(%%) T:* 7*$%'0%,%+ ' 6,+%' %. +:* %+*$4 + 9:%,: +:* &$7: $*7*+.

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(4) T:* .%* 0 ,'.%* ,6$4*. %*. ;*+9** -1 0 1 0 +:* :4* +:*

.%%$ .:7* ;*,6.* ,'. - .% (8 - )

7.0 REFERENCE

B, P6$* M+:*+%,. F'$ A04,*0

L*4*5 L'0'> B6++*$9'$+: 0 C' (76;%.:*$.) L+05

T;*$+/ J= F= G'0/ A 0 O&6/ G5 E5 O5 (12) A00%+%'

M+:*+%,. '$ W*.+ A$%, L'0'> L'& G$'67 UK L+05

T:* W*.+ A$%, E?%+%'. C'6,% (1#) M+:*+%, 0 .++%.+%,

T;*. 0 F'$6*5

V&'0. MR P6;%.:*$.

NOUN #3


74/131


UNIT "

TRIGONOMETRIC IDENTITIES AND EUATIONS

TABLE OF CONTENT

158 INTRODUCTION

258 OBJECTIVES

351 TRIGONOMETRIC IDENTITIES

35151 FUNDAMENTAL IDENTITIES

35152 TRIGONOMETRIC EUATION

352 COMPOUND ANGLES

35251 ADDITION FORMULAE (ADDITION AND SUBTRACTION)

35252 MULTIPLE AND HALF ANGLES

353 SUM AND DIFFERENCE FORMULAE (FACTOR FORMULAE)

35 PRODUCT FORMULAE

58 CONCLUSION

!58 SUMMARY

"58 TUTOR-MARKED ASSIGNMENT


1.0 INTRODUCTION

I +:* *$%*$ 6%+. ' +$%&''*+$%, $+%'./ +:*%$ $*,%7$',. 0 %4*$.*

+$%&''*+$%, 6,+%'. +:*$* 9$* '+. ' %7'$++ $*+%'. ;*+9**

+$%&''*+$%, 6,+%'.5 F'$ *?7*

sin = tan ; = cosec

cos sin

NOUN #


75/131


cos = cot ; = sec

sin cos

I +:*.* $*+%'. $* +$6* '$ &%4* 46* ' .6,: $*+%'. $* ,*0+$%&''*+$%, %0*+%+%*./ 7$'4%0*0 +:* 6,+%'. $* 0*%*05

T:%. 6%+ 9% ',6. ' +$%&''*+$%, %0*+%+%*./ 9:%,: .:'60 '$ +:* ;.%.

'$ 7$'4%& '+:*$ %0*+%+%*./ C'7'60 &*./ 0%*$*,* 0 7$'06,+

'$6*/ 6+%7* 0 : &*. 0 % +$%&''*+$%, *@6+%'./ 9:%,:

$* *;*00*0 % +:*5

.0 OB!ECTIVES

B +:* *0 ' +:%. 6%+/ +:* .+60*+. .:'60 ;* ;* +'>

D*%* +$%&''*+$%, %0*+%+%*. ,'$$*,+

P$'4* &%4* +$%&''*+$%, %0*+%+%*. ,'$$*,+ S%7 0 .'4* 7$';*. %4'4%& +$%&''*+$%, %0*+%+%*. 0

*@6+%'.5

E?7$*.. .6 0 0%*$*,* ' +9' &%4* &*. % +$%&''*+$%,

%0*+%+%*.

E?7$*.. 6+%7* 0 : &*. ' &%4* %0*+%+%*.

F,+'$%* +$%&''*+$%, *?7$*..%'.

3.1 TRIGONOMETRIC IDENTITIES (FUNDAMENTAL IDENTITIES)

3.1.1. TRIGONOMETRIC IDENTITIES (RIGHT-ANGLED TRIANGLE)

T$%&''*+$%, %0*+%+%*. $* $*+%'./ 9:%,: $* +$6* '$ &%4* 46* '

&%4* $%&:+-&*0 +$%&* ABC/ $%&:+-&*0 + B 0 &* C 9%+: +:*6.6 '++%'. .** F%& !515

F%&> !515

S% , -----= ; .% , (1)

;

C'. -------= ; ,'. (2) B

NOUN #!


76/131


B P+:&'$. +:*'$* 2 ,2 ;2/ .' .6;.+%+6+%& +:* 46*. ' 0 , $'

(1) 0 (2) 9* ';+%

( ; ,'. )2 (; .% )2 ;2/ .%7%%&

;2COS2 ;2.% ;2 /

0%4%0%& +:$'6&: ; ;2/ 9* :4*

S%2,'.2 1 OR ,'.2 .%2 1

A.'

S%2 1 - ,'.20

C'.2 1 .%2

E?7*>

15 W%+:'6+ +;*. '$ ,,6+'$. %0

() .%2"8 ,'.2"8 (;) .%2338 ,'.2338

S'6+%'>

() .%2"8' 32 0 ,'.2"8 ] 0 .6;.+%+6+%& %+' .%2"8 ,'.2"8/ &%4*.

(;) .% 338 -.% 38 -12 0 ,'. 338 ,'. 38 32 .6;.+%+6+%&%+' +:* &%4* *?7$*..%' .%2338 ,'.2338 +' ';+%

.%,* +:%. $*+%' .%2 ,'.21/ :'0. +$6* '$ 46*. ' / %+ %.

+:* +$%&''*+$%, %0*+%+5

F$' +:* ;'4* +$%&''*+$%, %0*+%+ S%2 ,'.2 1/ +:* ''9%&

+$%&''*+$%, %0*+%+%*. , ;* 0*06,*0>

S%2 ,'.21

NOUN #"

2


77/131


D%4%0* +:$'6&: ; ,'. / +:%. ;*,'*.

S% 2 ,'.2 1

C'.2 ,'.2 ,'.

2

;6+ .%2 + 0 1 .*,

,'.2 ,'.

.% 2 1 1 &%4*. .% 2 1 S*,2

,'.2 ,'. ,'.

T2 1 .*,2

.*,2 - 1 +2

A&%/ % 9* 0%4%0* .%2 ,'.2 1 ; .%2/ %+ ;*,'*.

S% 2 ,'. 1

.%2 .%2 .%2/ ;6+

,'. ,'+ 0 1 ,'.*,

.% .%

1 ,'+2 ,'.*,2

C'.*,2- 1 ,'+2

O+:*$ $*+%'. 9:%,: , ;* 0*06,*0 $*

(1) + ? ,'+ 1

(2) ,'. ? .*, 1

(3) .% ? ,'.*, 1()

1- ,'.2 N'+* +:+ 1 - ,'.2 .%2

.% 2

1- ,'.2 .%2

.%2 .%2

NOUN ##


78/131


55 1- ,'.2 1

H*,* $' +:*.* .%2*?7*. %+ , ;* 0*06,*0 +:+

(1) +*:

15 S'4* +:* *@6+%'/ &%4%& 46*. ' $' 8 +' 3"8 %,6.%4*5

() 3 - 3 ,'. 2 .%2 (;) ,'.2 .% 1 8

S+4'&+%

() 3-3,'. 2.%2/ :*$* +:* *;*$. ' +:* *@6+%' , ;* *?7$*..*0.

,'. .%,* .% 1 ,'.2

3 - 3,'. 2(1 - ,'.2)

3-3,'. 2-2,'.2

2,'.2- 3,'. 1 85

NOUN #


79/131


T:%. %. @60$+%, *@6+%' % ,'. 0 +:6. , ;* .'4*0 ; '+:* *+:'0. ' @60$+%, *@6+%'5

B ,+'$%+%' 2,'. - 3,'. 1 8 &%4*.

(2 ,'.- 1) (,'. - 1) 8

*%+:*$ 2,'.- 1 8 OR

,'.- 1

% 2,'.- 1 8 >= ,'. ] 0 ,'. %. 4*

,'.-1] "8 '$ 388

% ,'. 1/ ,'.-11 8 '$ 3"8

+:* 46*. ' 9:%,: .+%. +:* *@6+%' 9%+:% +:* &%4* $&* '

8 3"8 $* 85 "8/ 388 0 3"85

(;) C'.2

.% 1 8

I+ %. *.%*$ +' +$.'$ C'.2+' 1-.% +' '$ *@6+%' ' 7'9*$. '

.%25 T:6.

(1 - .%2) .% 1 8

1-.%2 .% 1 8

>= 1-.%2 .% 2 8

F,+'$%%&> (.% - 2)(.%) 8

55 *%+:*$ .%

-2 8 '$ .%

1 8% .%-2 8 = .%2 0 .%-1 2

T:%. 46* ' 0'*. '+ .+%. +:* &%4* *@6+%' ;*,6.* .%%*. ;*+9**

-1 0 1 +' .+%. +:* &%4* *@6+%'5 S' .% -12 %. '+ .'6+%' % .% 1 85

S% - 1 = .%-1(-1) 2#8

2#8 %. +:* $''+ ' +:* *@6+%' ;*,'*. %+ . 9%+:% +:* $&*

8 3"8

25 F%0 +:* .'6+%'. ' +:* *@6+%' % +:* %+*$4 8 3"8

1",'.2 2.% 13

S+4'&+%

C'.2 1 .%2/ +:%. 9% ;* .6;.+%+6+*0 %+' +:* *@6+%' +' &%4*

1"(1- .%2) 2.% 13

NOUN #


80/131


1" 1" .%22.% 13 8 >= 1".%2- 2.%- 1" 13 8 = 1".%2-2.% - 3 8

F,+'$%.%& &%4*. (.% 3)(2.% - 1) 8

55 *%+:*$ .% 3 8 OR

2.%- 1 8

.'/ % .%8 3 8 >= .% -3

.%-1(-3) .%-1(-853#!)F$' +:* +;*. 8 -225 T:%. %*. *%+:*$ % +:* +:%$0 '$ '6$+:

@60$+ .%,* .%%. *&+%4*

18 22 2 '$ 3"8 - 22 2 282 2 '$ 33# - !

% 2.% 1 = .% ] >= .%-1( ] )

55 38 .%,* .%%. 7'.%+%4*/

%. *%+:*$ % +:* %$.+ '$ .*,'0 @60$+

55 38 '$ 18 - 38 38 '$ 1!8

+:* .'6+%' ' +:* *@6+%'. '$ 8 3"8

%. 38/ 1!8/ 282 21 0 33# ! 5

35 F%0 9%+:'6+ +;*/ +:* 46* ' .*, / .%% + -!12

S+4'&+%

U.%& $%&:+ &* +$%&* %? +:* .%0*. ' +:* +$%&* 6.%& +:*


81/131


2!1?2,'.2 2.% '$ 8 3"8 % +:* %+*$4 ' 38

1" ?2

55 ? 1" [ 13

55 .% !13 0 ,'. [123/ ;6+ %. ';+6.*/ :*,* .% !13 0

,'. -1213

55 .% !13 0 .*, 1cos

&%4*. .*, -1 - -1312 12

13

5 P$'4* +:* ''9%& %0*+%+%*.>

.*,2 ,'.*,2 .*, ,'.*,

2

S'6+%'>

I 7$';*. ' +:%. .'$+/ .+$+ $' 9:+*4*$ *?7$*..%'. (*%+:*$ *+ :0 .%0*

'$ $%&:+ :0 .%0*) +' .:'9 +:+ %+ %. *@6 +' +:* '+:*$ ($%&:+ :0 .%0* '$ *+

:0 .%0*) 9:%,:*4*$ %. .%7*$ T:6. .+$+%& $' +:* *+ :0 .%0* (LHS)

S*,2 ,'.*,2 1 1

C'.2 .%2

S%7%%& &%4*. .%

2

,'.

C'.2.%2

B6+ .%2,'.2 1

.% 2 ,'. 1

C'.2.%2 C'.2.%2 1 1

C'.2 .%2

.*,2? ,'.*,2 .*,2? ,'.*,2 RHS

N'+* +:+ % *?7*. 1 0 2 9* ,',*+$+*0 ' ' &*. % +:* %$.+

$*4'6+%' '$ ;.%, &*.5 T:%. %. ;*,6.* % 77%,+%'. '

+$%&''*+$ +:* $* +:* '*. 6.6 $*@6%$*0

3.. COMPOUND ANGLES

NOUN 1


82/131


3..1. (A) ADDITION FORMULAE

FIG "..

I %& !52 ;'4*

PAR 8 - ARP

PRO RON (+ . PR| |ON)

S%(A B) A P AP

A A

RN AP RN AP

A

A

A

RN OR AP AR

OR OA AR OA

S% A C'. B C'. A S% B

S%(AB) S% A C'. B C'. A S% BS%%$ $' +:* .* %& !52

C'.(A B) 8 N - N

A A

ON PR ON - PR

OA OA OA

ON OR - PR MR

OR OA MR OA

NOUN 2


83/131


C'.AC'.B - S%AS%B

Q

55 C'.(A B) C'.AC'.B - S%AS%B

T (A B) S% (AB) .% ,* + S% C'.(A B) C'.

S% A C'. B C'.AS% B

C'.AC'.B S%AS%B

D%4%0%& ;'+: +:* 6*$+'$ 0 0*'%+'$ ; C'.AC'.B/

T(A B)

S%AC'.B C'.AS%B

C'.AC'.B C'.AC'.B

C'.AC'B S%AS%B

C'.AC'.B C'.AC'.B

S%7%%& &%4*. + A + B

1- + A + B

55 +(A B) + A + B

1 - + A + B

3..1. DIFFERENCE FORMULAE

T:* 0%*$*,* '$6* , ;* ';+%*0 $' +:* 00%+%' '$6 '$

$*7,%& B 9%+: (-B) % *,: ,.* +:6.

() S%(A - B) S%AC'.B - C'.AS%B

(;) C'.(A - B) C'.AC'.B S%AS%B

(,) T(A -B) + A - + B

1 +A + B

E;>*:

W%+:'6+ 6.%& +;*. '$ ,,6+'$. %0 +:* 46*. ' +:* ''9%& *4%&

'6$ .9*$. % .6$0 '$5

(%) C'.(! - 38) (%%) .%("8 !) (%%) + #!

S+4'&+%

(%) C'. (! - 38') %. % +:* '$ ' ,'. (A -B) 0 ; +:*

00%+%'0%*$*,* '$6 %+ %.

NOUN 3


84/131


(C'.(A - B) C'.AC'.B S%AS%B *?70%& C'.(! - 38) +:6./

9:*$* A ! 0 B 38 &%4*.

C'. !C'.38 S%! S%38 .' .6;.+%+6+%& +:* 46*. '$

C'. ! 12 0 .% ! 12

C'. 38 32 0 S% 38 ] %

C'. !C'.38 S%!S%38 &%4*.

(12) (32) (12)(12)

322 122

3 1 1 3 2 (1 3)

22 22

(%%) S% ("8 !) .%"8 C'.! C'."8S%!/ :*$*

S%(A B) S%AC'.B C'.AS%B . 77%*0

I A "8 0 B !

0 .6;.+%+6+%& +:* 46*. ' S% "8 32 / C'."8 ] /

S%! C'. 12 +' ';+%

(32)(12) (12)(12)

3 1

2

2 2

2

1 3 2 (1 3

22

(%%%) + #! + (!' 38') A77%& +:* '$6

+(A -B) + A - + B/ 9:*$*

1 + A + B

A ! 0 B 38 0 + ! 1 0 + 38 13

&%4*.1 1 3 1

3 31-1 1 1 1

3 3

3 1

3 - 1

NOUN


85/131


0 .%7%%& ; $+%'%%& +:* 0*'%+'$ &%4*.

(3 1) (3 1) 3 23 1

(3 1) (3 1) 3 3 3 1

23 2 32

E*&* ".1

15 P$'4* +:* ''9%& %0*+%+%*.

() + ,'+ 1

.% ,'.

(;) (.*, + ) (.*,- +) 1

(,) 2,'.2- 1 1 -2.%28 ,'.2- .%2

(0) ,'.*, + .*, ,'.*, .*,2

25 S'4* +:* ''9%& *@6+%'./ &%4%& 46*. ' $' 8 +' 3"8%,6.%4*

() .*,23+ - 1 A.> !/ "382"Z / 22!/2382"

(;) 3,'.2 #,'. " A.> 131 / 22 12

(,) 2.% 1 A.> 38/ 1!85

35 F%0 9%+:'6+ +;*.,,6+'$. +:* 46*. '

() S%

/ +

/ % ,'.

! 0

%. ,6+*A.> .% 3! 0 + a

(;) ,'. / ,'+ / % .% 1!1# 0 %. ,6+*

A.> ,'. 1#5 ,'+ 1!

(,) .* / .*, / % ,'+ 2821 0 %. $**?

A.> .% - 212 .*, -228

5 I .%A ! 0 ,'.B 1213/ 9:*$* A %. ';+6.* 0 B %.

,6+*/ %0 9%+:'6+ +;*.,,6+'$. +:* 46*. '

() .% (A B) A.> "3"!

(;) + (A B) A.> -"31"

(,) + (A B) A.> -33!"

NOUN !


86/131


3... MULTIPLE AND HALF ANGLE

3..;. MULTIPLE ANGLES (DOUBLE ANGLE)

T:%. %. *?+*.%' ' +:* 00%+%' '$6 I *,: ,.*/ 76++%& B A 9*

';+% '$ .% (A B) .% (A A) .% 2A .%,* .%(A B) .%A,'.B

,'.A.%B 0 $*7,%& B 9%+: A &%4*.>

.%(A A) .%A,'.A ,'.A.%A

.% 2A 2.%A,'.A 0,'.(A A) ,'.A,'.A - S%A.%A

,'.(2A) C'.2A - .%2A ;6+ .%2A 1 - ,'.2A

.6;.+%+6+%& &%4*. ,'.(2A) ,'.2A - 1 ,'.2A

55 ,'.2A 2,'.2A - 1 0 ,'.2 1 .%2A .

.' ,'.2A 1- .%2A - .%2A 1- 2.% A

55 ,'.2A ,'.2A- .%2A

2,'.2A - 1

+ 2A + A + A

1 - + A + A

2+A

1-+2A

3... HALF ANGLES

B .6;.+%+6+%& : &*. *?7* A2 '$ B2 %+' +:* 0'6;* &*./ +:*

'$6* ;'4* ;*,'*

() .%( A A ) .% A 2.% A C'. A

2 2 2 2

(;) COS( A A ) COSA COS2A - .%2A

2 2 2 2

2,'.2A -1

2

1- 2.%2A

2

(,) +( A A ) + A 2 + A2

2 2 1-+2A2

(0) .%2 A 1 - ,'. A / +:%. ,'*. $' .%2A 1- ,'. A /

2 2 2

NOUN "


87/131


(*) ,'.2A 1 ,'. A

2 2

() +2A - ,'. A

2 1 ,'. A

3535 SUM AND DIFFERENCE FORMULAE (FACTOR FORMULAE)

() .%(A B) .%A,'.B ,'.A.%B (1)

.%(A- B) .%A,'.B C'.A.%B (2)

00%& ;'+: (1) 0 (2)

.%(A B) .% (A -B) 2.%A,'.B

.6;+$,+%& ;'+: (1) 0 (2)

.%(A B) - .%(A -B) 2,'.A.%B

(;) ,'.(A B) ,'.A,'.B .%A.%B (3)

,'.(A - B) ,'.A,'.B .%A.%B ()

00%& ;'+: (3) 0 ()

,'.(AB) ,'.(A -B) 2,'.A,'.B

0 .6;.+$,+%& ;'+: (3) 0 () &%4*.>

,'.(A B) - ,'.(A - B) -2.%A.%B

9:%,: , ;* $*9$%++* .

,'.(A -B) - ,'.(AB) 2.%A.%B

(+:%. %. +' 4'%0 +:* %6. .%& &'++* % +:* %$.+ '*)5

3.3.1 PRODUCT FORMULAE

F$' +:* ;'4* .6 0 0%*$*,* '$6*/ '+:*$ %+*$*.+%& %0*+%+%*.

**$&*0

S%(AB) .%(A - B) 2.%A,'.B % AB %. *@6 +' ? %5*5 AB ? 0 A -

B / +:%. %7%*. +:+ 00%& ;'+: &%4*.

S%(AB) .%(A -B) 2.% ? C'. ? -

2 2

S%? .% 2.% ? C'. ?

2 2

S%? - .% 2,'. ? .% ?

2 2,'. ? C'. 2,'. ? ,'. ?

2 2

C'. ? C'. 2.% ? .% ?

2 2

C'. - C'.? 2.% ? .% ? -

2 2

NOUN #


88/131


T:*.* '$6* , .' ;* .++*0 % +:%. '$

C'.A,'.B ] b,'.(A B) ,'.(A -B) c

S%A.%B ] b.%(A -B) - ,'.(AB)c

S%A,'.B ] b .%(A B) .%(A -B) c

T:*.* $* ,*0 +:* 7$'06,+ '$6*5

E;>*

F%0 +:* 46* ' +:* ''9%& &*. 9%+:'6+ +;*. '$ ,,6+'$.5

() ,'. #! ,'. ! (;) .%#! .% 1! (,) ,'.3 -

,'. 1#

S+4'&+%

() +' .'4* +:* &%4* 7$';*/ 77 +:* 7$'06,+ + '$6 9:%,: .++*.

+:+ ,'.A,'.B ] b,'.(A B) ,'.(A -B) c .' +


89/131


2.% 8 C'. "8

2 2

2.%!,'.38

+:* 46*. ' .%! 12 ,'.38 32 / $*


90/131


% .% ? 8 .%-18 ?

.%,* +:*.* $* 46*. + 9:%,: 9:* .%? ? 8/ -18 0 18 +:* 8 ;*+9**

- 18 0 185

? -#28/ -18/8/ 18/ #28 (%,60%& +:* %+*$4.) ? - 18/ -!/ 8/ !/ 18 (0%4%0%& +:$'6&: ; )

.' +:* 46* ' ? 9:%,: .+%.%*. +:* *@6+%' $*

? -18/ -8/ -!/ 8/ !/ 8'1/ 185

E*&* ".

15 S'4* +:* *@6+%' .%(? 1#) ,'.(? -12) 85 # '$ 46*. ' ? $'

8 +' 3"8 %,6.%4*5

A.> ? 38 3#/ !823/ 21883#/ 23823

25 P$'4* +:* %0*+%+%*.

() ,'. B ,'. C ,'+ B - C

.% B - .% C 2

(;) .% ? - .%(?"8) .% (?128 8

(,) ,'.? ,'.(?128) ,'.(?28) 8

35 S'4* '$ +:* ''9%& *@6+%'./ '$ 46*. ' ? $' 8 +' 3"8

%,6.%4*5() ,'.? ,'. !? 8

A.> 38/ 8/ 1!8/ 28/ 2#8/ 338/ !/ 13!/ 22!/ 31!5

(;) .%3? ,'.2? 8 (:%+ ,'.2? .%(8 - 2?)

A.> !/ 12"/ 1/ 2#88/ 3285

5 E?7$*.. +:* ''9%& % ,+'$.5

() .%2 - .%2?

A.> 2,'.( ?) .% -?)

(;) .%(? 38) .%(? -38)

A. 3.%?(,) ,'.(8 - ?) ,'.

A.> 2,'.(! - ] ? ] )

(0) .%2(? 8) .%2(? -8)

2.%2? ,'.8

NOUN 8


91/131


$.0 CONCLUSION

I +:%. 6%+/ '6 :4* .** +:* ;*6+ ' +:* $*+%'. ' +$%&''*+$%,

%0*+%+%*. 0 :'9 *. +:* $* 77%*0 % .'4%& +$%&''*+$%, 6,+%'.

7$';*.5 F$' +:* 00%+%' '$6*/ 9* 9*$* ;* +' 0*%* +:* .6 00%*$*,* 0 7$'06,+ '$6* ; .%7* %76+%' ' '* ' +:* &*.

0 ; +:* '7*$+%'. ' 00%+%' 0 .6;+$,+%'5 T:%. 0* +$%&''*+$%,

%0*+%+%*. 65

".0 SUMMARY

I +:%. 6%+/ +:* ''9%& +$%&''*+$%, 6,+%'. %0*+%+%*. 9*$* 0*06,*0

$' +:* 60*+ %0*+%+%*. %5*5

S%2 ,'.2 1

1 .%2 ,'.2

1 -,'.2 .%2

1 +2 .*,2

,'+2 ,'.*,2

(+) (,'+ ) 1

(,'. ) (.*, 8) 1

(.% ) (,'.*, ) 1

F$' +:* 00%+%' '$6*/ (00%+%' 0 .6;+$,+%')5

S%( AB) .% A,'.B ,'.A.%B

S%(A -B) .%A,'.B - ,'.A.%BC'.(AB) ,'.A,'.B - .%A.%B

C'.(A-B) ,'.A,'.B - .%A.%B

T (AB) + A + B 0

1+ A + B

T(A -B) + A - + B

1 + A + B

T:* ''9%& 6+%7* &*. (0'6;* &*.)/ : &*./ .6 0

0%*$*,* 0 7$'06,+ '$6* 9*$* 0*06,*05

S%2A 2.%A,'.A

1- 2.%2A

T 2A 2+A

1- +2A

NOUN 1


92/131


C'.2A ,'.2A .%2A 2,'.2A 1

(M6+%7* &*. (0'6;* &*.)

H;, A%?*

S%A 2.% A C'. A

2 2

COSA COS2A - I

2

1 - 2S%2A

2

T A 2 + A

1-+2A

S6 0 D%*$*,* '$6* (,+'$ '$6*)

2,'.A.%B .%(AB) - .%(A -B)

2.%A,'.B .%(AB - .%(A -B)

2,'.A,'.B C'. (AB) ,'.(A-B)

2.%A.%B ,'.(A-B) - ,'.(AB)

A0 % +:*

P+54' ,+4;*

C'.A,'.B ] b,'.(A B) ,'.(A -B) c

S%A.%B ] b.%(A -B) - ,'.(AB)c

S%A,'.B ] b .%(A B) .%(A -B) c

NOUN 2


93/131


8.0 TUTOR-MARKED ASSIGNMENT

15 %0 +:* 46*. ' +:* ''9%& 9%+:'6+ +;*. '$ ,,6+'$./ *4%& '6$

.9*$. % .6$0 '$5

() (%) + 18! (%%) ,'. 1! (%%%) ,'. 3!

(%4) .% 1"!A.> (%) - 2 -3 (%%) 3 1 (%%%) 3 1 (%4) 3 - 1

22 22 22(;) % ,'.A ! 0 ,'. B 1213 (A 0 B $* ;'+: ,6+*) F%0 +:*

46*. '

(%) .% (A B) A. !""!

(%%) ,'.(A -B) A. "3"!

(%%%) +(AB) A. !"33

25 S'4* +:* *@6+%'. '$ 8 3"8

() .%2 + A.> 8/ !/ 13!/ 18/ 22!/ 38/ 3"8(;) ,'.2 2,'. A.> 115 # '$ 25 !3

35 F%0 9%+:'6+ +;*. '$ ,,6+'$./ +:* 46*. '

() 2.%1! ,'.1! A.> ]

(;) !8 ,'. !8 A.> 1

22(,) 2 + !8'

A.-1

1- +2!8 '

(0) .%222 ] 8 ,'.222 ] A.> - -1 2

7.0 FURTHER READING AND OTHER RESOURCES

REFERENCES

A%&'/ J5C5 (*0) (11) > I+$'06,+'$ U%4*$.%+ M+:*+%,. 1> A&*;$/

+$%&''*+$5 C'7*?5 N6;*$.5 O%+.: > A$%, - *7 P6;%.:*$.5

L+05

B, L'& G$'67 UK L+05

NOUN 3


94/131


V&'0. MR P6;%.:*$.

NOUN


95/131


UNIT 8

SOLUTION OF TRIANGLES AND HEIGHTS AND DISTANCES

TABLES OF CONTENT

158 INTRODUCTION

258 OBJECTIVES

351 SOLUTION OF TRIANGLES

351515 SINE RULE

35152 COSINE RULE

352 HEIGHTS AND DISTANCES

352515 ANGLES OF ELEVATION AND DEPRESSION

352525 AREAS OF TRIANGLES AND PARALLELOGRAM

58 CONCLUSION !58 SUMMARY

"58 TUTOR-MARKED ASSIGNMENT

#58 REFERENCES

1.0 INTRODUCTION

T:%. 6%+ %. ''9-67 ' +:* 7$*4%'6. 6%+. ' +$%&''*+$%, 6,+%'. 0

+:*%$ $*+%'. 9%+: +:* $%&:+-&*0 +$%&*5 I +:%. 6%+/ 9* .: 0%.,6../ +:*

.'6+%' ' +$%&*. ( +7*. ' +$%&*.) % 9:%,: ,.* %0%& +:*

&*. 0 .%0*. ' +$%&* 9:* +:* ''9%& $*


96/131


+:*.* %'$+%' :*7 % +:* %?%& ' +:* .:7* ' +$%&*(+:* &*

'77'.%+* +:* &$*+*$ .%0* %. ;%&&*$ +: +:* &* '77'.%+* +:* .*$ .%0*)5

T:* &*. ' **4+%' 0 0*7$*..%'. $* 0*%*0 0 %. 6.*0 % +:*

,,6+%' ' :*%&:+. 0 0%.+,*. % 7$,+%, 7$';*.

.0 OB!ECTIVES

B +:* *0 ' +:%. 6%+/ +:* .+60*+. .:'60 ;* ;* +'>

0*$%4* +:* .%* 0 ,'.%* $6*.

77 +:* .%* 0 ,'.%* $6*. ,'$$*,+ +' .'6+%'. ' +$%&*.

0*06,* +:* ,'$$*,+ $* ' +$%&* 6.%& +$%&''*+$%, $+%'. %.

] ; S% C '$ ] ; , .% A '$ ] , .% B

0*%* &*. ' **4+%' 0 0*7$*..%'

77 +$%&''*+$%, $+%'. +' &*. ' **4+%' 0 0*7$*..%' %

%0%& :*%&:+. 0 0%.+,*.5

3.1 SOLUTION OF TRIANGLES

3.1.1 SINE RULE

S' ' '6 6.+ :4* 6.*0 +:* .%* $6* 9%+:'6+


97/131


B D

F&? 8.1 ()

F%& "51 ) %. ,6+* &*0 +$%&* 0 %&5 "51 (;) %. ';+6.* - &*0

+$%&*>

G%4*ABC . .:'9 % %&. "51 ( 0 ;) ;'4*

R*@6%$*0 +' 7$'4* (R5T5P) ; , (.% $6*)

.% A .% B .% C

C'.+$6,+%'> D$9 AD 7*$7*0%,6$ +' BC (%& "51 )

O$ D$9 AD 7*$7*0%,6$ +' BC 7$'06,*0 (%& "51 ;)

P$''> % ABD % %& "515 (1)

S% B ?,/ +:%. *. +:+

C .% B ? (2)

A.' % ADC % %& "51 5

S% C ?; = ; .% C ?/ +:%. *. +:+ % ;'+: (1) 0 (2)

X , .% B ; .% C 0 0%4%0%& ;'+: .%0*. ; S% B S% C

&%4*.5

, ; (3)

.% C .% B

I %& "51 ; +:* ';+6.* &*0 +$%&* I ADC

S% B ?, = , .% B ? ()

NOUN #


98/131


S% (18 -C) ? ;6+ .% (18 - C) .% C (+:* .% ' ';+6.*B

&* %. *@6 +' +:* .%* ' %+. .677**+ %5*5 ;'+: &*. .6 67 +' 18)

55 S% C ?; = ; .% C ? (!)

0%4%0%& ;'+: .%0*. , S% B ; S% C ? ; S% B S% C &%4*.

, ;

.% C .% B

S' % ;'+: +:* ,6+* 0 ';+6.* &*0 +$%&*. +:* .* $*.6+. 9*$*

';+%*0 % +:* 7*$7*0%,6$ %. 0$'77*0 $' C +' AB/ %+ , .' ;* 7$'4*0

+:+ ('6 , +$ +:%.)

; ,.% A .% B .% C5

T:%. %. +:* S%* R6*5 H*$* A %. &* + A 0 %. +:* .%0* '77'.%+* +:* &*

+ A/ +:* .* 77%*. +' B 0 ; 0 C 0 ,5

N'+* +:+ +:%. .%* $6* , ;* 9$%++* .

.% A .% B .% C

; ,

T:* .%* $6* %. 6.*0 +' .'4* +$%&* 9:*

(%) +9' .%0*. 0 '* .%0* %. &%4* 0

(%%) +9' .%0*. 0 '-%,60*0 &* %. &%4*5

E;>* 1

15 S'4* +:* ''9%& +$%&*. ABC 9:%,: :4*

() A 2! 2!/ B "2 !1 0 35 2,5

(;) A 112 2/ !523 , 0 ; #5"!,

(,) C 12! 3/ 52, 0 , 52,5

S+4'&+%:

R**;*$ +'


99/131


.6;.+%+6+%& +:* 46*. ' A 0 B %+' +:* *@6+%' &%4*.

2! !2 "2 1! C 18

C 18 - (2! !2 - "2 1!)18 '#

1'!3

C 1 !3

F%& "> 25

T' &*+ +:* .%0*. ; 0 , 6.%& +:* .%* $6* ; ,

/

S% A .% B .% C

+9' ' +:*.* *@6+%'. , ;* 6.*0 +:6.>

352 ;

.%(2!!) .%("21! )

; .% 2! !2 352.%("2 1!)

; 352 .%("21!) 352 .%("252!).% 2!!2 .%(2!5#)

Y'6 , 6.* '6$ ,,6+'$ '$ '&$%+: +;*. :*$* '$ *. ,,6+%'.5 I

+:* *?7* ,,6+%'. 9. 6.*05

; 3538"!2#!! #5#! ,

853"338#212

55 ; #5#! ,

'$ ,/ ' +:* +9' *@6+%'. , .' ;* 77%*05

NOUN


100/131


,

.% A .% C

352 ,

.% 2!5#8 .% 1>

, .% 2! 5 # 352 .%1>

352 .% 1 >

= .% 2!5#

351#382

C 853"338#212

C 5#!,

, 15/;#5#! , 0 ,5#!,5

N'+* +:%. 46*. $* % &$***+ 9%+: +:* ,'0%+%' +:+ &$*+*$ &*. ,*

&$*+*$ .%0*.5

(;) A 112 2 0 !523 , 0 ; #5"!, ';.*$4* :*$* +:+ +:*

.%0* !523 ,%& &* A %. .*$ +: +:* .%0* ; #5"!, 0 %+

%. '+ 7'..%;* +' :4* +$%&* 9%+: +9' ';+6.* &*./ :*,* .%0*

.:'60 ;* &$*+*$ +: ; 0 .%,* +:%. %. '+ +:* ,.*5 T:* +$%&* :.

;* '77'.%+* +:* $&*$ &* (+$ +' .'4* +:%. +$%&*/ 9:+ 0%0 '6

';.*$4*0)

(,) C 12! 3/ 52 , 0 , 52,5

S+4'&+%

T:*$* %. 7'..%;* .'6+%' :*$* .%,* C 9:%,: %. $&*$ :. .%0* , &$*+*$

+: .%0* 5

F&?. 8.3

NOUN 188


101/131


U.%& +:* .%* $6*

,

.% A .% C

0 .6;.+%+6+%& +:* &%4* 52, 0 , 52, &%4*.

52 52

.% A .% 12!3

52 .%A 52.% 12! 3

S% A 52.% 2!3 52.%5#2

52 52

S%A 358!8 851!8"!52

A .%- 85 1!!8"! 25!#

B 18 - (12!5 #2 25!#) 18 - 1!85 #2 25#1

B 25 #1+' &*+ ;5

; ,

.% B .% C

; 52 .% 25#1 .%12!85#2

; .% 12!5#2 52.% 25#15

; 5 .% 25#1

.% 12!5#2

; 58"881#

8511##31

; !588!" , !588",

55 A 25 !#/ B 25#1/ ; !588",

3.1. THE COSINE RULE

L%


102/131


F&? 8.3;.

F&?. 8.3..

G%4*> DABC 9%+: +:* 6.6 '++%'. .** %& > "535 (,6+* - &*0) 0

"53;5 (';+6.* &*0 +$%&*.5

R*@6%$*0 +' P$'4*> ,2 2 ;2- 2 ; ,'. C (,'.%* $6*)

C'.+$6,+%'> D$9 7*$7*0%,6$ $' A +' B C (%& "53) 0 $' A +'

BC

7$'06,*0 (%& "53;5)

P$''> % %&>"53 %ABC5

;2 ?2 2 (1) (P+:&'$. +:*'$*) % ABC/ ,22 ( -?)2 2/;

.%7%%,+%'/

,2-2-2??22

;6+ ?2 2 ;2% (1)

.6;.+%+6+%& ,2 2- 2? ;2 (2)

I ABC/ ,'. C ?; = ; ,'. C ?

NOUN 182


103/131


.' .6;.+%+6+%& '$ ? % (2) &%4*.

,2 2 ;2- 2; ,'. C

55 C2- 2 ;2- 2;,'. C

P$''> % %& "53;5 (O;+6.* &*0 +$%&*)

I ADC/ ;2X2 2 (%%) (P+:&'$. +:*'$*5)

I ABD/ C2 ( ?)2 2(P+:&'$. +:*'$*5)

S%7%%& &%4*./ ,2 2 2? Y (2)

B6+ ?2 2 ;2 (1)

S' .6;.+%+6+%& % (2) '$ ;2&%4*. ,2 2 ;2 2?

I ADC/ ,'.(18 - ,) ?; / ;6+ ,'. (I8 - C)

S%,* C %. ';+6.* %. - ,'. C +:*$*'$*

-,'. C ?;

= -;,'.C ?

S' .6;.+%+6+%& '$ +:* 46* ' ? % (2) / '6 ';+%>

C2 2 ;2 2 ( - ; ,'. C)

C2 2 ;22- 2; ,'. C5

T:%. %. +:* .* $*.6+ . '$ +:* ,6+* &*0 +$%&*5 A0 +:%. %. +:* C'.%*

R6*>

C2 2 ;22- 2; ,'. C

A.' ; $*%& +:* &*. 0 '$ $*-0$9%& +:* 7*$7*0%,6$. +:*

''9%& ,'.%* $6*. , .' ;* 7$'4*0 (+$ %+)

2 ;2 ,2- 2? ,'.A

OR

;2 2 ,2 - 2, ,'. B

T:* ,'.%* $6* %. 77%*0 % +:* .'6+%' ' +$%&*. 9:* +:* ''9%& $*

&%4*>

(1) +9' .%0*. 0 %,60*0 &* '$

(2) +:$** .%0*.5

NOUN 183


104/131


E;>*:

15 S'4* +$%&* ABC/ 9%+:

() A 25 3/ ; #523 , 0 C !5",5

(;) B 1!8 53/ 51 , 0 , !52" ,5(,) C 58!/ 252!, 0 ; "52 ,

S+4'&+%:

N'+* 9. $**;*$ +'

A2 ;2 ,2- 2;, ,'. A 0 .6;.+%+6+%& ; #523 ,/

C !5", 0 A 253'

9:%,: 9*$* &%4* %+' +:* '$6 0 .%7%%& &%4*.>

A2 (#523)2 (!5")2 - 2 (#523)(!5") ,'. 253

!252#2 2511" - (15")(!5") (85#33) !252#2 2511" - !#581!

258! - !#581!

258! - !#581! 2513

2 2513

2513 51#"55 52,

NOUN 18


105/131


H*$* +:* 7'.%+%4* .@6$* $''+ ' 9. +


106/131


#15132

55 C ,'.-185"!!"!2!!

58"#

581'

S' +:* +9' *+:'0. () 0 (;) &4* +:* .* 46* '$ ;6+ %+ %. 6.6*.%*$ +' .6* +:* .%* $6* % %0%& +:* %..%& &*.5

T:* dB 18 - (e ,) (.6 . ')

18 - (253 581)

18 - 15 dB 51"/

+:*$*'$* +:* %..%& 7$+. $*>

52,/ dB 51" 0 581'

F&?: 8.$

U.%& +:* ,'.%* $6*

;2 2 ,2 - 2, ,'. B

0 .6;.+%+6+%& 46*. '>

51,/ ,!52",0dB

%+' +:* '$6/ 9* ';+%

;2 (51)2 (!52")2 - 2(51) (!52") ,'. 1!85

#531 2#5""#" - 35#332 ? ,'. (18 - 1!8)

#531 2#5""#" 35#332 ,'. 25# #531 2#5""#" 151"

;2 15#!3 0 ; 15#!3; 135#3 ,

N'+* ,'. 1!853 - ,'. 25# 0 9:* .6;.+%+6+*0 %+' +:* '$6 %+

;*,'*.

NOUN 18"


107/131


;2 2 ,2 2, ,'. B (9:*$* B %. ,6+*)

T' %0 &* A 6.%& +:* .%* $6*

.% A .% B ;

.% A .% 1!853 ,$'.. 6+%7

51 135#3

S%7%%& 135#3 .%A 51 .% 1!853 &%4*.

.% A 51.%1!853 85321!2#

135#3

.%A 85321!2#0 A .%-1 85321!2# 1 5#"

A&* B 9% +:* ;* *@6 +'>

18-(A ) (.6 ' 8)

B 18 - (15#" 1!853)

18 - 1"58"

185

55 ; 135 #3,/ A 15#" 0 dB 185

F&? 8."

S+4'&+%:

T:* +:$** .%0*. ' +:* +$%&* $* &%4* +' ';+% +:* &*./ +:* ,'.%*'$6 %. 6.*0 + +:6.>

C'. A ;2 ,2 - 2

2;,

NOUN 18#

; "52,

, 58!


108/131


.6;.+%+6+%& '$ 252!/ ; "52 0 , 58! %+' +:* ;'4* '$6/ 9*

';+%

C'. A ("-2)2 (58!)2 -(252!)2 0

2("52)(58!)

S%7%%& 353#" 1"5 - !58"2

!85!

SO ,'. A !!5381 - !58"2! !852##"

!85! ! 85!

55 ,'. A 85#23#

0 A ,'.-1(85#) !5

&% 6.%& ,'.%* $6* +' ';+% +:6.>

C'. A ;2 ,2 2 .6;.+%+6+%& +:* 46*. ' / ;/ , . ;'4* %+' +:*

'$6 &%4*.

2;,

C'. (252!)2 ("52)2 - (58!)2

2(252!)("52)

!58"2! 353#" - 1"582!

258

,'. C 85228!12

.' C ,'.-1

852228!12C 185"

B +:*

15 G%4* +:$** .%0*. ' +:* +$%&* . % *?7* 2(C)

NOUN 18


109/131


C'. A ;2 ,2 - 2

2;,

OR

C'. B 2 ,2 - ;2

2,

OR

C'. C 2 ;2 -,2

2;

C ;* 6.*0 +' %0 ' +:* +9' &*. 0 +:* +:%$0 &*. %. '60/ ; +:*

6.* ' +:* .6 ' +:* %+*$%'$ &*. ' +$%&* %5*5

ABC18

25 G%4* +9' &*. 0 '* .%0*

I *?7* 1()/ +9' &*. 0 '* .%0* %. &%4*/ +:* .%* $6* %. 6.*0 :*$*/

+' %0 +:* *&+: ' +:* .%0*. +:6.>

;*,6.* C , *.% ;* '60 ; 18 - "252! -2!5# 1 5

'$ , +' %0 +:* .%0* C5

.% A S% C

35 G%4* +9' .%0*. 0 %,60*0 &* % +:%. %&6$*/

NOUN 18

;

.% A .% B


110/131


B %. &%4* 9:%,: %*. ;*+9** +:* +9' &%4* .%0*. 0 ,5

T:* ,'.%* $6* %. 6.*0 % *?7* 2() ;'4* +' %0 +:* .%0* +' ; +:* +:*

.%* $6* %. 6.*0 +' %0 *%+:*$ A '$ C +:* +:* +:%$0 &* , ;* '60 ;

+:* .6 ' &*. ' +$%&* +:*'$*

/5 G%4* +9' .%0*. 0 ' - %,60*0 &*. +:+ %. +:* &* 0'*. '+ %*.

;*+9** +:* +9' &%4* .%0*.5

T9' ,.*. $* +$*+*0 :*$*5

C;* 1:

W:* +:* &%4* &* %. ,6+* +9' 7'..%;* +$%&*. , ;* 0$9 . ''9.>

F&?8.8

I +:* .%0* '77'.%+* +:* &%4* &*. B % +:%. ,.* %. *.. +: +:* '+:*$ &%4*

&* C %5*5 ; , . % F%& "5"5 ;'4*5 T:%. *. +:+ C 9% :4* +9'

46*.(,6+* 0 ';+6.*)

C;* II

W:* +:* .%0* '77'.%+* +:* &%4* &* %. &$*+*$ . % F%& "5# ;*'9/ '

'* +$%&* %. 7'..%;* (; = , 0 dB %. ,6+*)

F&?: 8.7

NOUN 118


111/131


I ;'+: ,.*. +:* .%* $6* %. 6.*0 +' %0 '* ' +:* &*. 0 +:* .%0* , ;*

'60 ; *%+:*$ +:* .%* '$ ,'.%* $6*

E;>*:

S'4* +:* +$%&* A BC 9%+:

C 185"/ , 252!, 0 "52,5

;2

S'6+%'

I +:* %&6$* ;'4*/ +:*$* $* +9' 7'..%;* +$%&*. .%,* , %5 *5

252! "52

6.%& +:* .%* $6* +' %0 A

.% A .% C

,

S% A .% C

,

"52.%185" 85!83"883"

252!

.% A 85!83"883"

A .%-1( 85!83"883")

A 38523

B6+ A %. *%+:*$ ,6+* '$ ';+6.*5

55 A 38523 '$ (18 - 38523)

NOUN 111

A2


112/131


38523 '$ 15##

.' 9:* A 38523

B 18 - 38523 -185" (.6 . ' )

18 - 85" 13531' 0 9:* A38523

B 18 - (15## 185") 18 - 1"8523 18 5 ##

+:* 6.%& +:* .%* $6* &% +' ';+% +:* .%0*. ; ; 0 ;2

;2 ,

.% B .% C 9:*$* B 13531

;2 - 252!

.% 13 531 .% 1185"

;2 252!.%3531 15""23""

.% 185" 8511!83#;2 58,

A.'/ %0%& ;9:* B 15##/ 6.%& +:* .%* $6*

;1 ,

.% B .% C (B 15##)

;1 , .% B

.% C

0 .6;.+%+6+%& +:* 46*. ' ,/ dB 0

;1 252! .% 15##

.% 185"

85#"18!1"#1 512

8511!83#

55 ; 5 12, 51, (2 0*,5 7,*.)

ABC :. *%+:*$

A 38523/ B 13531 0 ;2 58,

OR

A 15##/ B 15##0 ;1 51,5

T:* +9' 7'..%;* .9*$. $* +:*

NOUN 112

Documents

27101574 Mth 133 Trigonometry