Mathematical formula: several area formulas for converting the circle center angle radian size angle and radian into a sector in an arbitrary circle with the same end edge

Mathematical formula: several area formulas for converting the circle center angle radian size angle and radian into a sector in an arbitrary circle with the same end edge

α Absolute value of = L / R

All the math formulas in high school?

Parabola: y = ax * + BX + C
Y equals the square of ax plus BX plus C
When a > 0, the opening is upward
When a < 0, the opening is downward
When C = 0, the parabola passes through the origin
When B = 0, the axis of symmetry of the parabola is the y-axis
And the vertex formula y = a (x + H) * + K
Y is equal to a times the square of (x + H) + K
-H is the X of the vertex coordinates
K is the y of the vertex coordinates
It is generally used to calculate the maximum and minimum values
Parabolic standard equation: y ^ 2 = 2px
It means that the focus of the parabola is on the positive half axis of X, the focus coordinate is (P / 2,0), and the Quasilinear equation is x = - P / 2
Since the focus of the parabola can be on any half axis, there is a common standard equation y ^ 2 = 2px y ^ 2 = - 2px x ^ 2 = 2PY x ^ 2 = - 2PY
Edit the formula about circle in this paragraph
Volume = 4 / 3 (PI) (R ^ 3)
Area = (PI) (R ^ 2)
Perimeter = 2 (PI) r
The standard equation of a circle (x-a) 2 + (y-b) 2 = R2 note: (a, b) are the coordinates of the center of the circle
General equation of circle x2 + Y2 + DX + ey + F = 0 note: D2 + e2-4f > 0
（1） Calculation formula of ellipse perimeter
Ellipse perimeter formula: l = 2 π B + 4 (a-b)
Ellipse perimeter theorem: the circumference of an ellipse is equal to the circumference of a circle (2 π b) with the short semi axis of the ellipse as the radius plus four times the difference between the long semi axis of the ellipse (a) and the short semi axis (b)
（2） Calculation formula of ellipse area
Elliptic area formula: S = π ab
Ellipse area theorem: the area of an ellipse is equal to the product of the circumference (π) multiplied by the length of its long half axis (a) and the length of its short half axis (b)
Although there is no ellipse perimeter t in the above ellipse perimeter and area formulas, these two formulas are derived from ellipse perimeter T. the constant is the body and the formula is for use
Volume calculation formula of elliptical object long radius of ellipse * short radius * Pai * height
Edit trigonometric function of this section
Two angle sum formula
sin(A+B)=sinAcosB+cosAsinB sin(A-B)=sinAcosB-sinBcosA
cos(A+B)=cosAcosB-sinAsinB cos(A-B)=cosAcosB+sinAsinB
tan(A+B)=(tanA+tanB)/(1-tanAtanB) tan(A-B)=(tanA-tanB)/(1+tanAtanB)
cot(A+B)=(cotAcotB-1)/(cotB+cotA) cot(A-B)=(cotAcotB+1)/(cotB-cotA)
Angle doubling formula
tan2A=2tanA/(1-tan2A) cot2A=(cot2A-1)/2cota
cos2a=cos2a-sin2a=2cos2a-1=1-2sin2a
sin α+ sin( α+ 2π/n)+sin( α+ 2π*2/n)+sin( α+ 2π*3/n)+……+sin[ α+ 2π*(n-1)/n]=0
cos α+ cos( α+ 2π/n)+cos( α+ 2π*2/n)+cos( α+ 2π*3/n)+……+cos[ α+ 2 π * (n-1) / N] = 0 and
sin^2( α)+ sin^2( α- 2π/3)+sin^2( α+ 2π/3)=3/2
tanAtanBtan(A+B)+tanA+tanB-tan(A+B)=0
Quadruple angle formula:
sin4A=-4*(cosA*sinA*(2*sinA^2-1))
cos4A=1+(-8*cosA^2+8*cosA^4)
tan4A=(4*tanA-4*tanA^3)/(1-6*tanA^2+tanA^4)
Quintuple angle formula:
sin5A=16sinA^5-20sinA^3+5sinA
cos5A=16cosA^5-20cosA^3+5cosA
tan5A=tanA*(5-10*tanA^2+tanA^4)/(1-10*tanA^2+5*tanA^4)
Hexagonal formula:
sin6A=2*(cosA*sinA*(2*sinA+1)*(2*sinA-1)*(-3+4*sinA^2))
cos6A=((-1+2*cosA^2)*(16*cosA^4-16*cosA^2+1))
tan6A=(-6*tanA+20*tanA^3-6*tanA^5)/(-1+15*tanA^2-15*tanA^4+tanA^6)
Sevenfold angle formula:
sin7A=-(sinA*(56*sinA^2-112*sinA^4-7+64*sinA^6))
cos7A=(cosA*(56*cosA^2-112*cosA^4+64*cosA^6-7))
tan7A=tanA*(-7+35*tanA^2-21*tanA^4+tanA^6)/(-1+21*tanA^2-35*tanA^4+7*tanA^6)
Octagonal formula:
sin8A=-8*(cosA*sinA*(2*sinA^2-1)*(-8*sinA^2+8*sinA^4+1))
cos8A=1+(160*cosA^4-256*cosA^6+128*cosA^8-32*cosA^2)
tan8A=-8*tanA*(-1+7*tanA^2-7*tanA^4+tanA^6)/(1-28*tanA^2+70*tanA^4-28*tanA^6+tanA^8)
Nine fold angle formula:
sin9A=(sinA*(-3+4*sinA^2)*(64*sinA^6-96*sinA^4+36*sinA^2-3))
cos9A=(cosA*(-3+4*cosA^2)*(64*cosA^6-96*cosA^4+36*cosA^2-3))
tan9A=tanA*(9-84*tanA^2+126*tanA^4-36*tanA^6+tanA^8)/(1-36*tanA^2+126*tanA^4-84*tanA^6+9*tanA^8)
Tenfold angle formula:
sin10A=2*(cosA*sinA*(4*sinA^2+2*sinA-1)*(4*sinA^2-2*sinA-1)*(-20*sinA^2+5+16*sinA^4))
cos10A=((-1+2*cosA^2)*(256*cosA^8-512*cosA^6+304*cosA^4-48*cosA^2+1))
tan10A=-2*tanA*(5-60*tanA^2+126*tanA^4-60*tanA^6+5*tanA^8)/(-1+45*tanA^2-210*tanA^4+210*tanA^6-45*tanA^8+tanA^10)
·Universal formula:
sin α= 2tan( α/ 2)/[1+tan^2( α/ 2)]
cos α= [1-tan^2( α/ 2)]/[1+tan^2( α/ 2)]
tan α= 2tan( α/ 2)/[1-tan^2( α/ 2)]
Half angle formula
sin(A/2)=√((1-cosA)/2) sin(A/2)=-√((1-cosA)/2)
cos(A/2)=√((1+cosA)/2) cos(A/2)=-√((1+cosA)/2)
tan(A/2)=√((1-cosA)/((1+cosA)) tan(A/2)=-√((1-cosA)/((1+cosA))
cot(A/2)=√((1+cosA)/((1-cosA)) cot(A/2)=-√((1+cosA)/((1-cosA))
Sum difference product
2sinAcosB=sin(A+B)+sin(A-B) 2cosAsinB=sin(A+B)-sin(A-B)
2cosAcosB=cos(A+B)-sin(A-B) -2sinAsinB=cos(A+B)-cos(A-B)
sinA+sinB=2sin((A+B)/2)cos((A-B)/2 cosA+cosB=2cos((A+B)/2)sin((A-B)/2)
tanA+tanB=sin(A+B)/cosAcosB tanA-tanB=sin(A-B)/cosAcosB
cotA+cotBsin(A+B)/sinAsinB -cotA+cotBsin(A+B)/sinAsinB
Sum of the first n items of some series
1+2+3+4+5+6+7+8+9+…+n=n(n+1)/2 1+3+5+7+9+11+13+15+…+(2n-1)=n2
2+4+6+8+10+12+14+…+(2n)=n(n+1) 1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2+…+n^2=n(n+1)(2n+1)/6
1^3+2^3+3^3+4^3+5^3+6^3+…n^3=(n(n+1)/2)^2 1*2+2*3+3*4+4*5+5*6+6*7+…+n(n+1)=n(n+1)(n+2)/3
Sine theorem a / Sina = B / SINB = C / sinc = 2R note: where R represents the radius of the circumscribed circle of the triangle
Cosine theorem B2 = A2 + c2-2accosb note: angle B is the angle between edge a and edge C
Multiplication and factorization A2-B2 = (a + b) (a-b) A3 + B3 = (a + b) (a2-ab + B2) a3-b3 = (a-b (A2 + AB + B2)
Trigonometric inequality | a + B ≤ | a | + | B | A-B ≤ | a | + | B | a | ≤ B-B ≤ a ≤ B
|a-b|≥|a|-|b| -|a|≤a≤|a|
Edit the solution of the univariate quadratic equation in this section
-b+√(b2-4ac)/2a -b-√(b2-4ac)/2a
Relationship between root and coefficient X1 + x2 = - B / a X1 * x2 = C / a note: Weida theorem
Discriminant b2-4a = 0 note: the equation has two equal real roots
B2-4ac > 0 note: the equation has two unequal real roots
b2-4ac0
Parabolic standard equation y2 = 2px y2 = - 2px x2 = 2PY x2 = - 2PY
Side area of straight prism s = C * h side area of oblique prism s = C '* h
Square pyramid side area s = 1 / 2C * H 'square pyramid side area s = 1 / 2 (c + C') H '
Side area of round table s = 1 / 2 (c + C ') l = pi (R + R) l surface area of ball s = 4Pi * R2
Cylindrical side area s = C * H = 2pi * h conical side area s = 1 / 2 * c * l = pi * r * l
Arc length formula L = a * r a is the radian number of circle center angle R > 0 sector area formula s = 1 / 2 * L * r
Cone volume formula v = 1 / 3 * s * h cone volume formula v = 1 / 3 * pi * r2h
Volume of inclined prism v = s'L note: where s' is the area of straight section and l is the length of side edge
Cylinder volume formula v = s * h cylinder v = pi * r2h
Figure perimeter area volume formula
Perimeter of rectangle = (length + width) × two
Perimeter of square = side length × four
Area of rectangle = length × wide
Square area = side length × Side length
Edit the area of this triangle
If the triangle base a and height h are known, then s = ah / 2
If the triangle has three sides a, B, C and half circumference P, then s = √ [P (P - a) (P - b) (P - C)] (Helen formula) (P = (a + B + C) / 2)
And: (a + B + C) * (a + B-C) * 1 / 4
Given the angle c between two sides a and B of the triangle, then s = absinc / 2
Let the three sides of the triangle be a, B and C respectively, and the radius of the inscribed circle be r
Then triangle area = (a + B + C) R / 2
Let the three sides of the triangle be a, B and C respectively, and the radius of the circumscribed circle be r
Then triangle area = ABC / 4R
If the triangles a, B and C are known, then s = √ {1 / 4 [C ^ 2A ^ 2 - ((C ^ 2 + A ^ 2-B ^ 2) / 2) ^ 2]} ("three oblique quadrature" Southern Song, Qin and Jiushao)
| a b 1 |
S△=1/2 * | c d 1 |
| e f 1 |
【| a b 1 |
|C D 1 | is the third-order determinant. The triangle ABC is in the plane rectangular coordinate system a (a, b), B (C, d), C (E, f), where ABC
| e f 1 |
It's best to take the selected area from the upper right corner in counterclockwise order, because the results obtained are generally positive. If you don't take it according to this rule, you may get a negative value, but it doesn't matter. As long as you take the absolute value, it won't affect the size of the triangle area!]
Edit the middle line area formula of qinjiushao triangle in this section
S=√[(Ma+Mb+Mc)*(Mb+Mc-Ma)*(Mc+Ma-Mb)*(Ma+Mb-Mc)]/3
Where Ma, MB and MC are the midline length of the triangle
Area of parallelogram = bottom × high
Area of trapezoid = (upper bottom + lower bottom) × High ÷ 2
Diameter = radius × 2 radius = diameter ÷ 2
Circumference of a circle = Pi × Diameter=
PI × radius × two
Area of circle = Pi × radius × radius
Surface area of the box=
(long) × Width + length × Height + width × High) × two
Volume of box = length × wide × high
Surface area of cube = edge length × Edge length × six
Volume of cube = edge length × Edge length × Edge length
Side area of cylinder = perimeter of bottom circle × high
Surface area of cylinder = upper and lower bottom surface area + side area
Volume of cylinder = bottom area × high
Volume of cone = bottom area × High ÷ 3
Box (cube, cylinder)
Volume = bottom area × high
Edit the plan drawing of this section
Name symbol perimeter C and area s
Square a - side length C = 4A
S＝a2
Rectangle A and B - side length C = 2 (a + b)
S＝ab
Triangle a, B, C - length of three sides
Height on H-A side
S - half of the perimeter
A. B, C - internal angle
Where s = (a + B + C) / 2 s = ah / 2
＝ab/2?sinC
＝[s(s-a)(s-b)(s-c)]1/2
＝a2sinBsinC/(2sinA)
Edit the inference and theorem in this paragraph
1 there is only one straight line through two points
2. The shortest line segment between two points
3 the complementary angles of the same angle or equal angle are equal
The remainder of the same angle or equal angle is equal
5 there is and only one straight line perpendicular to the known straight line
6 among all the line segments connected by a point outside the straight line and each point on the straight line, the vertical line segment is the shortest
The parallel axiom passes through a point outside the line, and there is and only one line parallel to the line
8 if both lines are parallel to the third line, the two lines are parallel to each other
The equipotential angles are equal and the two straight lines are parallel
10. The offset angles are equal and the two straight lines are parallel
11. The internal angles of the same side are complementary, and the two straight lines are parallel
12 the two straight lines are parallel and the equipotential angles are equal
13. The two straight lines are parallel and the internal offset angles are equal
14. The two straight lines are parallel and complementary to each other
Theorem 15 the sum of two sides of a triangle is greater than the third side
16 infer that the difference between the two sides of the triangle is less than the third side
17 sum theorem of internal angles of a triangle the sum of the three internal angles of a triangle is equal to 180 °
18 inference 1 the two acute angles of a right triangle are complementary to each other
19 inference 2 an outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it
20 inference 3 an external angle of a triangle is greater than any internal angle that is not adjacent to it
21 the corresponding sides and angles of congruent triangles are equal
22. The edge angle axiom (SAS) has two congruent triangles whose two sides are equal to their included angles
The 23 angle, edge and angle axiom (ASA) has two congruent triangles whose two angles are equal to their clamping edges
Inference (AAS) has two angles and the opposite side of one of them corresponds to two equal triangles congruent
The 25 side axiom (SSS) has two congruent triangles whose three sides correspond to the same
The hypotenuse and right angle axiom (HL) two right triangles with hypotenuse and one right angle are congruent
Theorem 1 the distance from the point on the bisector of an angle to both sides of the angle is equal
Theorem 2 points with the same distance from both sides of an angle are on the bisector of the angle
The bisector of an angle is the collection of all points with equal distance to both sides of the angle
Property theorem of isosceles triangle two base angles of isosceles triangle are equal (i.e. equilateral equal angle)
31 inference 1 the bisector of the top angle of an isosceles triangle bisects the bottom edge and is perpendicular to the bottom edge
The bisector of the top angle, the center line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide with each other
33 inference 3 the angles of an equilateral triangle are equal, and each angle is equal to 60 °
34 judgment theorem of isosceles triangle if two angles of a triangle are equal, the opposite sides of the two angles are also equal (isometric to equilateral)
Inference 1 a triangle whose three angles are equal is an equilateral triangle
36 inference 2 an isosceles triangle with an angle equal to 60 ° is an equilateral triangle
37 in a right triangle, if an acute angle is equal to 30 °, the right side opposite it is equal to half of the hypotenuse
38 the center line on the hypotenuse of a right triangle is equal to half of the hypotenuse
Theorem 39 the distance between the point on the vertical bisector of a line segment and the two endpoints of the line segment is equal
40 inverse theorem

Calculation formula of sector area and arc length I haven't learned it yet. I want to learn it in advance. You'd better give an example

S=∏R^2
L=2∏R
S = Π (R ^ 2) x / 360 x is the sector angle
l=∏RX/180
Respondent: Yu sunspot - Assistant Level 2 9-8 08:44

What are the arc length formula and sector area formula expressed in radian system? What are Secant and cosecant?

l=|a|r s=1/2lr=1/2*ar ²= 1/2 *l ²/ A where l is the arc length, R is the radius and a is the radian
Secant: properties of y = secx: (1) definition domain, {x|x ≠ π / 2 + K π, K ∈ Z} (2) value domain, ｜ secx ｜ ≥ 1. That is, secx ≥ 1 or secx ≤ - 1; (3) Y = secx is an even function, that is, sec (- x) = secx. The image is symmetrical to the y-axis; Thick lines are secant functions and thin lines are cosecant functions
(4) Y = secx is a periodic function. The period is 2K π (K ∈ Z, and K ≠ 0), and the minimum positive period T = 2 π. (5) the Secant and cosine are reciprocal to each other; Cosecant and sine are reciprocal to each other; (6) The secant function tends to a straight line infinitely, x = π / 2 + K π; (7) Secant function is unbounded; (8) Derivative of secant function: (secx) ′ = secx × tarx； (9) Indefinite integral of secant function: ∫ secxdx = ln ∣ secx + TaNx ∣ + C
The cosecant function is recorded as y = CSC α Properties: 1. In the definition of trigonometric function, CSC α= r/y ； 2. Cosecant function and sine are reciprocal to each other; 3. Definition domain: {x|x ≠ K π, K ∈ Z}; 4. Value range: {y | y ≤ - 1 or Y ≥ 1}, i.e. ▏ y ▏ ≥ 1; 5. Periodicity: the minimum positive period is 2 π; 6. Parity: odd function. (image asymptote is: x = k π cosecant function and sine function are reciprocal to each other)

What is the formula of sector arc length and area, expressed in radian system

Area = (1 / 2) LR = (1 / 2) θ r2

In a circle, the conversion formula between the length of the chord and the length of the arc to which it is opposite

Let the center angle be a degree and the radius be r
Express the chord length by trigonometric function
Chord length is R * sin (1 / 2a)
Formula for arc length
L = (a, R Square) / 360
The conversion formula can be obtained by dividing the two