1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k=

(1)

(2)

=

fp_k= =VVQVNMTTQN=

=

`çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë=oÉëÉêîÉÇK=

(3)

(4)

Preface

=

qÜáë= Ü~åÇÄççâ= áë= ~= ÅçãéäÉíÉ= ÇÉëâíçé= êÉÑÉêÉåÅÉ= Ñçê= ëíì- ÇÉåíë= ~åÇ= ÉåÖáåÉÉêëK= fí= Ü~ë= ÉîÉêóíÜáåÖ= Ñêçã= ÜáÖÜ= ëÅÜççä=

ã~íÜ=íç=ã~íÜ=Ñçê=~Çî~åÅÉÇ=ìåÇÉêÖê~Çì~íÉë=áå=ÉåÖáåÉÉêáåÖI=

ÉÅçåçãáÅëI=éÜóëáÅ~ä=ëÅáÉåÅÉëI=~åÇ=ã~íÜÉã~íáÅëK=qÜÉ=ÉÄççâ=

Åçåí~áåë= ÜìåÇêÉÇë= çÑ= Ñçêãìä~ëI= í~ÄäÉëI= ~åÇ= ÑáÖìêÉë= Ñêçã=

kìãÄÉê= pÉíëI= ^äÖÉÄê~I= dÉçãÉíêóI= qêáÖçåçãÉíêóI= j~íêáÅÉë=

~åÇ= aÉíÉêãáå~åíëI= sÉÅíçêëI= ^å~äóíáÅ= dÉçãÉíêóI= `~äÅìäìëI=

aáÑÑÉêÉåíá~ä=bèì~íáçåëI=pÉêáÉëI=~åÇ=mêçÄ~Äáäáíó=qÜÉçêóK==

qÜÉ= ëíêìÅíìêÉÇ= í~ÄäÉ= çÑ= ÅçåíÉåíëI= äáåâëI= ~åÇ= ä~óçìí= ã~âÉ=

ÑáåÇáåÖ= íÜÉ= êÉäÉî~åí= áåÑçêã~íáçå= èìáÅâ= ~åÇ= é~áåäÉëëI= ëç= áí=

Å~å=ÄÉ=ìëÉÇ=~ë=~å=ÉîÉêóÇ~ó=çåäáåÉ=êÉÑÉêÉåÅÉ=ÖìáÇÉK===

=

(5)

C h a p t e r 1

Number Sets

=

1.1 Set Identities

=

pÉíëW=^I=_I=`=

råáîÉêë~ä=ëÉíW=f=

`çãéäÉãÉåí=W=^′ = mêçéÉê=ëìÄëÉíW=^ ⊂ ==_ bãéíó=ëÉíW=∅=

råáçå=çÑ=ëÉíëW=^ ∪ =_ fåíÉêëÉÅíáçå=çÑ=ëÉíëW=^ ∩ =_ aáÑÑÉêÉåÅÉ=çÑ=ëÉíëW=^y_=

=

= 1. ^ ⊂ =f

=

2. ^ ⊂ =^

=

3. ^ = =áÑ=_ ^ ⊂ =~åÇ=_ _ ⊂ .=^

=

4. bãéíó=pÉí=

⊂^

∅ =

=

5. råáçå=çÑ=pÉíë==

{

ñöñ ^çêñ _

}

_

^

`= ∪ = ∈ ∈ =

=

(12)

===== =

=

Figure 1.

=

6. `çããìí~íáîáíó=

^ _ _

^∪ = ∪ =

=

7. ^ëëçÅá~íáîáíó=

(

_ `

) (

^ _

)

`

^∪ ∪ = ∪ ∪ =

=

8. fåíÉêëÉÅíáçå=çÑ=pÉíë=

{

ñöñ ^~åÇñ _

}

_

^

`= ∪ = ∈ ∈ = =

=

===== =

=

Figure 2.

=

9. `çããìí~íáîáíó=

^ _ _

^∩ = ∩ =

=

10. ^ëëçÅá~íáîáíó=

(

_ `

) (

^ _

)

`

^∩ ∩ = ∩ ∩ =

(13)

^∩ ∪ = ∩ ∪ ∩ K=

=

12. fÇÉãéçíÉåÅó=

^

^∩ = I==

^

^∪ = =

=

13. açãáå~íáçå=

∅

=

∅

∩

^ I=

f f

^∪ = =

=

14. fÇÉåíáíó=

^

^∪∅= I==

^ f

^∩ =

= 15. `çãéäÉãÉåí=

{

ñ föñ ^

}

^′= ∈ ∉

=

16. `çãéäÉãÉåí=çÑ=fåíÉêëÉÅíáçå=~åÇ=råáçå f

^

^∪ ′= I==

∅

′=

∩ ^

^ =

= 17. aÉ=jçêÖ~å∞ë=i~ïë

(

^∪_

)

′=^′∩_′I==

(

^∩_

)

′=^′∪_′=

=

18. aáÑÑÉêÉåÅÉ=çÑ=pÉíë

{

ñöñ _~åÇñ ^

}

^ y _

`= = ∈ ∉ =

=

(14)

===== =

=

Figure 3.

=

19. _y^=_y

(

^∩_

)

= 20. _y^=_∩^′

= 21. ^y^=∅

= 22. ^y_=^=áÑ=^∩ _=∅.

=

===== =

=

Figure 4.

=

23.

(

^y_

)

∩`=

(

^∩`

) (

y _∩`

)

24. ^ =′ fy^

25. `~êíÉëá~å=mêçÇìÅí

( )

{

ñIó öñ ^~åÇó _

}

_

^

`= × = ∈ ∈

=

(15)

1.2 Sets of Numbers

=

k~íìê~ä=åìãÄÉêëW=k=

tÜçäÉ=åìãÄÉêëW=k_M= fåíÉÖÉêëW=w=

mçëáíáîÉ=áåíÉÖÉêëW=w =⁺ kÉÖ~íáîÉ=áåíÉÖÉêëW=w =⁻ o~íáçå~ä=åìãÄÉêëW=n=

oÉ~ä=åìãÄÉêëW=o==

`çãéäÉñ=åìãÄÉêëW=`==

=

26. k~íìê~ä=kìãÄÉêë

`çìåíáåÖ=åìãÄÉêëWk =

{

NIOIPI_K

}

{

_KI PI OI NIMINIOIPI_K

}

w

w= ⁻∪ ∪ ⁺ = − − − K=

=

29. o~íáçå~ä=kìãÄÉêë

oÉéÉ~íáåÖ=çê=íÉêãáå~íáåÖ=ÇÉÅáã~äëW==







 = ∈ ∈ ≠

= ~åÇ ~ w ~åÇ Ä w ~åÇ Ä M Ä

ñ ~ ö ñ

n K=

=

30. fêê~íáçå~ä=kìãÄÉêë

kçåêÉéÉ~íáåÖ=~åÇ=åçåíÉêãáå~íáåÖ=ÇÉÅáã~äëK

=

(16)

31. oÉ~ä=kìãÄÉêë==

råáçå=çÑ=ê~íáçå~ä=~åÇ=áêê~íáçå~ä=åìãÄÉêëW=oK=

=

32. `çãéäÉñ=kìãÄÉêë

{

ñ áóöñ o ~åÇ ó o

}

`= + ∈ ∈ I==

ïÜÉêÉ=á=áë=íÜÉ=áã~Öáå~êó=ìåáíK

=

33. k⊂w⊂n⊂o⊂`=

=

=== =

=

Figure 5.

=

(17)

1.3 Basic Identities

=

oÉ~ä=åìãÄÉêëW=~I=ÄI=Å=

=

34. ^ÇÇáíáîÉ=fÇÉåíáíó=

~ M

~+ = =

= 35. ^ÇÇáíáîÉ=fåîÉêëÉ=

( )

~ M

~+ − = =

= 36. `çããìí~íáîÉ=çÑ=^ÇÇáíáçå=

~ Ä Ä

~+ = + =

= 37. ^ëëçÅá~íáîÉ=çÑ=^ÇÇáíáçå=

(

~+Ä

)

+Å=~+

(

Ä+Å

)

=

38. aÉÑáåáíáçå=çÑ=pìÄíê~Åíáçå=

( )

Ä

~ Ä

~− = + − =

= 39. jìäíáéäáÅ~íáîÉ=fÇÉåíáíó=

~ N

~ =⋅ =

=

40. jìäíáéäáÅ~íáîÉ=fåîÉêëÉ=

~ N

~⋅N= I=~ ≠ M

=

41. jìäíáéäáÅ~íáçå=qáãÉë=M M

M

~⋅ =

= 42. `çããìí~íáîÉ=çÑ=jìäíáéäáÅ~íáçå=

~ Ä Ä

~⋅ = ⋅

=

(18)

43. ^ëëçÅá~íáîÉ=çÑ=jìäíáéäáÅ~íáçå=

( )

~⋅Ä ⋅Å=~⋅

( )

Ä⋅Å

= 44. aáëíêáÄìíáîÉ=i~ï=

(

Ä Å

)

~Ä ~Å

~ + = + =

=

45. aÉÑáåáíáçå=çÑ=aáîáëáçå=

Ä

~ N Ä

~ = ⋅ =

=

1.4 Complex Numbers

=

k~íìê~ä=åìãÄÉêW=å=

fã~Öáå~êó=ìåáíW=á=

`çãéäÉñ=åìãÄÉêW=ò=

oÉ~ä=é~êíW=~I=Å=

fã~Öáå~êó=é~êíW=ÄáI=Çá=

jçÇìäìë=çÑ=~=ÅçãéäÉñ=åìãÄÉêW=êI=ê I=_N ê =_O

^êÖìãÉåí=çÑ=~=ÅçãéäÉñ=åìãÄÉêW= ϕ I=ϕ I=_N ϕ =O

=

= á

á^N= = á^R = =á á^Q^å⁺^N=á= N

á^O=− = á^S=−N= á^Q^å⁺^O=−N= á

á^P= =− á^T =−á= á^Q^å⁺^P=−á= 46.

N

á^Q = = á^U = =N á^Q^å = =N

=

47. ò=~+Äá=

=

48. `çãéäÉñ=mä~åÉ=

=

(19)

===== =

=

á=

=

52. á

Ç Å

~Ç ÄÅ Ç Å

ÄÇ

~Å Çá Å

Äá

~

O O O

O ⋅

+ + − +

= + +

+ =

=

53. `çåàìÖ~íÉ=`çãéäÉñ=kìãÄÉêë=

Äá

~ Äá

~^|||||||+ = − =

=

54. ~= Åçëê ϕI=Ä= ëáåê ϕ==

=

(20)

= =

Figure 7.

=

55. mçä~ê=mêÉëÉåí~íáçå=çÑ=`çãéäÉñ=kìãÄÉêë=

(

ϕ+ ϕ

)

=

+Äá ê Åçë áëáå

~ =

=

56. jçÇìäìë=~åÇ=^êÖìãÉåí=çÑ=~=`çãéäÉñ=kìãÄÉê=

fÑ=~ +Äá=áë=~=ÅçãéäÉñ=åìãÄÉêI=íÜÉå=

O

O Ä

~

ê= + =EãçÇìäìëFI==

~

~êÅí~åÄ

=

ϕ =E~êÖìãÉåíFK=

=

57. mêçÇìÅí=áå=mçä~ê=oÉéêÉëÉåí~íáçå=

(

_N _N

) (

_O _O _O

)

N O

N ò ê Åçë áëáå ê Åçë áëáå

ò ⋅ = ϕ + ϕ ⋅ ϕ + ϕ =

( ) ( )

[

_N _O _N _O

]

O

Nê Åçë áëáå

ê ϕ +ϕ + ϕ +ϕ

= =

=

58. `çåàìÖ~íÉ=kìãÄÉêë=áå=mçä~ê=oÉéêÉëÉåí~íáçå=

(

Åçëϕ+áëáåϕ

)

=ê

[

Åçë

( )

−ϕ +áëáå

( )

−ϕ

]

ê ^||||||||||^|

||||||||||

=

59. fåîÉêëÉ=çÑ=~=`çãéäÉñ=kìãÄÉê=áå=mçä~ê=oÉéêÉëÉåí~íáçå=

( )

=

[ ( )

−ϕ +

( )

−ϕ

]

ϕ +

ϕ Åçë áëáå

ê N ëáå á Åçë ê

N =

(21)

60. nìçíáÉåí=áå=mçä~ê=oÉéêÉëÉåí~íáçå=

( )

( ) [ (

_N _O

) (

_N _O

) ]

O N

O O

O

N N

N

O

N Åçë áëáå

ê ê ëáå

á Åçë ê

ëáå á Åçë ê ò

ò = ϕ −ϕ + ϕ −ϕ

ϕ + ϕ

ϕ +

= ϕ =

=

61. mçïÉê=çÑ=~=`çãéäÉñ=kìãÄÉê=

( )

[

ϕ+ ϕ

]

=

[ ( )

ϕ +

( )

ϕ

]

= ê Åçë áëáå ê Åçë å áëáåå

ò^å ^å ^å =

=





 



 ϕ+ π + ϕ+ π

= ϕ + ϕ

= å

â ëáå O

å á â Åçë O

ê ëáå

á Åçë ê

ò ^å ^å

å I==

ïÜÉêÉ==

N å I I O I N I M

â= K − K==

=

64. bìäÉê∞ë=cçêãìä~=

ñ ëáå á ñ Åçë

É^áñ = + =

=

(22)

C h a p t e r 2

Algebra

=

2.1 Factoring Formulas

=

oÉ~ä=åìãÄÉêëW=~I=ÄI=Å==

k~íìê~ä=åìãÄÉêW=å=

=

65. ~^O−Ä^O=

(

~+Ä

)(

~−Ä

)

(23)

( ) (

^å ^N ^å Ô ^å ^P Ô ^å Ô ^å ^N

)

å

å Ä ~ Ä ~ ~ Ä ~ Ä ~Ä Ä

~ + = + ⁻ − ⁻ + ⁻ −K+ ⁻ − ⁻ K=

=

2.2 Product Formulas

oÉ~ä=åìãÄÉêëW=~I=ÄI=Å==

^å= ^å`_M~^å+ ^å`_N~^å⁻^NÄ+^å`_O~^å⁻^OÄ^O+K+^å`_å₋_N~Ä^å⁻^N+^å`_åÄ^åI ïÜÉêÉ=

(

å â

)

>

â

>

`_â å

å

= − =~êÉ=íÜÉ=Äáåçãá~ä=ÅçÉÑÑáÅáÉåíëK=

=

80.

(

~+Ä+Å

)

Ô=~Ô+ÄÔ+ÅÔ+O~Ä+O~Å+OÄÅ=

=

81.

(

~+Ä+Å+K+ì+î

)

Ô=~Ô+ÄÔ+ÅÔ+K+ìÔ+îÔ+=

(

~Ä ~Å ~ì ~î ÄÅ Äì Äî ìî

)

O + + + + + + + + + +

+ K K K =

(24)

2.3 Powers

=

_~ëÉë=EéçëáíáîÉ=êÉ~ä=åìãÄÉêëFW=~I=Ä==

mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã=

=

82. ~^ã~^å=~^ã⁺^å=

=

83. _å ^ã ^å

ã

~ ~

~ = − =

=

84.

( )

~Ä ^ã=~^ãÄ^ã=

=

85. _ã

ã ã

Ä

~ Ä

~ =



 



 =

=

86.

( )

^~^ã ^å⁼^~^ãå⁼

=

87. ~^M = I=N ~ ≠ =M

=

88. ~^N= =N

=

89. ^ã _ã

~

~⁻ = N =

=

90. ^å ^å ^ã

ã

~

~ = =

=

(25)

2.4 Roots

=

_~ëÉëW=~I=Ä==

mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã=

M Ä I

~ ≥ =Ñçê=ÉîÉå=êççíë=Eå =OâI=â ∈kF=

=

91. ^å ~Ä =^å ~^å Ä=

=

92. ^å ~ ^ãÄ=^åã~^ãÄ^å =

=

93. ^å _å^å Ä

~

Ä~ = I=Ä ≠M=

=

94. ^åã _å

ã

åã å

åã ã

ã å

Ä

~ Ä

~ = = I=Ä ≠MK=

=

95.

( )

^å ^~^ã ^é⁼^å ^~^ãé ⁼

=

96.

( )

^å^~ ^å^{= =}^~

=

97. ^å ~ =^ã ^åé~^ãé =

=

98. ^å

ã å ~ =ã ~ =

=

99. ^{ã å} ~ =^ãå~=

=

100.

( )

^å^~ ^ã⁼^å ^~^ã ⁼

=

(26)

101.

~

N ^å ^å ^N

å

= − I=~ ≠ K=M

= 102.

O Ä

~

~ O

Ä

~ Ä ~

~

O

O− ± − −

= +

± =

= 103.

Ä

~ Ä

~ N

= −

±

m =

=

2.5 Logarithms

=

mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=ñI=óI=~I=ÅI=â=

k~íìê~ä=åìãÄÉêW=å==

=

104. aÉÑáåáíáçå=çÑ=içÖ~êáíÜã=

ñ äçÖ

ó= _~ =áÑ=~åÇ=çåäó=áÑ=ñ =~^óI=~ > I=M ~ ≠ K=N

=

105. äçÖ_~N=M=

=

106. äçÖ_~~=N=

= 107.





<

∞ +

>

∞

= −

N

~ áÑ

N

~ M áÑ

äçÖ_~ =

=

108. äçÖ_~

( )

ñó =äçÖ_~ñ+äçÖ_~ó=

=

109. ñ äçÖ ñ äçÖ ó

äçÖ = − =

(27)

110. ^äçÖ

( )

^ñ ^å^äçÖ~^ñ å

~ = =

=

111. äçÖ ñ

å ñ N

äçÖ_~^å = _~ =

=

112. äçÖ ñ äçÖ Å

~ äçÖ

ñ ñ äçÖ

äçÖ _Å _~

Å Å

~ = = ⋅ I=Å > I=M Å ≠ K=N

= 113.

~ äçÖ Å N äçÖ

Å

~ = =

=

114. ñ =~^äçÖ^~^ñ=

=

115. içÖ~êáíÜã=íç=_~ëÉ=NM=

ñ äçÖ ñ

äçÖ_NM = =

=

116. k~íìê~ä=içÖ~êáíÜã=

ñ äå ñ

äçÖ_É = I==

ïÜÉêÉ= OKTNUOUNUOUK

â N N äáã É

â

â  =



 

 +

= →∞ =

=

117. äåñ MKQPQOVQäåñ NM

äå ñ N

äçÖ = = =

=

118. äçÖñ OKPMORURäçÖñ É

äçÖ ñ N

äå = = =

=

(28)

2.6 Equations

=

oÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI=éI=èI=ìI=î=

pçäìíáçåëW=ñ I=_N ñ I=_O ó I=_N ó I=_O ó_P=

=

119. iáåÉ~ê=bèì~íáçå=áå=låÉ=s~êá~ÄäÉ=

M Ä

~ñ+ = I=

~ ñ −= ÄK==

=

120. nì~Çê~íáÅ=bèì~íáçå=

M Å Äñ

~ñ^O+ + = I=

~ O

~Å Q Ä ñ Ä

O O

I N

−

±

=− K=

=

121. aáëÅêáãáå~åí=

~Å Q Ä

a= ^O− =

=

122. sáÉíÉ∞ë=cçêãìä~ë=

fÑ=ñ^O+éñ+è=MI=íÜÉå==





=

−

= +

è ñ ñ

é ñ ñ

O N

O

N K=

=

123. ~ñ^O+Äñ=MI=ñ_N= I=M

~ ñ_O =−Ä K=

=

124. ~ñ^O+Å=MI=

~ ñ_N_I_O=± −Å K=

=

125. `ìÄáÅ=bèì~íáçåK=`~êÇ~åç∞ë=cçêãìä~K==

M è éó

ó^P+ + = I==

(29)

î ì

ó_N= + I=

( ) (

ì î

)

á

O î P O ì

ó_O_I_P=−N + ± + I==

ïÜÉêÉ==

P

O O

P é O

è O

ì è 



 

 +



 

 + 

−

= I= ^P

O O

P é O

è O

î è 



 

 +



 



− 

−

= K==

=

2.7 Inequalities

s~êá~ÄäÉëW=ñI=óI=ò=

oÉ~ä=åìãÄÉêëW=





å P O

NI~ I~ I I~

~ Ç I Å I Ä I

~

K I=ãI=å=

aÉíÉêãáå~åíëW=aI=a I=_ñ a I=_ó a ==_ò

=

126. fåÉèì~äáíáÉëI=fåíÉêî~ä=kçí~íáçåë=~åÇ=dê~éÜë==

=

fåÉèì~äáíó= fåíÉêî~ä=kçí~íáçå= dê~éÜ=

Ä ñ

~≤ ≤ =

[ ]

~IÄ = Ä =

ñ

~< ≤ =

(

~IÄ

]

= Ä =

ñ

~≤ < =

[

~IÄ

)

= Ä =

ñ

~< < =

( )

~IÄ = Ä =

<ñ ≤

∞

− I=

Ä ñ ≤ =

(

⁻^∞^I^Ä

]

⁼

= Ä

=

(30)

127. fÑ=~ >ÄI=íÜÉå=Ä <~K=

=

128. fÑ=~ >ÄI=íÜÉå=~−Ä>M=çê=Ä−~<MK=

=

129. fÑ=~ >ÄI=íÜÉå=~+Å>Ä+ÅK=

=

130. fÑ=~ >ÄI=íÜÉå=~−Å>Ä−ÅK=

=

131. fÑ=~ >Ä=~åÇ=Å >ÇI=íÜÉå=~+Å>Ä+ÇK=

=

132. fÑ=~ >Ä=~åÇ=Å >ÇI=íÜÉå=~−Ç>Ä−ÅK=

=

133. fÑ=~ >Ä=~åÇ=ã > I=íÜÉå=M ã~ >ãÄK=

=

134. fÑ=~ >Ä=~åÇ=ã > I=íÜÉå=M

ã Ä ã~ > K=

=

135. fÑ=~ >Ä=~åÇ=ã < I=íÜÉå=M ã~ <ãÄK=

=

136. fÑ=~ >Ä=~åÇ=ã < I=íÜÉå=M

ã Ä ã~ < K=

=

137. fÑ=M<~<Ä=~åÇ=å > I=íÜÉå=M ~ <^å Ä^åK=

=

138. fÑ=M<~<Ä=~åÇ=å < I=íÜÉå=M ~ >^å Ä^åK=

=

139. fÑ=M<~<ÄI=íÜÉå=^å ~ <^å ÄK=

= 140.

O Ä

~Ä ~+

≤ I==

ïÜÉêÉ=~ > =I=M Ä >MX=~å=Éèì~äáíó=áë=î~äáÇ=çåäó=áÑ=~ =ÄK==

=

141. N O

~+ ≥ I=ïÜÉêÉ=~ > X=~å=Éèì~äáíó=í~âÉë=éä~ÅÉ=çåäó=~í=M ~ = K=N

(31)

142.

å

~

~ ~

~

~ ^N ^O ^å

å N O å

+ +

≤ + K

K I=ïÜÉêÉ=~_NI~_OIKI~_å>MK=

=

143. fÑ=~ñ+Ä>M=~åÇ=~ > I=íÜÉå=M

~ ñ −> Ä K=

=

144. fÑ=~ñ+Ä>M=~åÇ=~ < I=íÜÉå=M

~ ñ −< ÄK==

=

145. ~ñ^O+Äñ+Å>M=

=

= ~ > =M ~ < =M

=

= M a > =

=

= ñN

ñ < I=ñ >ñ_O=

=

= =

O

N ñ ñ

ñ < < =

=

= M a = =

=

ñ

ñ_N< I=ñ >ñ_N=

= =

∅

∈

ñ =

=

= M a < =

=

∞

<

∞

− ñ =

= =

=∅

∈

ñ =

=

(32)

146. ~+Ä ≤ ~ + Ä =

=

147. fÑ=ñ < I=íÜÉå=~ −~<ñ<~I=ïÜÉêÉ=~ > K=M

=

148. fÑ=ñ > I=íÜÉå=~ ñ −< ~=~åÇ=ñ >~I=ïÜÉêÉ=~ > K=M

=

149. fÑ=ñ^O< I=íÜÉå=~ ñ < ~I=ïÜÉêÉ=~ > K=M

=

150. fÑ=ñ^O > I=íÜÉå=~ ñ > ~ I=ïÜÉêÉ=~ > K=M

=

151. fÑ=

( ) ( )

ñ ^M Ö

ñ

Ñ > I=íÜÉå=

( ) ( )



( )



≠

>

⋅ M ñ Ö

M ñ Ö ñ

Ñ K=

=

152.

( ) ( )

ñ ^M Ö

ñ

Ñ < I=íÜÉå=

( ) ( )



( )



≠

<

⋅ M ñ Ö

M ñ Ö ñ

Ñ K=

=

2.8 Compound Interest Formulas

=

cìíìêÉ=î~äìÉW=^=

fåáíá~ä=ÇÉéçëáíW=`=

^ååì~ä=ê~íÉ=çÑ=áåíÉêÉëíW=ê=

kìãÄÉê=çÑ=óÉ~êë=áåîÉëíÉÇW=í=

kìãÄÉê=çÑ=íáãÉë=ÅçãéçìåÇÉÇ=éÉê=óÉ~êW=å=

=

153. dÉåÉê~ä=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~=

åí

å N ê

`

^ 



 

 +

= =

(33)

154. páãéäáÑáÉÇ=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~=

fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=çåÅÉ=éÉê=óÉ~êI=íÜÉå=íÜÉ=éêÉîáçìë=

Ñçêãìä~=ëáãéäáÑáÉë=íçW=

(

N ê

)

^í

`

^= + K=

=

155. `çåíáåìçìë=`çãéçìåÇ=fåíÉêÉëí=

fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=Åçåíáåì~ääó=Eå→∞FI=íÜÉå==

`Éêí

^ = K=

=

(34)

C h a p t e r 3

Geometry

=

3.1 Right Triangle

=

iÉÖë=çÑ=~=êáÖÜí=íêá~åÖäÉW=~I=Ä=

eóéçíÉåìëÉW=Å=

^äíáíìÇÉW=Ü=

jÉÇá~åëW=ã_~I=ã_ÄI=ã_Å=

^åÖäÉëW=αI β =

o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=

o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=

^êÉ~W=p=

=

= =

Figure 8.

=

156. α+β=VM =°

=

(35)

157. α= =Åçëβ Å

ëáå ~ =

=

158. α= =ëáåβ Å

Åçë Ä =

= 159. α= =Åçíβ

Ä

í~å ~ =

= 160. α= =í~åβ

~

Åçí Ä =

= 161. α= =ÅçëÉÅβ

Ä

ëÉÅ Å =

= 162. α= =ëÉÅβ

~ ÉÅ Å

Åçë =

= 163. móíÜ~ÖçêÉ~å=qÜÉçêÉã=

O O

O Ä Å

~ + = =

= 164. ~^O = I=ÑÅ Ä^O =ÖÅI==

ïÜÉêÉ= Ñ= ~åÇ= Å= ~êÉ= éêçàÉÅíáçåë= çÑ= íÜÉ= äÉÖë= ~= ~åÇ= ÄI= êÉëéÉÅ- íáîÉäóI=çåíç=íÜÉ=ÜóéçíÉåìëÉ=ÅK=

=

===== =

=

Figure 9.

=

(36)

165. Ü^O= I===ÑÖ

ïÜÉêÉ=Ü=áë=íÜÉ=~äíáíìÇÉ=Ñêçã=íÜÉ=êáÖÜí=~åÖäÉK==

= 166.

Q Ä ~ ã

O O O

~= − I=

Q

~ Ä ã

O O O

Ä= − I===

ïÜÉêÉ=ã_~=~åÇ=ã_Ä=~êÉ=íÜÉ=ãÉÇá~åë=íç=íÜÉ=äÉÖë=~=~åÇ=ÄK==

=

= =

Figure 10.

= 167.

O ã_Å = I==Å

ïÜÉêÉ=ã_Å=áë=íÜÉ=ãÉÇá~å=íç=íÜÉ=ÜóéçíÉåìëÉ=ÅK=

=

168. ã_Å

O o=Å = =

= 169.

Å Ä

~

~Ä O

Å Ä ê ~

+

= +

−

= + =

= 170. ~Ä =ÅÜ=

=

(37)

171.

O ÅÜ O p=~Ä= =

=

3.2 Isosceles Triangle

=

_~ëÉW=~=

iÉÖëW=Ä=

_~ëÉ=~åÖäÉW=β = sÉêíÉñ=~åÖäÉW=α=

^äíáíìÇÉ=íç=íÜÉ=Ä~ëÉW=Ü=

mÉêáãÉíÉêW=i=

^êÉ~W=p=

=

= =

Figure 11.

= 172.

VM αO

−

°

=

β =

= 173.

Q Ä ~ Ü

O O

O= − =

(38)

174. i=~+OÄ=

= 175. = = ëáåα

O Ä O p ~Ü

O

=

3.3 Equilateral Triangle

=

páÇÉ=çÑ=~=Éèìáä~íÉê~ä=íêá~åÖäÉW=~=

^äíáíìÇÉW=Ü=

o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=

mÉêáãÉíÉêW=i=

^êÉ~W=p=

=

= =

Figure 12.

= 176.

O P Ü =~ =

=

(39)

177.

P P Ü ~ P

o=O = =

= 178.

O o S

P Ü ~ P

ê=N = = =

= 179. i =P~=

= 180.

Q P

~ O p ~Ü

= O

= =

=

3.4 Scalene Triangle

E^=íêá~åÖäÉ=ïáíÜ=åç=íïç=ëáÇÉë=Éèì~äF=

=

páÇÉë=çÑ=~=íêá~åÖäÉW=~I=ÄI=Å=

pÉãáéÉêáãÉíÉêW=

O Å Ä é=~+ + ==

^åÖäÉë=çÑ=~=íêá~åÖäÉW=α IIβ γ=

^äíáíìÇÉë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW=Ü_~IÜ_ÄIÜ_Å= jÉÇá~åë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW=ã_~Iã_ÄIã_Å= _áëÉÅíçêë=çÑ=íÜÉ=~åÖäÉë=α IIβ γW=í_~Ií_ÄIí_Å= o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=

^êÉ~W=p=

=

(40)

===== =

=

Figure 13.

=

181. α+β+γ=NUM =°

= 182. ~+Ä>ÅI==

~ Å Ä+ > I==

Ä Å

~+ > K=

=

183. ~−Ä <ÅI==

~ Å Ä− < I==

Ä Å

~− < K=

= 184. jáÇäáåÉ=

O

è = I=~ èöö~K=

=

===== =

=

Figure 14.

(41)

185. i~ï=çÑ=`çëáåÉë=

α

− +

=Ä Å OÄÅÅçë

~Ô Ô Ô I=

β

− +

=~ Å O~ÅÅçë

ÄÔ Ô Ô I=

γ

− +

=~ Ä O~ÄÅçë

ÅÔ Ô Ô K=

=

186. i~ï=çÑ=páåÉë=

o ëáå O

Å ëáå

Ä ëáå

~ =

= γ

= β

α I==

ïÜÉêÉ=o=áë=íÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉK==

= 187.

p Q

~ÄÅ Ü

O

~Ä Ü

O

~Å Ü O

ÄÅ ëáå

O Å ëáå

O Ä ëáå

O o ~

Å Ä

~

=

= γ =

β= α =

= =

=

188.

( )( )( )

é

Å é Ä é

~

ê^O= é− − − I==

Å Ä

~ Ü

N Ü

N ê

N= + + K=

=

189.

( )( )

ÄÅ Å é Ä é

ëáå αO = − − I=

( )

ÄÅ

~ é é

Åçë O −

α=

I=

( )( )

(

é ~

)

é

Å é Ä é í~å O

é Å

)

Å é

Ü_Å =O − − − K=

(42)

191. Ü_~=Äëáåγ=ÅëáåβI=

α

= γ

=~ëáå Åëáå

Ü_Ä I=

α

= β

=~ëáå Äëáå

Ü_Å K=

= 192.

Q

~ O

Å ã Ä

O O O O

~= + − I==

Q Ä O

Å ã ~

O O O O

Ä + −

= I==

Q Å O

Ä ã ~

O O O O

Å = + − K=

=

===== =

=

Figure 15.

=

193. ã_~

P

^j =O I= ã_Ä

P

_j =O I= ã_Å

P

`j =O =EcáÖKNRFK=

=

194.

( )

^O

O

~ Ä Å

~ é ÄÅé í Q

+

= − I==

( )

^O

O

Ä ~ Å

Ä é

~Åé í Q

+

= − I==

( )

^O

O

Å ~ Ä

Å é

~Äé í Q

+

= − K=

1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k=

Preface

Contents

C h a p t e r 1

Number Sets

1.1 Set Identities

{

}

(

) (

)

{

}

(

) (

)

(

) (

) (

)

(

) (

) (

)

{

}

(

)

(

)

{

}

(

)

(

)

(

) (

)

( )

{

}

1.2 Sets of Numbers

{

}

{

}

{

}

{

}

{ }

{

}

{

}

1.3 Basic Identities

( )

(

)

(

)

( )

( )

( )

(

)

1.4 Complex Numbers

(

) (

) (

) (

)

(

) (

) (

) (

)

(

)(

⁽

⁾⁽

⁾ (