Shoppers in mathematics - formulas, mathematical symbols in geometry, trigonometry

Collection of cheat sheets in mathematics.

Mathematics cheat sheets - mathematical symbols

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

Shoppers in mathematics - Formulas in geometry

Shoppers in mathematics - Formulas in geometry:

• Formulas for the area of \u200b\u200bthe circle and its parts

Numerical characteristics	Picture	Formula
Area of \u200b\u200ba circle		, where R - The radius of the circle, D. - The diameter of the circle
Sector Square		, if the size of the angle α expressed in radiances
		, if the size of the angle α expressed in degrees
The area of \u200b\u200bthe segment		, if the size of the angle α expressed in radiances
		, if the size of the angle α expressed in degrees

Formulas for the length of the circle and its arcs

Numerical characteristics	Picture	Formula
Circumference		C \u003d2π R \u003dπ  D., where R - The radius of the circle, D. - The diameter of the circle
The length of the arc		L.(α) = α R, if the size of the angle α expressed in radiances
		, if the size of the angle α expressed in degrees

• Proper polygons

Used designations

Formulas for the side, perimeter and area of \u200b\u200bthe correct n. - Ugulnik

Value	Picture	Formula	Description
Perimeter		P \u003d an	Perimeter expression across the side
Square			Expression of the area through the side and radius of the inscribed circle
Square			Expression of the area across the side
Side			The expression of the side through the radius of the inscribed circle
Perimeter			The expression of the perimeter through the radius of the inscribed circle
Square			Expression of the area through the radius of the inscribed circle
Side			The expression of the side through the radius of the described circle
Perimeter			The expression of the perimeter through the radius of the described circle
Square			Expression of the area through the radius of the described circle

Formulas for the side, perimeter and area of \u200b\u200bthe correct triangle

Value	Picture	Formula	Description
Perimeter		P \u003d 3a	Perimeter expression across the side
Square			Expression of the area across the side
Square			Expression of the area through the side and radius of the inscribed circle
Side			The expression of the side through the radius of the inscribed circle
Perimeter			The expression of the perimeter through the radius of the inscribed circle
Square		View the output of the formula	Expression of the area through the radius of the inscribed circle
Side			The expression of the side through the radius of the described circle
Perimeter			The expression of the perimeter through the radius of the described circle
Square			Expression of the area through the radius of the described circle

Formulas for the side, perimeter and area of \u200b\u200bthe correct hexagon

Value	Picture	Formula	Description
Perimeter		P \u003d 6A	Perimeter expression across the side
Square			Expression of the area across the side
Square		S \u003d 3ar	Expression of the area through the side and radius of the inscribed circle
Side			The expression of the side through the radius of the inscribed circle
Perimeter			The expression of the perimeter through the radius of the inscribed circle
Square			Expression of the area through the radius of the inscribed circle
Side		a \u003d r	The expression of the side through the radius of the described circle
Perimeter		P \u003d 6R	The expression of the perimeter through the radius of the described circle
Square			Expression of the area through the radius of the described circle

Formulas for the side, perimeter and square area

Value	Picture	Formula	Description
Perimeter		P \u003d 4A	Perimeter expression across the side
Square		S \u003da2	Expression of the area across the side
Side		a \u003d 2R	The expression of the side through the radius of the inscribed circle
Perimeter		P \u003d 8R	The expression of the perimeter through the radius of the inscribed circle
Square		S \u003d4r2	Expression of the area through the radius of the inscribed circle
Side			The expression of the side through the radius of the described circle
Perimeter			The expression of the perimeter through the radius of the described circle
Square		S \u003d2R2	Expression of the area through the radius of the described circle

• Formulas for the area of \u200b\u200bthe triangle

Figure	Picture	Formula of the area	Designations
Arbitrary triangle			a - Any side h a - The height lowered on this side
			a and b. - Any two sides, FROM - The angle between them
		.	a, b, C- Parties, p. - A semi -perimeter The formula is called "Formula Heron"
			a - Any side B, s - adjacent angles
			a, b, C - Parties, r - radius of an inscribed circle, p. - A semi -perimeter
			a, b, C - Parties, R - radius of the described circle
		S \u003d2R2 sin A sin B. sin C.	A, b, C - Corners, R - radius of the described circle
Equilateral (correct) triangle			a - side
			h - height
			r - radius of the inscribed circle
			R - radius of the described circle
Right triangle			a and b. - Katets
			a - Katet, φ - adjacent sharp corner
			a - Katet, φ - opposite sharp corner
			c. - Hypotenuse, φ - any of the sharp corners

• Formulas for quadrangle areas

Quadrangle	Picture	Formula of the area	Designations
Rectangle		S \u003d AB	a and b. - adjacent sides
			d.- Diagonal, φ - any of the four angles between the diagonals
		S \u003d2R2 sin φ It turns out from the upper formula substitution D \u003d 2R	R - radius of the described circle, φ - any of the four angles between the diagonals
Parallelogram		S \u003d A H a	a - side, h a - The height lowered on this side
		S \u003d ABsin φ	a and b. - adjacent sides, φ - The angle between them
			d.1,  d.2 - Diagonals, φ - any of the four angles between them
Square		S \u003d a2	a - side of a square
		S \u003d4r2	r - radius of the inscribed circle
		View the output of the formula	d. - The diagonal of the square
		S \u003d2R2 It turns out from the upper formula substitution d \u003d 2R	R - radius of the described circle
Rhombus		S \u003d A H a	a - side, h a - The height lowered on this side
		S \u003da2 sin φ	a - side, φ - any of the four corners of the rhombus
			d.1,  d.2 - Diagonal
		S \u003d2ar View the output of the formula	a - side, r - radius of the inscribed circle
			r - radius of an inscribed circle, φ - any of the four corners of the rhombus
Trapezius			a and b. - grounds, h - height
		S \u003d m h	m - middle line, h - height
			d.1,  d.2 - Diagonals, φ - any of the four angles between them
			a and b. - grounds, c. and d. - Side sides
Deltoid		S \u003d ABsin φ	a and b. - unequal aspects, φ - The angle between them
			a and b. - unequal aspects, φ 1 - angle between sides equal a , φ 2 - angle between sides equal b..
		S \u003d(a + b) r	a and b. - unequal aspects, r - radius of the inscribed circle
		View the output of the formula	d.1,  d.2 - Diagonal
Arbitrary convex quadrangle			d.1,  d.2 - Diagonals, φ - any of the four angles between them
Inscribed quadrangle		,	a, b, c, d - the lengths of the sides of the quadrangle, p. - semi -perimeter, The formula is called "Formula Brahmagupta"

• Coordinate method

The distance between the points BUT(x1; u1)  and AT(x2; u2)
Coordinates ( x;  u) The middle of the segment AB with ends BUT(x1;  u1) and AT(x2;  u2)
The equation is direct
Circular equation with radius R and with the center at the point ( x0;  u0)
If a BUT ( x1;  u1) and AT ( x2;  u2), then the coordinates of the vector	(X2-X1; u2-Wh1}
The addition of vectors	{x1; y1} +  {x2; y2} =  { xone  x2; yone  y2} {x1; y1}    {x2; y2} =  {xone  x2; yone  y2}
The multiplication of the vector {x; y} on the number k.	k.  {x; y} = k. { k.  x; k.   y}
The length of the vector
Scalar work of vectors and	∙    =  ∙ where — the angle between the vectors    and
Scalar work of vectors in coordinates	{x1; y1} and {x2; y2} ∙  = xone· x2 + yone· y2
The scales of the vector {x; y}
Cosine of the angle between vectors {x1; y1} and {x2; y2}
A necessary and sufficient condition for the perpendicularity of vectors	{x1; y1} ┴  {x2; y2} ∙  = 0 or  xone· x2 + yone· y2= 0

Mathematics cheat sheets - formulas in trigonometry

Shoppers in mathematics - formulas in trigonometry:

• The main trigonometric identities

$$\mathrm{s.}i{\mathrm{n.}}^{2}x+\mathrm{c.}o{\mathrm{s.}}^{2}x=1$$

$$tgx=\frac{\mathrm{s.}i\mathrm{n.}x}{\mathrm{c.}o\mathrm{s.}x}$$

$$\mathrm{c.}tgx=\frac{\mathrm{c.}o\mathrm{s.}x}{\mathrm{s.}i\mathrm{n.}x}$$

$$tgx\mathrm{c.}tgx=1$$

$$t{g}^{2}x+1=\frac{1}{\mathrm{c.}o{\mathrm{s.}}^{2}x}$$

$$\mathrm{c.}t{g}^{2}x+1=\frac{}{}$$

• Double argument formulas (angle)

$$\mathrm{s.}i\mathrm{n.}2x=2\mathrm{c.}o\mathrm{s.}x\mathrm{s.}i\mathrm{n.}x$$

$$\begin{array}{cc}\mathrm{s.}i\mathrm{n.}2x& =\frac{2tgx}{1+t{g}^{2}x}\\ & =\frac{2\mathrm{c.}tgx}{1+\mathrm{c.}t{g}^{2}x}\\ & =\frac{2}{tgx+\mathrm{c.}tgx}\end{array}$$

$$\begin{array}{cc}\mathrm{c.}o\mathrm{s.}2x& ={\mathrm{cOS}}^{2}x-\mathrm{s.}i{\mathrm{n.}}^{2}x\\ & =2\mathrm{c.}o{\mathrm{s.}}^{2}x-1\\ & =1-2\mathrm{s.}i{\mathrm{n.}}^{2}x\end{array}$$

$$\begin{array}{cc}\mathrm{c.}o\mathrm{s.}2x& =\frac{1-t{g}^{2}x}{1+t{g}^{2}x}\\ & =\frac{\mathrm{c.}t{g}^{2}x-1}{\mathrm{c.}t{g}^{2}x+1}\\ & =\frac{\mathrm{c.}tgx-tgx}{\mathrm{c.}tgx+tgx}\end{array}$$

$$\begin{array}{cc}tg2x& =\frac{2tgx}{1-t{g}^{2}x}\\ & =\frac{2\mathrm{c.}tgx}{\mathrm{c.}t{g}^{2}x-1}\\ & =\frac{2}{\mathrm{c.}tgx-tgx}\end{array}$$

$$\begin{array}{cc}\mathrm{c.}tg2x& =\frac{\mathrm{c.}t{g}^{2}x-1}{2\mathrm{c.}tgx}\\ & =\frac{2\mathrm{c.}tgx}{\mathrm{c.}t{g}^{2}x-1}\\ & =\frac{\mathrm{c.}tgx-tgx}{2}\end{array}$$

• Triple argument formulas (angle)

$$\mathrm{s.}i\mathrm{n.}3x=3\mathrm{s.}i\mathrm{n.}x-4\mathrm{s.}i{\mathrm{n.}}^{3}x$$

$$\mathrm{c.}o\mathrm{s.}3x=4\mathrm{c.}o{\mathrm{s.}}^{3}x-3\mathrm{c.}o\mathrm{s.}x$$

$$tg3x=\frac{3tgx-t{g}^{3}x}{1-3t{g}^{2}x}$$

$$\mathrm{c.}tg3x=\frac{\mathrm{c.}t{g}^{3}x-3\mathrm{c.}tgx}{3\mathrm{c.}t{g}^{2}x-1}$$

• Formulas of the sum of trigonometric functions

$$\mathrm{s.}i\mathrm{n.}\alpha +\mathrm{s.}i\mathrm{n.}\beta =2\mathrm{s.}i\mathrm{n.}\frac{\alpha +\beta }{2}\cdot \mathrm{c.}o\mathrm{s.}\frac{\alpha -\beta }{2}$$

$$\mathrm{c.}o\mathrm{s.}\alpha +\mathrm{c.}o\mathrm{s.}\beta =2\mathrm{c.}o\mathrm{s.}\frac{\alpha +\beta }{2}\cdot \mathrm{c.}o\mathrm{s.}\frac{\alpha -\beta }{2}$$

$$tg\alpha +tg\beta =\frac{\mathrm{s.}i\mathrm{n.}(\alpha +\beta )}{\mathrm{c.}o\mathrm{s.}\alpha \mathrm{c.}o\mathrm{s.}\beta }$$

$$\mathrm{c.}tg\alpha +\mathrm{c.}tg\beta =\frac{\mathrm{s.}i\mathrm{n.}(\alpha +\beta )}{\mathrm{c.}o\mathrm{s.}\alpha \mathrm{c.}o\mathrm{s.}\beta }$$

$$(\mathrm{s.}i\mathrm{n.}\alpha +\mathrm{c.}o\mathrm{s.}\alpha {)}^2=1+\mathrm{s.}i\mathrm{n.}2\alpha$$

• Reverse trigonometric functions