Shoppers in mathematics - for an exam in mathematics, to prepare for the exam

Mathematics cheat sheets that will help to pass exams without any problems.

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.

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

Formulas in mathematics - cheat sheet in pictures

Formulas in mathematics - cheat sheet in pictures:

Video: cheat sheet on the first part of the profile exam

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