Shoppers in mathematics - formulas, mathematical symbols

Collection of cheat sheets in mathematics.

Mathematics cheat sheets - mathematical symbols

Mathematics cheat sheets - mathematical symbols:

• The main mathematical symbols

Symbol	The name of the symbol	Meaning / definition	example
=	equal sign	equality	5 = 2 + 3 5 equal 2 + 3
≠	the sign is not equal	inequality	5 ≠ 4 5 is not equal to 4
≈	about equal	approximation	sin (0.01) ≈ 0.01, x ≈  y means that x approximately equal y
/	strict inequality	more than	5/ 4 5 more than 4
<	strict inequality	less than	4 ‹5 4 less than 5
≥	inequality	more or equal	5 ≥ 4, x ≥  y means that x more or equal y
≤	inequality	less or equal	4 ≤ 5, x ≤ y means that x less or equal y
()	round brackets	first calculate the expression inside	2 × (3 + 5) \u003d 16
[]	brackets	first calculate the expression inside	[(1 + 2) × (1 + 5)] \u003d 18
+	plus sign	addition	1 + 1 = 2
—	minus sign	subtraction	2 — 1 = 1
±	plus - minus	operations plus and minus	3 ± 5 \u003d 8 or -2
±	minus plus	both minus and plus surgery	3 ∓ 5 \u003d -2 or 8
*	star	multiplication	2 * 3 = 6
×	a sign of times	multiplication	2 × 3 \u003d 6
⋅	the point of multiplication	multiplication	2 ⋅ 3 = 6
÷	division	division	6 ÷ 2 \u003d 3
/	the dividing oblique feature	division	6/2 = 3
—	horizontal line	division / fraction
maud	according to the module	calculation of the remainder	7 mod 2 \u003d 1
.	period	decimal point, tenant	2,56 = 2 + 56/100
a b	strength	exponent	2 3= 8
a ^ b	carriage	exponent	2 ^ 3 \u003d 8
√  a	square root	√  and ⋅ √  a \u003d a	√ 9 \u003d ± 3
3 √ a	cubic root	3 √ A ⋅3 √ A ⋅3 √ A \u003d A	3 √ 8 \u003d 2
4 √ a	the fourth root	4 √ A ⋅4 √ A ⋅4 √ A ⋅4 √ A \u003d A	4 √ 16 \u003d ± 2
p √ a	nth degree root (radical)		for n. \u003d 3, n. √ 8 \u003d 2
%	percent	1% = 1/100	10% × 30 \u003d 3
‰	pMILLE	1 ‰ \u003d 1/1000 \u003d 0.1%	10 ‰ × 30 \u003d 0.3
pPM	for a million	1 parts per million \u003d 1/1000000	10 parts per million × 30 \u003d 0.0003
pPB	per billion	1PPB \u003d 1/1000000000	10PPB × 30 \u003d 3 × 10-7
pPT	to trillion	1PPT \u003d 10 -12	10PPT × 30 \u003d 3 × 10-10

Symbols of geometry

Symbol	The name of the symbol	Meaning / definition	example
∠	corner	formed by two rays	∠ABC \u003d 30 °
	measured angle		ABC \u003d 30 °
	spherical angle		AOB \u003d 30 °
∟	right angle	\u003d 90 °	α \u003d 90 °
°	degree	1 turnover \u003d 360 °	α \u003d 60 °
grad	degree	1 turnover \u003d 360 degrees	α \u003d 60 degrees
′	prime Minister	angular minute, 1 ° \u003d 60 ′	α \u003d 60 ° 59 ′
″	double stroke	corner second, 1 ′ \u003d 60 ″	α \u003d 60 ° 59′59 ″
	line	endless line
AB	line segment	line from point A to point b
	ray	line that starts from point A
	arc	arc from point a to point b	\u003d 60 °
⊥	perpendicular	perpendicular lines (angle 90 °)	AC ⊥ BC
∥	parallel	parallel lines	Ab ∥ CD
≅	corresponds	the equivalence of geometric shapes and sizes	∆ABC≅ ∆XYZ
~	similarity	the same forms, different sizes	∆ABC ~ ∆XYZ
Δ	triangle	the shape of the triangle	ΔABC≅ δbcd
|  x —  u |	distance	distance between points X and Y	|  x —  u | \u003d 5
π	constant pi	π \u003d 3.141592654 ... The ratio of the length of the circle to the diameter of the circle.	c. =  π ⋅  d. \u003d 2⋅ π ⋅  r
glad	radians	radiana angular unit	360 ° \u003d 2π Rad
c.	radians	radiana angular unit	360 ° \u003d 2π with
grad	gradians / Gonons	corner block	360 ° \u003d 400 degrees
g	gradians / Gonons	corner block	360 ° \u003d 400 g

• Symbols of algebra

Symbol	The name of the symbol	Meaning / definition	example
x	variable x	unknown meaning for searching	when 2 x \u003d 4, then x \u003d 2
≡	equivalence	identically
≜	equal by definition	equal by definition
\u003d	equal by definition	equal by definition
~	about equal	weak approach	11 ~ 10
≈	about equal	approximation	sin (0.01) ≈ 0.01
∝	proportionally	proportionally	y ∝  x, when y =  kX, K constant
∞	lemniscat	a symbol of infinity
≪	much less than	much less than	1 1000000 ≪
≫	much more than	much more than	1000000 ≫ 1
()	round brackets	first calculate the expression inside	2 * (3 + 5) = 16
[]	brackets	first calculate the expression inside	[(1 + 2) * (1 + 5)] = 18
{}	suspenders	kit
⌊  x ⌋	floor brackets	rounds the number to a smaller whole	⌊4.3⌋ = 4
⌈  x ⌉	ceiling brackets	rounds the number to the upper whole	⌈4.3⌉ = 5
x !	exclamation point	factorial	4! = 1 * 2 * 3 * 4 = 24
|  x |	vertical stripes	absolute value	| -5 | = 5
f (  x )	function X	displays values \u200b\u200bx in f (x)	e (  x ) \u003d 3 x +5
(  and ∘  g )	functional composition	(  e ∘  g ) (  x ) =  e (  g (  x ))	f (  x ) \u003d 3 x ,  g (  x ) =  x -1 ⇒ ( f ∘  g ) (  x ) \u003d 3 ( x -one)
(  a ,  b )	open interval	(  a ,  b. ) = {  x |  a <  x <  b }	x ∈ (2.6)
[  a ,  b ]	closed interval	[  a ,  b. ] = {  x |  a ≤  x ≤  b }	x ∈ [2.6]
∆	delta	change / difference	∆  t =  t1 —  t0
∆	discriminant	Δ =  b.2 - four alternating current
∑	sigma	summation - the sum of all values \u200b\u200bin the range	Σ  x i \u003d xone+ x2+ ... + xp
∑∑	sigma	double summation
∏	title pi	product - a work of all values \u200b\u200bin the series range	∏  x i \u003d xone∙ x2∙ ... ∙ xn.
e	e constant/ Euler's number	e \u003d 2.718281828 ...	e \u003d lim (1 + 1 / x )  x ,  x → ∞
γ	Permanent Euler-Masqueeroni	γ \u003d 0.5772156649 ...
φ	Golden section	golden section constant
π	constant pi	π \u003d 3.141592654 ... The ratio of the length of the circle to the diameter of the circle.	c. =  π ⋅  d. \u003d 2⋅ π ⋅  r

• Symbols of linear algebra

Symbol	The name of the symbol	Meaning / definition	example
·	dot	scalar product	a ·  b
×	cross	vector product	a ×  b
BUT ⊗  B	tensor work	tensor work a and b	BUT ⊗  B
	internal product
[]	brackets	matrix of numbers
()	round brackets	matrix of numbers
|  BUT |	determinant	the determinant of the matrix a
det ( BUT )	determinant	the determinant of the matrix a
||  x ||	double vertical stripes	norm
BUTT	transpose	the matrix is \u200b\u200btransparent	(  AT )  iJ = (  A )  ji
A†	Hermitova matrix	the matrix conjugated transparent	(  A† )  iJ = (  A )  ji
BUT*	Hermitova matrix	the matrix conjugated transparent	(  A* )  iJ = (  A )  ji
BUT-1	inverse matrix	AA-1 =  I
rank ( BUT )	the rank of matrix	the rank of matrix a	rank ( BUT ) \u003d 3
dull ( U )	measurement	the dimension of the matrix a	dim ( U ) \u003d 3

• Symbols of probability and statistics

Symbol	The name of the symbol	Meaning / definition	example
P. (  BUT )	probability function	the probability of event a	P. (  A ) \u003d 0.5
P. (  A ⋂  B. )	the probability of intersection of events	the likelihood that the events a and b	P. (  A ⋂  B. ) \u003d 0.5
P. (  A ⋃  B. )	the probability of combining events	the likelihood that the events a or b	P. (  A ⋃  B. ) \u003d 0.5
P. (  A |  B. )	the function of conditional probability	the probability of event a this event B has occurred	P. (  A | B. ) \u003d 0.3
f (  x )	probability density function (PDF)	P. (  a ≤  x ≤  b. ) =  ∫ f (  x )  dX
F (  x )	cumulative distribution function (CDF)	F (  x ) =  R (  X ≤  x )
μ	The average population	the average value of the totality	μ = 10
E. (  X )	expected value	the expected value of the random value X	E. (  X ) \u003d 10
E. (  X | Y )	conditional expectation	the expected value of the random value X, taking into account Y	E. (  X | Y \u003d 2 ) \u003d 5
var (  X )	deviation	dispersion of random size X	var (  X ) \u003d 4
σ  2	deviation	a dispersion of the set of set	σ  2 \u003d 4
sTD (  X )	standard deviation	standard deviation of random value X	sTD (  X ) \u003d 2
σ  X	standard deviation	the value of the standard deviation of the random value X	σ  X  =  2
	median	the average value of the random value X
cOV (  X ,  Y )	coaring	coarration of random values \u200b\u200bX and Y	cOV (  X, Y. ) \u003d 4
corr (  X ,  Y )	correlation	correlation of random values \u200b\u200bX and Y	corr (  X, Y. ) \u003d 0.6
ρ X ,  Y	correlation	correlation of random values \u200b\u200bX and Y	ρ X ,  Y \u003d 0.6
∑	summation	summation - the sum of all values \u200b\u200bin the range
∑∑	double summation	double summation
Mon	Mode	the value that is most often found in the population
Mr	the average range	Mr = (  x max +  x min ) / 2
Mkr	median sample	half of the population below this value
Q. 1	nizhny / First Road	25% of the population below this value
2 quarter	mediana / second ten	50% of the population below this value \u003d median sample
3 quarter	upper / third ten	75% of the population below this value
x	selective average	arithmetic mean / average	x \u003d (2 + 5 + 9) / 3 \u003d 5,333
with2	selective dispersion	evaluator of the dispension of the sample of the population	s.2 \u003d 4
with	standard sampling deviation	Assessment of a standard deviation of the sample of the population	s. \u003d 2
z x	standard assessment	z x = (  x - x) / s. x
X ~	distribution X	distribution of random value X	X ~  N. (0.3)
N. (  μ ,  σ 2 )	normal distribution	gAUSOVO Distribution	X ~  N. (0.3)
U (  a ,  b )	uniform distribution	equal probability in the range A, b	X ~  U (0.3)
ehr (λ)	exponential distribution	f (  x )  \u003d λE—  λx ,  x ≥0
gamma (  c. , λ)	gamma distribution	f (  x )  \u003d λ cxc-1e.—  λx / Γ (  c. ),  x ≥0
χ  2 (  to )	distribution of chi-square	f (  x )  \u003d x k. / 2-1e.—  x / 2 / (2 k / 2 Γ (  k. / 2))
F (  k.1 , k2 )	F Distribution
Basket (  n. ,  p. )	binomial distribution	f (  k. )  =  n. C. k. P. k. (one -p )  nK
Poisson (λ)	poisson distribution	e (  To )  sign equals λ  To e—  λ /  To !
Goom (  p. )	geometric distribution	f (  k. )  \u003d p (one -p )  k.
Hg (  N. ,  K. ,  n. )	hypergeometric distribution
Berne (  p. )	Distribution of Bernoulli

• Symbols of calculus and analysis

Symbol	The name of the symbol	Meaning / definition	example
	limit	the limit value of the function
ε	epsilon	is a very small number close to zero	ε →  0
e	e constant/ Euler's number	e \u003d 2.718281828 ...	e \u003d lim (1 + 1 / x )  x ,  x → ∞
y ‘	derivative	derivative - designation of Lagrange	(3 x3 ) ‘\u003d 9 x2
u »	the second derivative	derivative from the derivative	(3 x3 ) "\u003d 18 x
u(  p )	n-I derivative	n times conclusion	(3 x3 )  (3) \u003d 18
	derivative	derivative - designation of Leibniz	d. (3 x3 ) /  dX \u003d 9 x2
	the second derivative	derivative from the derivative	d.2 (3 x3 ) /  dX2 \u003d 18 x
	n-I derivative	n times conclusion
	time derivative	time derivative - Newton's designation
	second time derivative	derivative from the derivative
D. x y	derivative	derivative - designation of Euler
D. x2 u	the second derivative	derivative from the derivative
	private derivative		∂ (  x2 +  y2 ) / ∂  x \u003d 2 x
∫	integral	opposite to origin	∫  f (x) dx
∫∫	double integral	integrating the function of two variables	∫∫  f (x, y) dxdy
∫∫∫	triple integral	integration of function 3 variables	∫∫∫  f (x, y, z) dxdydz
∮	closed circuit / linear integral
∯	integral with a closed surface
∰	integral of a closed volume
[  a ,  b ]	closed interval	[  a ,  b. ] = {  x |  a ≤  x ≤  b }
(  a ,  b )	open interval	(  a ,  b. ) = {  x |  a <  x <  b }
i	imaginary unit	i ≡ √ -1	g \u003d 3 + 2 i
z *	comprehensively conjugated	z =  a +  bI →  z * =  a —  bI	g * \u003d 3 - 2 i
z	comprehensively conjugated	z =  a +  bI →  z =  a —  bI	g \u003d 3 - 2 I
Re ( z )	actual part of the complex number	z =  a +  bI → Re ( z ) =  a	Re (3 - 2 i ) \u003d 3
IM ( z )	imaginary part of the complex	z =  a +  bI → IM ( z ) =  b.	IM (3 - 2 i ) \u003d -2
|  z |	absolute value / value of a complex number	|  z | = |  a +  bi | = √ (  a2 +  b.2 )	| 3 - 2 i | \u003d √13
arg ( z )	the argument of the integrated number	Radius angle in a complex plane	arg (3 + 2 i ) \u003d 33.7 °
∇	nabla / del	gradient operator / divergence	∇  e (  x ,  u ,  g )
	vector
	a single vector
x *  u	convolution	u (  t ) =  x (  t ) *  h (  t )
	Laplace transformation	F (  s. ) =  {  f (  t )}
	fourier transformation	X (  ω ) =  {  f (  t )}
δ	delta-function
∞	lemniscat	a symbol of infinity

Mathematics cheat sheet for primary school

Mathematics cheat sheet for primary school:

• Multiplication table from 1 to 20

Cheatheller in profile mathematics

Scarling in specialized mathematics:

• F-lla of a half argument.

sin² ern /2 \u003d (1 - cos ern) /2

cos² ern /2 \u003d (1 + cosement) /2

tG ern /2 \u003d sinorn /(1 + cosement) \u003d (1-cos ern) /sin isp

Μ   + 2 n, n  z

• F-li transformation of the amount into the production.

sin x + sin y \u003d 2 sin ((x + y)/2) cos ((x-y)/2)

sin x-sin y \u003d 2 cos ((x+y)/2) sin ((x-y)/2)

cOS X + COS Y \u003d 2COS (X + Y)/2 COS (X-Y)/2

cOS X -COS Y \u003d -2SIN (X+Y)/2 SIN (X -Y)/2

• Formulas preobr. production. In the amount

sin x sin y \u003d ½ (cos (x-y)-cos (x+y))

cos x cos y \u003d ½ (cos (x-y)+ cos (x+ y))

sin x cos y \u003d ½ (sin (x-y)+ sin (x+ y))

• The ratio between functions

sin x \u003d (2 tg x/2)/(1+tg ²x/2)

cOS X \u003d (1-TG ² 2/x)/(1+ tg² x/2)

sin2x \u003d (2TGX)/(1+TG ²x)

sin² ern \u003d 1 /(1+ctg² mon) \u003d tg² mics /(1+tg² isp)

cos² ern \u003d 1 / (1+tg² isp) \u003d ctg² √ / (1+ctg² isp)

cTG2 piped

sin3 pipes \u003d 3sinorn -4sin³ √ \u003d 3cos² ern sinorn -sin³

cOS3P \u003d 4COS³ Š -3 COSP \u003d COS³ Š -3COSPorn ML

tG3MER \u003d (3TGHPER -TG³ M)/(1-3TG² M)

cTG3P \u003d (CTG³ ispg mill)/(3CTG² isp)

sin ern /2 \u003d   ((1-cosement) /2)

cOS ern /2 \u003d   ((1+COSP) /2)

tGHP /2 \u003d   ((1-COSP) /(1+COSP)) \u003d

sinorn /(1+cosement) \u003d (1-cosement) /sinising

ctg mill /2 \u003d   ((1+COSM) /(1-cosement)) \u003d

sinorn /(1-cosising) \u003d (1+cosement) /sinising

sin (arcsin isp) \u003d ₽

cOS (arccos isp) \u003d ₽

tg (arctg isp) \u003d ₽

cTG (arcctg isp) \u003d ₽

arcsin (sinoff) \u003d ern; Μ  [- /2;  /2]

arccos (cos isp) \u003d Š;   [0; ]

arctg (tg isp) \u003d √; Μ  [- /2;  /2]

arcctg (ctg isp) \u003d ₽;   [0; ]

arcsin (sin )=

arccos (COS ) =

arctg (TG )=  — K.

Μ  (- /2 + k;  /2 + k)

arcctg (CTG ) =  — K.

Μ  ( k; (k+1) )

arcsinorn \u003d -Arcsin (—oft) \u003d  /2-arcosoff \u003d

\u003d arctg ern / (1-pan ²)

arccosoff \u003d  -arccos (-M) \u003d  /2-assin ern \u003d

\u003d arc ctg pipes / (1-pan ²)

arctgovern \u003d -arctg (-M) \u003d  /2 -arcctg pan \u003d

\u003d arcsin ern / (1+ ²)

arc ctg √ \u003d  -arc cctg (—off) \u003d

\u003d arc cos mon / (1-pan ²)

arctg ern \u003d arc ctg1/√ \u003d

\u003d arcsin ern / (1+ ²) \u003d arccos1 / (1+isp)

arcsin ern + arccos \u003d  /2

arcctg ern + arctg pipes \u003d  /2

• Indicative equations.

Inequality: if a ^{f (x)}›(‹) A ^{a (h)}

Logarithms: inequalities:

log _af (x) ›(‹) log _a  (x)

1. a ›1, then: f (x)› 0

 (x) ›0

f (x) › (x)

2. 0 ‹a‹ 1, then: \u003d "" f (x) \u003d "" ›0

 (x) ›0

f (x) ‹ (x)

3. log _{f (x)}  (x) \u003d a

ODZ:  (x) ›0

f (x) ›0

f (x)  1

Trigonometry:

1. Decomposition into multipliers:

sin 2x -  3 cos x \u003d 0

2sin x cos x -3 cos x \u003d 0

cos X (2 sin x -  3) \u003d 0

2. Solutions by replacement

3.Sin² x - sin 2x + 3 cos² x \u003d 2

sin² x - 2 sin x cos x + 3 cos ² x \u003d 2 sin² x + cos² x

Then it is written if sin x \u003d 0, then cos x \u003d 0,

and this is impossible, \u003d ›can be divided into COS X

• Trigonometric nervous:

sin  m

2 K+ ₁ =  =  ₂+ 2 K.

2 K+ ₂ =  = ( ₁+2 )+ 2 K.

Example:

I cos ( /8+x) ‹ 3/2

 k + 5 /6  /8 + x ‹7 /6 + 2 K

2 k+ 17 /24 ‹x  /24+ 2 k ;;;;

II sin ern \u003d 1/2

2 k + 5 /6 \u003d √ \u003d 13 /6 + 2 K

cOS  (= ) m

2 K + ₁ <  <  ₂+2 K.

2 K+ ₂<  < ( ₁+2 ) + 2 K.

cos mon  -  2/2

2 k +5 /4 \u003d √ \u003d 11 /4 +2 K

tG  (= ) m

 K+ Arctg M=  = Arctg M + K.

cTG (= ) m

 K+Arcctg M ‹ <  + K.

• Integrals:

 x ^n.dX \u003d x ⁿ⁺¹/(n + 1) + C

 a ^xdX \u003d AX/LN A + C

 e ^x DX \u003d E ^x + C.

 cos x dx \u003d sin x + cos

 sin x dx \u003d - cos x + c

 1/x dx \u003d ln | x | + C.

 1/cos² x \u003d tg x + c

 1/sin² x \u003d - ctg x + c

 1/ (1-x²) dx \u003d arcsin x +c

 1/ (1-x²) dx \u003d-arccos x +c

 1/1 + x² dx \u003d arctg x + c

 1/1 + x² dx \u003d - arcctg x + c

Mathematics cheat sheets - fractions

Mathematics cheat sheets - fractions:

Rule:	Sample solution
1. At addition (subtraction)  fraction with  identical denominators We coil (subtract) their numerators, and leave the denominator the same. - If the fraction is reduced, then we reduce it. - If the fraction is wrong, then we highlight the whole part, dividing the numerator into a denominator with the remainder.
2. At addition (subtraction)  fraction with  different denominators First, bring them to the common denominator, and then rule 1.
3. At addition  mixed numbers with the same denominators We coil their entire parts and fractional parts. The fractional parts are coordinated by rule 1. - If the fractional part is reduced, then we reduce it. - If the fractional part is the wrong fraction, then we distinguish the whole part from it and add it to the existing whole part.
4. At subtraction  mixed numbers with the same denominators We subtract their whole parts and fractional parts. We subtract the fractional parts by rule1. - If the fractional part of the first number is less than the fractional part of the second number, then we separate from the whole part 1 And we translate it along with the fractional part into the wrong fraction, then we subtract the entire parts and fractional parts. - If the fractional part of the first number is absent, then we separate from the whole number 1 And we write it down in the form of a fraction with the same numbers in the numerator and denominator (the numbers should be equal to the denominator of the second number), then we subtract entire parts and fractional parts.
5. At addition (subtraction)  mixed numbers with different denominators First, we bring their fractional parts to the common denominator, and then rules 3 ( according to the rule 4).
Rule:	Sample solution
7.At multiplication  fractions for the number Only the numerator is multiplying this number, and leave the denominator the same. - If the fraction is reduced, then we reduce it. - If the fraction is wrong, then we highlight the whole part, dividing the numerator into a denominator with the remainder.
eight.At multiplication  fraction We multiply the numerator by the numerator, and the denominator by the denominator. - If you can reduce, then first reduce, and then multiply. - If the fraction is wrong, then we highlight the whole part, dividing the numerator into a denominator with the remainder.
9.At multiplication  mixed numbers We transfer them to the wrong fraction, and then rules 8.
ten.At division  fraction The division is replaced by multiplication, while we turn the second shot, then rules 6.
eleven.At division  fractions for the number you need to write this number in the form of a frax with a denominator 1, then rules 10.
12.At division  mixed numbers We transfer them to the wrong fraction, and then rules 10.
13.At division  mixed number for an integer number We translate the mixed number into irregular fraction, and then along rules 11.
fourteen.To mixed number  translate in incorrect fraction You need to multiply the denominator by the whole part and add the numerator. Record the resulting number in the numerator, and leave the denominator the same.

Examination cheat sheets

Examination cheat sheets:

• Geometry

Trigonometry:	$\sin A=\frac{a}{\mathrm{c.}}$    $\mathrm{cOS}A=\frac{\mathrm{b.}}{\mathrm{c.}}$
	$\text{tG}A=\frac{\sin A}{\mathrm{cOS}A}=\frac{a}{\mathrm{b.}}$
Cosine theorem:	${\mathrm{c.}}^2=a^2+{\mathrm{b.}}^2-2a\mathrm{b.}\cdot \mathrm{cOS}\mathrm{C.}$
Sinus theorem:	$\frac{a}{\sin A}=\frac{\mathrm{b.}}{\sin \mathrm{B.}}=\frac{\mathrm{c.}}{\sin \mathrm{C.}}=2R$		where r is the radius of the described circle
The equation of the circle:	${(x-{}_0)}^2+{(y-{}_0)}^2=R^2$	where $(x_0;y_0)$ Coordinates of the center of the circle
The ratio of inscribed and central angles:	$\beta =\frac{\alpha }{2}=\frac{\cup \alpha }{2}$
The described circle, triangle:	$R=\frac{a\mathrm{b.}\mathrm{c.}}{4\mathrm{S.}}$		See also the theorem of sinuses. The center lies at the intersection of median perpendiculars.
Inscribed circle, triangle:	$r=\frac{\mathrm{S.}}{\mathrm{p.}}$		where P is the semi -perimeter of the polygon. The center lies at the intersection of bisector.
The described circle, quadrangle:	$\alpha +\gamma =\beta +\delta ={180}^{\circ }$
Inscribed circle, quadrangle:	$a+\mathrm{c.}=\mathrm{b.}+\mathrm{d.}$
Bisectress property:	$\frac{a}{x}=\frac{\mathrm{b.}}{y}$
The intersecting chords theorem:	$AM\cdot \mathrm{B.}M=\mathrm{C.}M\cdot \mathrm{D.}M$		These theorems must be able to display
The coal theorem between the tangent and the chord:	$\alpha =\frac{1}{2}\cup A\mathrm{B.}$
The theorem about the tangent and secant:	$\mathrm{C.}{M}^2=AM\cdot \mathrm{B.}M$
Tangular segments theorem:	$A\mathrm{B.}=A\mathrm{C.}$

• Square of figures:

• Probability

• Functions graphs, functions studied at school

The name of the function	Formula of function	Function schedule	The name of the graphics	Note
Linear	y \u003d kx		Straight	Linear dependence - direct proportionality y \u003d kx, where k. ≠ 0 - proportionality coefficient.
Linear	y =  kX +  b.		Straight	Linear dependence: coefficients k. and b. - Any real numbers. (k. \u003d 0.5, b. \u003d 1)
Quadratic	y \u003d x2		Parabola	Quadratic dependence: Symmetric parabola with the top at the beginning of the coordinates.
Quadratic	y \u003d xn.		Parabola	Quadratic dependence: n. - Natural even number ›1
Steep	y \u003d xn.		Cuban parabola	Odd degree: n. - natural odd number ›1
Steep	y \u003d x1/2		Function schedule y = √ x	Steep dependence ( x1/2 = √ x).
Steep	y \u003d k/x		Hyperbola	Case for a negative degree (1/x \u003d x-1). Opend-proportional dependence. (k. \u003d 1)
Indicative	y =  a x		A schedule of indicative function	Indicative function for a \u003e one.
Indicative	y \u003d a x		A schedule of indicative function	Indicative function for 0 ‹ a \u003c one.
Logarithmic	y \u003d log ax		Schedule of logarithmic function	Logarithmic function: a \u003e one.
Logarithmic	y \u003d log ax		Schedule of logarithmic function	Logarithmic function: 0 ‹ a \u003c one.
Sinus	y \u003d sin x		Sinusoid	Trigonometric function sinus.
Cosine	y \u003d cos x		Cosinusoid	The trigonometric function is cosine.
Tangent	y \u003d tg x		Tangensoid	Trigonometric function of tangent.
Cotangent	y \u003d CTG x		Kotangensoid	Trigonometric function of cotangenes.

• Formulas of the work.

• Division formula with residual a \u003d b c + r,r B.
• 
• Perimeter formula p \u003d a 4 p \u003d (a + b) 2
• 
• a \u003d p: 4 (side of the square) a \u003d (p - b 2): 2 (side of the rectangle)
• Volume formula:
• 
• - rectangular parallelepiped v \u003d a b C (a- day, b-width, c- height)
• 
• a \u003d v: (a b) (side of a rectangular parallelepiped)
• 
• - Cuba v \u003d a a a a a
• 
• a \u003d v: (a a) (side of the cube)

Trigonometric formulas for high school students

• Trigonometric functions of one angle

• Trigonometric functions of the amount and difference of two angles

• Trigonometric functions of the double angle

Formulas of lowering degrees for squares of trigonometric functions

• Formulas of lowering degree for cubes of sinus and cosinea

• Tangens expression through a sinus and a double angle mowing

• Transformation of the amount of trigonometric functions into a work

• Transformation of the work of trigonometric functions in the amount

• Expression of trigonometric functions through a half angle tangent

• Trigonometric functions of the triple angle

Mathematics cheat sheets to prepare for the exam

Mathematics cheat sheets to prepare for the exam:

Formulas of abbreviated multiplication

(a+b) ² \u003d a ² + 2AB + B ²

(a-b) ² \u003d a ² - 2AB + B ²

a ² - b ² \u003d (a-b) (a+b)

a ³ - b ³ \u003d (a-b) (a ² + ab + b ²)

a ³ + b ³ \u003d (a+b) (a ² - AB + B ²)

(a + b) ³ \u003d a ³ + 3a ²b+ 3AB ²+ b ³

(a - b) ³ \u003d a ³ - 3a ²b+ 3AB ²- b ³

The properties of degrees

a ⁰ \u003d 1 (a ≠ 0)

a ^m/N \u003d (a≥0, n ε n, m ε n)

a ^{- R} \u003d 1/ A ^r (a ›0, r ε q)

a ^m · A ^n. \u003d a ^{m + N}

a ^m : a ^n. \u003d a ^{m - N} (a ≠ 0)

(a ^m) ^N. \u003d a ^mn

(AB) ^N. \u003d a ^n. B. ^n.

(a/b) ^n. \u003d a ^N./ b ^N.

The first -shaped

If f ’(x) \u003d f (x), then f (x) - the primary

for f (x)

Functionf(x) \u003d PrimaryF(x)

k \u003d kx + c

x ^n. \u003d x ^n.+1/n + 1 + C

1/x \u003d ln | x | + C.

e. ^x \u003d E ^x + C.

a ^x \u003d a ^x/ ln a + c

1/√x \u003d 2√x + c

cos x \u003d sin x + c

1/ sin ² x \u003d - ctg x + c

1/ cos ² x \u003d tg x + c

sin x \u003d - cos x + c

1/ x ² \u003d - 1/x

Geometric progression

b.  _n.+1 \u003d b _n. · Q, where n ε n

q - denominator of progression

b.  _n. \u003d b ₁ · Q.  ^{n. - one} -N-th member of the progression

Sumn-s members

S.  _n. \u003d (b _N. Q - b _one )/Q-1

S.  _n. \u003d b _one (Q. ^N. -1)/Q-1

Module

| A | \u003d a, if a favor

-a, if a ‹0

Formulas COSand sin

sin (-x) \u003d -sin x

cos (-x) \u003d cos x

sin (x + π) \u003d -sin x

cos (x + π) \u003d -cos x

sin (x + 2πk) \u003d sin x

cOS (x + 2πk) \u003d COS X

sin (x + π/2) \u003d cos x

Volumes and surfaces of bodies

1. Prism, straight or inclined, parallelepipedV \u003d s · h

2. Direct prism S. _SIDE\u003d p · h, p is the perimeter or circumference length

3. The parallelepiped is rectangular

V \u003d a · b · c; P \u003d 2 (a · b + b · c + c · a)

P is the full surface

4. Cube: V \u003d a ³ ; P \u003d 6 A ²

5.  Pyramid, correct and wrong.

S \u003d 1/3 S · H; S - base area

6.The pyramid is correct S \u003d 1/2 P · A

A - Apofem of the correct pyramid

7. Circular cylinder V \u003d s · h \u003d πr ²h

8. Circular cylinder: S. _SIDE \u003d 2 πrh

9. Circular cone: V \u003d 1/3 sh \u003d 1/3 πr ²h

ten. Circular cone:S. _SIDE \u003d 1/2 pl \u003d πrl

Trigonometric equations

sin x \u003d 0, x \u003d πn

sin x \u003d 1, x \u003d π/2 + 2 πn

sin x \u003d -1, x \u003d -π/2 + 2 πn

cos x \u003d 0, x \u003d π/2 + 2 πn

cos x \u003d 1, x \u003d 2πn

cos x \u003d -1, x \u003d π + 2 πn

Addition Theorems

cos (x +y) \u003d cosx · cosy - sinx · siny

cos (x -y) \u003d cosx · cosy + sinx · siny

sin (x + y) \u003d sinx · cosy + cosx · siny

sin (x -y) \u003d sinx · cosy -cosx · siny

tg (x ± y) \u003d tg x ± tg y/ 1 ^—₊ tg x · tg y

ctg (x ± y) \u003d tg x ^—₊ tg y/ 1 ± tg x · tg y

sin x ± sin y \u003d 2 cos (x ± y/2) · cos (x ^—₊y/2)

cOS X ± COSY \u003d -2 sin (x ± y/2) · sin (x ^—₊y/2)

1 + cos 2x \u003d 2 cos ² x; cOS ²x \u003d 1+cos2x/2

1 - cos 2x \u003d 2 sin ² x; sin ²x \u003d 1- COS2X/2

6.Trapezius

a, b - bases; h - height, c - the middle line s \u003d (a+b/2) · h \u003d c · h

7.Square

a - side, d - diagonal s \u003d a ² \u003d D ²/2

8. Rhombus

a - side, D ₁, d ₂ - diagonals, α is the angle between them s \u003d D ₁d. ₂/2 \u003d A ²sinα

9. The correct hexagon

a - side s \u003d (3√3/2) a ²

ten.A circle

S \u003d (l/2) r \u003d πr ² \u003d πd ²/4

eleven.Sector

S \u003d (πr ²/360) α

Differentiation rules

(f (x) + g (x) ’\u003d f’ (x) + g ’(x)

(k (f (x) ’\u003d kf’ (x)

(f (x) g (x) ’\u003d f’ (x) g (x) + f (x) · g ’(x)

(f (x)/g (x) ’\u003d (f’ (x) g (x) - f (x) · g ’(x))/g ² (x)

(X ^n.) ’\u003d Nx ^n-1

(tg x) ’\u003d 1/ cos ² x

(ctg x) ’\u003d - 1/ sin ² x

(f (kx + m)) ’\u003d kf’ (kx + m)

Tangent equation to function graphics

y \u003d f ’(a) (x-a) + f (a)

SquareS. figures limited by straightx=a, x=b.

S \u003d ∫ (f (x) - g (x)) dx

Newtonian formula

∫_a^b. f (x) dx \u003d f (b) - f (a)

t  π/4  π/2  3π/4  π  cOS √2/2 0 --√2/2 1 sin √2/2 1 √2/2 0 t  5π/4  3π/2  7π/4  2π  cOS --√2/2 0 √2/2 1 sin --√2/2 -1 --√2/2 0 t  0  π/6  π/4  π/3  tG 0 √3/3 1 √3 cTG - √3 1 √3/3
in x \u003d b x \u003d (-1) ^n. arcsin b + πn

cos x \u003d b x \u003d ± arcos b + 2 πn

tg x \u003d b x \u003d arctg b + πn

ctg x \u003d b x \u003d arcctg b + πn

Theorem sinusov: a/sin α \u003d b/sin β \u003d c/sin γ \u003d 2r

Cosine theorem: With ²\u003d a ²+b ²-2ab cos y

Uncertain integrals

∫ dx \u003d x + c

∫ x ^n. DX \u003d (x  ^n. +1/n + 1) + C

∫ dx/x ² \u003d -1/x + c

∫ dx/√x \u003d 2√x + c

∫ (kx + b) \u003d 1/k f (kx + b)

∫ sin x dx \u003d - cos x + c

∫ cos x dx \u003d sin x + c

∫ dx/sin ² x \u003d -ctg + c

∫ dx/cos ² x \u003d tg + c

∫ x ^r DX \u003d x ^R+1/r + 1 + C

Logarithms

1. LOG _a A \u003d 1

2. log _a 1 \u003d 0

3. log _a (b ^n.) \u003d n log _a B.

4. log _A^n. b \u003d 1/n log _a B.

5. log _a B \u003d log _C. B/ log _c. a

6. log _a B \u003d 1/ log _B. a

Degree  0  30  45  60  sin 0 1/2 √2/2 √3/2 cOS 1 √3/2 √2/2 1/2 tG 0 √3/3 1 √3 t  π/6  π/3 2π/3 5π/6 cOS √3/2 1/2 -1/2 --√3/2 sin 1/2 √3/2 √3/2 1/2 90  120  135  150  180 1 √3/2 √2/2 1/2 0 0 -1/2 -√2/2 --√3/2 -1 - -√3 -1 √3/3 0 t  7π/6  4π/3  5π/3  11π/6  cOS --√3/2 -1/2 1/2 √3/2 sin -1/2 --√3/2 --√3/2 -1/2

Double argument formulas

cOS 2X \u003d COS ²x - sin ² x \u003d 2 cos ² x -1 \u003d 1 -2 sin ² x \u003d 1 - tg ² X/1 + TG ² x

sin 2x \u003d 2 sin x · cos x \u003d 2 tg x/ 1 + tg ²x

tG 2X \u003d 2 TG X/ 1 - TG ² x

cTG 2x \u003d CTG ² X - 1/2 CTG X

sin 3x \u003d 3 sin x - 4 sin ³ x

cOS 3X \u003d 4 COS ³ x - 3 cos x

tG 3X \u003d 3 TG X - TG ³ X / 1 - 3 TG ² x

sin s cos t \u003d (sin (s+t)+sin (s+t))/2

sin s sin t \u003d (cos (s-t)-cos (s+t))/2

cOS S COS T \u003d (COS (S + T) + COS (S-T))/2

Differentiation formulas

c ’\u003d 0 ()’ \u003d 1/2

x ’\u003d 1 (sin x)’ \u003d cos x

(kx + m) ’\u003d k (cos x)’ \u003d - sin x

(1/x) ’\u003d - (1/x ²) (ln x) ’\u003d 1/x

(E. ^x) ’\u003d E ^x; (X ^n.) ’\u003d Nx ^N-1; (log _a x) ’\u003d 1/x ln a

Square of flat figures

1. A rectangular triangle

S \u003d 1/2 a · b (a, b - cuttings)

2. An isosceles triangle

S \u003d (A/2) · √ B ² - a ²/4

3. An equilateral triangle

S \u003d (A ²/4) · √3 (a - side)

four.Arbitrary triangle

a, b, c - sides, a - base, h - height, a, b, c - angles lying against the sides; p \u003d (a+b+c)/2

S \u003d 1/2 A · H \u003d 1/2 A ²b sin c \u003d

a ²sinb sinc/2 sin a \u003d √p (p-a) (p-b) (p-c)

5. Parallelogram

a, b - sides, α - one of the corners; h - height s \u003d a · h \u003d a · b · sin α

cos (x + π/2) \u003d -sin x

Formulas TGand CTG

tg x \u003d sin x/ cos x; Ctg x \u003d cos x/sin x

tg (-x) \u003d-tg x

cTG (-x) \u003d-ctg x

tg (x + πk) \u003d tg x

ctg (x + πk) \u003d ctg x

tg (x ± π) \u003d ± tg x

ctg (x ± π) \u003d ± ctg x

tg (x + π/2) \u003d - ctg x

cTG (x + π/2) \u003d - tg x

sin ² X + COS ² x \u003d 1

tg x · ctg x \u003d 1

1 + TG ² x \u003d 1/ cos ² x

1 + CTG ² x \u003d 1/ sin ²x

tG ² (x/ 2) \u003d 1 - cos x/ 1 + cos x

cOS ² (x/ 2) \u003d 1 + cos x/ 2

sin ² (x/ 2) \u003d 1 - cos x/ 2

eleven.Ball: V \u003d 4/3 πr ³ \u003d 1/6 πd ³

P \u003d 4 πr ² \u003d πd ²

12.Ball segment

V \u003d πh ² (R-1/3h) \u003d πh/6 (h ² + 3r ²)

S. _SIDE \u003d 2 πrh \u003d π (r ² + h ²); P \u003d π (2R ² + h ²)

13.Ball layer

V \u003d 1/6 πh ³ + 1/2 π (r ² + h ²) · H;

S. _SIDE \u003d 2 π · R · H

14. Ball sector:

V \u003d 2/3 πr ² h ’where h’ is the height of the segment containing in the sector

Formula of the roots of the square equation

(A a a a azeals, b≥0)

(a≥0)

aX ² + bx + c \u003d 0 (a ≠ 0)

If d \u003d 0, then x \u003d -b/2a (d \u003d b ²-4ac)

If d ›0, then x _1,2 \u003d -b ± /2a

Vieta theorem

x ₁ + x ₂ \u003d -b/a

x ₁ · X ₂ \u003d C/A

Arithmetic progression

a _n.+1\u003d a  _n. + D, where n is a natural number

d is the difference in progression;

a _n. \u003d a _one + (n-1) · D-formula of the nth penis

Sum N.members

S.  _n. \u003d (a _one + a _N. )/2) n

S.  _n. \u003d ((2A _one + (n-1) d)/2) n

Radius of the described circle near the polygon

R \u003d A/ 2 SIN 180/ N

The radius of the inscribed circle

r \u003d A/ 2 TG 180/ N

Circle

L \u003d 2 πr s \u003d πr ²

The area of \u200b\u200bthe cone

S. _SIDE \u003d πrl

S. _Con \u003d πr (l+r)

Tangent angle- The attitude of the opposing leg to the adjacent. Kotangenes - on the contrary.

Formulas in mathematics - cheat sheet in pictures

Formulas in mathematics - cheat sheet in pictures:

Video: Musical quiz for elementary grades

Read also on our website:

• Ecology quiz with answers: questions for elementary grades
• Poems for children for a reader contest - touching, humorous, funny
• Fands for children in poetry - funny tasks for a fun pastime
• Stencils for children - for drawing, cutting, coloring
• Songs about professions for children
• Literary quiz for poetry