Arithmetics various set of numbers
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Inha-math-info-book
- Bu sahifa navigatsiya:
- Reduced quadratic equation
- Properties of inequalities
- Quadratic inequalities
- Infinitely decreasing geometric progression
- FUNCTIONS
- Absolute value functions
- Definition of inverse function
- Inverse trigonometric functions
- PLANE GEOMETRY
- Inscribed and circumscribed circles
- Main theorems in triangles
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Properties of roots:
c x x a b x x 2 1 2 1 , ; 2 2 2 1 2 2 1 2 2 2 1 2 2 a ac b x x x x x x ; 2 3 2 1 2 1 3 2 1 3 2 3 1 3 ) ( 3 a bc a b x x x x x x x x ; 3 3 3 2 3 1 2 2 2 2 2 1 3 1 1
; 2 1 1 c abc b x x c ac b x x ; Reduced quadratic equation 0 2 q px x
q x x p x x 2 1 2 1
; .
0 2
c bx ax x If х 1 , х 2 , х 3 are roots of the equation, then
) ( 3 2 1
x x a
2 3 1 2 1
x x x x x b
3 2 1 x x x c Biquadrate
0 2 4
bx ax
Sum of roots is equal to zero; The quotient of the greatest solutions for the lowest is -1. 11
System of equations with two unknown variables
General form: 2 22 21 1 12 11 b y a x a b y a x a
If 2 1 22 12 21 11 b b a a a a , then the system has no roots. If 22 12 21 11 a a a a , then the system has unique root. If
2 1 22 12 21 11 b b a a a a , then the system has infinitely many roots. Unique solution 1) system
2 22 21 1 12 11
y a x a b y a x a at
22 12 2 1 a a b b has unique root. 2) system 2 22 21 1 12 11 b y a x a b y a x a at
21 11 2 1 a a b b has unique root. Properties of inequalities 1) If
) ( ) ( x g x f , then at 0
)
) (
cg x cf , and at 0
)
) (
cg x cf . 2) If ) ( ) (
g x f
) ( ) ( x g x f , then ) ( ) ( 2 2 x g x f n n
) ( ) ( 2 2
g x f n n . 3) If c x f ) (
c x f ) ( , then
c x f c ) ( c x f c ) ( . 4) The arithmetic mean is not less than the geometric mean
b a 2 . 5) 0 n
1 1 2 2
n n . 6) 0
1 1 n n .
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Non-strict 0 2 c bx ax (
0 2
bx ax ) 1) if 0 , 0
a , then
; ; 2 1 x x (
2 1 ; x x x ); 2) if 0 , 0 D a , then
; x
1
x ; 3) if 0 , 0 D a , then
; x
x ; 4) if 0 , 0
a , then
2 1 ; x x x (
; ; 2 1 x x );
5) if 0 , 0 D a , then
1 x x ( ; x );
6) if 0 , 0 D a , then
(
;
).
0 2 c bx ax (
0 2
bx ax ) 1) if 0 , 0
a , then
; ; 2 1 x x ( 2 1 ; x x x ); 2) if 0 , 0 D a , then
1 x x
x ; 3) if 0 , 0
a , then
; x (
); 4) if
0 , 0
a , then
2 1 ; x x x (
; ; 2 1 x x );
5) if 0 , 0 D a , then
( 1
x ); 6) if 0 , 0 D a , then
(
;
).
1) difference: p n a a a a d p n n n 1 ; 2) n th term:
n a a n 1 1 , md a a m n n ; 3) middle term: 2 1 2 1 1 ... 2 2
n n mt n a a a a a a , 2
n k mt n a a a a ; 4)
l k n m a a a a l k m n , ; 5) Sum of first n term:
1 1 2 1 2 2
n mt a n d a a S n a n n
6) kd n S S S k n k n ; 7)
m S S n m n m S n m n m , ; 8) ) 1 ( k dn S S n k n , here
k n S is sum of numbers from n up to k ;
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1) denominator:
1 ; 2) n th term: 1 1
n q b b ,
m n n q b b ; 3) middle term: 1 2 1 2 1 ... mt n n n n b b b b b b
,
n n k n k b b b b ; 4) l k n m b b b b l k m n , ; 5) Sum of first n term: 1 1 1 1 1
q b q b q b S n n n
6) ) ( 2 m n n m S S S ; 7) 1 k n k n q S S , here
k n S is a sum of numbers from n up to k ;
q b S 1 1 , 1
.
1)
2 ) 1 ( ...
3 2 1 n n n ;
2)
2 ) 1 2 ( ... 5 3 1 n n ; 3) ) 1 ( 2 ...
6 4 2 n n n ;
4) 1 ) 1 ( 1 ... 3 2 1 2 1 1 n n n n ; 5) 1 2 ) 1 2 ( ) 1 2 ( 1 ... 5 3 1 3 1 1 n n n n ; 6) 6 ) 1 2 )( 1 ( ...
3 2 1 2 2 2 2
n n n ; 7) 2 3 3 3 3 2 ) 1 ( ... 3 2 1 n n n .
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N x a log
0 , 1 , 0
a a . 1) 0 1 log a ; 1 log
a ; 2) if 1 0
, 1
a or
1 , 1 0 b a , then
0 log
b a ; 3) if 1 , 1
a or
1 0 , 1 0 b a , then
0 log
b a ; 4) if 0 , 1 c b a , then
c b a a log
log ; 5) if c b a , 1 0 , then
c b a a log
log ; 6) 2 1 2 1 log
log log
N N N N a a a ; 7) 2 1 2 1 log
log log
N N N N a a a ; 8)
b n m b a m a n log
log ; 9) a b b c c a log
log log
;
y y c b a x b a log
log log
log ...
; 10) 1 2 2 1 log
log log
log N N N N b a b a ; 11) N a N a log ; a c b b c a log
log ; 12) x x lg log 10 is decimal logarithm; 13) x x e ln log is natural logarithm; 14) Let 1 a b . If
1 0 p , then
p p b a log
log , if 1
, then
log
log
15) Let 1 b a . If
1
, then
log
log , if 1 0 p , then
p p b a log
log
16) Let 0 b a . If
1 0 p , then
b a p p log
log , if 1
, then
log
log
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1) if
n x f y 2 ) ( , then the domain of a function 0 ) ( x f . 2) if ) ( ) ( x f x g y , then the domain of a function 0 ) ( x f . 3) if log
( ) g x y f x , then the domain of a function 1 ) ( 0 ) ( 0 ) ( x g x g x f .
4) if ) ( arcsin x f y or ) ( arccos x f y , then the domain of a function 1 ) ( x f . 5) if tan( ) y x , then the domain of a function Z n n x , 2 . 6) if cot( )
y x , then the domain of a function Z n n x , . Range of function 1) in
c bx ax y 2 if 0
, then the range of function – ; ) ( 0 y y E , if
0
, then
0 ; 0 ) (
y E . Here a b ac y 4 4 2 0 . 2) if x a y , then the range of function ; 0 ) ( y E . 3) if kx b kx a y sin
cos , then the range of function 2 2 2 2 ; ) ( b a b a y E . Absolute value functions
0
, 0
, x x x x x
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x y
a x y
a x y
a x y
a x y Quadratic function
General form:
0 2 0 2 ) (
x x a c bx ax y ; Domain of function:
; ) ( y D ; Range of function: ; ) ( 0 0 y y E a ,
0 ; ) ( 0 y y E a ; Vertex of the parabola: a b ac y a b x 4 4 , 2 2 0 0 ; The axis of symmetry: a b x 2 .
General form: x a y
1 , 0 a a ; Domain of function:
; ) ( y D ; Range of function: ; 0 ) ( y E ; At 1 a it increases; At
1 0 a it decreases; Graph always passes through (0;1). 17
Logarithmic function
General form: x y a log
, 0 , 1 , 0 x a a ; Domain of function: ; 0 ) ( y D ; Range of function: ; ) ( y E : At 1 a it increases; At
1 0 a it decreases; Graph always passes through (1;0). Definition of inverse function In order to find inverse function of ) (x f y : 1) We solve equation regarding the variable x and find x ; 2) In the obtained presentation we replace x into
y ,
into
. 3) We need to consider domain of the original function to write inverse function in final form. For example: Let us find inverse function of 3 1 2 x y , 1 x . 1) 1 3 2 3 2 1 3 1 2
x y x y x ; 2) y x
1 3 2 x y .
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1)
arcsin
: funct.dom.: 1 ; 1 x , funct.range:
2 ; 2 y
2) x y arccos
: funct.dom.: 1 ; 1 x , funct.range:
0
3)
arctan y x : funct.dom.: R x , funct.range: 2 ; 2 y
4) cot y arc x : funct.dom.: R x , funct.range: ; 0
.
1) Measure: 180
1 rad 57 17 45
,
1 rad 0,017453 rad 180
; 2) Type: acute: 90 0 right: 90
obtuse:
180
90
straight –
180
, are adjacent angles, 2 1 , and 2 1 , are vertical angles
180
2 1 2 1
;
corresponding: 1,5; 2,6; 3,7; 4,8; internal cross lying: 3,5; 4,6; external cross lying: 1,7; 2,8; adjacent: 3,6; 4,5; 19
external adjacent: 1,8; 2,7;
180 Triangle 1 1 1 , , are external angles, , , are inner angles. 1)
180 ; 2)
360
1 1 1 ;
3) 1 1 1 ;
; b c a a c b c b a
;
; ; 4) b c a a c b c b a
;
; ;
Altitude is the length of the perpendicular from the vertex to the opposite side or on its continuation. 1)
cSin bSin a S h a 2 ;
aSin b S h b 2 ;
aSin c S h c 2 ; 2) ab ac bc c b a h h h c b a : : 1 : 1 : 1 : : ; 3)
c b a h h h r 1 1 1 1 , r is the radius of the inscribed circle;
a h h h 2 3 3 2 1
20
mn pq xy
2 2 2 2 2 2 m q y p n x Median The median is a segment connecting the vertex to the middle of the opposite side. 1)
c b a c b m AA a 2 2 1 2 2 1 2 2 2 2 2 1 ; acCos c a b c a m BB b 2 2 1 2 2 1 2 2 2 2 2 1 ; abCos b a c b a m CC c 2 2 1 2 2 1 2 2 2 2 2 1 ; 2) 2 2 2 2 2 2 4 3
b a m m m c b a ; 3)
2 2 2 2 3 2
c b m m m a ;
2 2 2 2 3 2
c a m m m b ;
2 2 2 2 3 2
b a m m m c ;
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4) Coordinates of the point of intersection of the medians: ) ; ( ), ; ( ), ; ( 3 3 2 2 1 1 y x C y x B y x A
) ; (
x O
3 , 3 3 2 1 3 2 1 y y y y x x x x ;
c b m AP m CQ m BD , , , BD OE 6 1
EOQ EOP S S S 24 1
ABC BEP BQE S S S 8 1
5) a c b x 2 2 2 Bisector Bisector is a segment of the corresponding angles’ bisector lying inside the triangle. 1)
2 1
c b a ; 2 1
c a b ;
b S S 2 1 ;
1 2 ; 2) c b bcCos c b a c b a bc c b 2 2 1 ;
c a acCos c b a c b a ac c a 2 2 1 ; 22
b a abCos c b a c b a ab b a 2 2 1 ; 3) The angle between the bisectors of adjacent angles equal 90° ;
1
2 b a ab x , x is a bisector; dc yb a c d b y ,
2
OP OB OD OA PC AP BC AC , , ; Inscribed and circumscribed circles 1) The center of the inscribed circle is at the point of intersection of the bisectors of the inner angles; 2) The center of the circumscribed circle is located at the intersection of the average perpendicular of sides; 3)
p c p b p a p p p S r ) )( )( ( , 2 c b a p ,
r is the radius of inscribed circle 4) )
)( ( 4 4 c p b p a p p abc S abc R , R is the radius of circumscribed circle; 23
5) 2 2 2 4 Cos Cos Cos p R ; 6) ( ) tan ( ) tan
2 2 ( ) tan tan
tan tan
4 ; 2 2 2 2 2 2 2 r p a p b p c p RSin Sin Sin
Area of a triangle
1) 2 2 2 c b a h c h b h a S ; 2) ) )( )( ( c p b p a p p S ; 3) pr R abc S 4 ; 4) ) )( )( ( 3 4 c b a m m m m m m m S , 2 c b a m m m m ; 5) Sin Sin Sin a abSin S 2 2 1 2 ;
2 3 2 1 S S S S ;
n m S
S n m q p mp S ) )( ( 1
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1.
Law of sinus: R Sin c Sin b Sin a 2
R is the radius of circumscribed circle. 2.
; 2 ; 2 2 2 2 2 2 2
c a b bcCos c b a
; 2 2 2 2
b a c
;
bCos a
aCos c cCos aCos b ;
3. Mollweide's formula: 2 2 Sin Cos c b a
4. Law of tangents: tan
cot 2 2 tan cot
2 2
b a b
5. bc c p b p Sin 2 ; bc a p p Cos 2 . Download 1.81 Mb. Do'stlaringiz bilan baham: 
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