How to calculate the radius of an arc with known chord length and arch height? Given the chord length and arch height, the formula for finding the radius? Given the arc length and arch height, the formula for calculating the chord length?

For chord length d and arch height h, the formula for calculating radius R: r = D ^ 2 / (8h) + H / 2. Given arc length C and arch height h, it is difficult to calculate chord length L. let me talk about the next method and introduce the parameter @ = C / R sin @ = (R-H) / R, so that R can be obtained

Find the radius of an arc with known chord length and height For example, the chord length is 100, the height between the midpoint of the chord and the midpoint of the arc is 30, and the radius of the arc is calculated. I hope to have the simplest formula

Let the radius be r, the chord length be m, and the height be n
R＾2＝（M/2）＾2+（R-N）＾2

Formula for radian?

l=ra

Formula of triangle area (about trigonometric function) Senior one compulsory 4

Let the three sides of the triangle be a, B and C respectively, and the angles of each side are a, B and C respectively
The area of a triangle is: 1 / 2 * a * b * sinc = 1 / 2 * b * c * Sina = 1 / 2 * a * c * SINB

Trigonometric function area formula It uses trigonometric function to calculate the area There are three

·Square relation:
sin^2 α＋ cos^2 α＝ one
1＋tan^2 α＝ sec^2 α
1＋cot^2 α＝ csc^2 α
·Product relationship:
sin α= tan α× cos α
cos α= cot α× sin α
tan α= sin α× sec α
cot α= cos α× csc α
sec α= tan α× csc α
csc α= sec α× cot α
·Reciprocal relationship:
tan α · cot α＝ one
sin α · csc α＝ one
cos α · sec α＝ one
Relationship with suppliers:
sin α/ cos α＝ tan α＝ sec α/ csc α
cos α/ sin α＝ cot α＝ csc α/ sec α
In right triangle ABC,
The sine of angle a is equal to the opposite side of angle a than the oblique side,
The cosine is equal to the adjacent edge of angle a than the hypotenuse
The tangent is equal to the opposite side next to the adjacent side,
·[1] Constant deformation formula of trigonometric function
·Trigonometric function of sum and difference of two angles:
cos( α+β)= cos α· cos β- sin α· sin β
cos( α-β)= cos α· cos β+ sin α· sin β
sin( α ± β)= sin α· cos β ±cos α· sin β
tan( α+β)= (tan α+ tan β)/ (1-tan α· tan β)
tan( α-β)= (tan α- tan β)/ (1+tan α· tan β)
·Trigonometric function of trigonometric sum:
sin( α+β+γ)= sin α· cos β· cos γ+ cos α· sin β· cos γ+ cos α· cos β· sin γ- sin α· sin β· sin γ
cos( α+β+γ)= cos α· cos β· cos γ- cos α· sin β· sin γ- sin α· cos β· sin γ- sin α· sin β· cos γ
tan( α+β+γ)= (tan α+ tan β+ tan γ- tan α· tan β· tan γ)/ (1-tan α· tan β- tan β· tan γ- tan γ· tan α)
·Auxiliary angle formula:
Asin α+ Bcos α= (A ²+ B ²)^ (1/2)sin( α+ t) , where
sint=B/(A ²+ B ²)^ (1/2)
cost=A/(A ²+ B ²)^ (1/2)
tant=B/A
Asin α- Bcos α= (A ²+ B ²)^ (1/2)cos( α- t),tant=A/B
·Angle doubling formula:
sin(2 α)= 2sin α· cos α= 2/(tan α+ cot α)
cos(2 α)= cos ² ( α)- sin ² ( α)= 2cos ² ( α)- 1=1-2sin ² ( α)
tan(2 α)= 2tan α/ [1-tan ² ( α)]
·Triple angle formula:
sin(3 α)= 3sin α- 4sin ³ ( α)
cos(3 α)= 4cos ³ ( α)- 3cos α
·Half angle formula:
sin( α/ 2)=±√((1-cos α)/ 2)
cos( α/ 2)=±√((1+cos α)/ 2)
tan( α/ 2)=±√((1-cos α)/ (1+cos α))= sin α/ (1+cos α)= (1-cos α)/ sin α
·Power reduction formula
sin ² ( α)= (1-cos(2 α))/ 2=versin(2 α)/ two
cos ² ( α)= (1+cos(2 α))/ 2=covers(2 α)/ two
tan ² ( α)= (1-cos(2 α))/ (1+cos(2 α))
·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)]
·Integration sum difference formula:
sin α· cos β= (1/2)[sin( α+β)+ sin( α-β)]
cos α· sin β= (1/2)[sin( α+β)- sin( α-β)]
cos α· cos β= (1/2)[cos( α+β)+ cos( α-β)]
sin α· sin β=- (1/2)[cos( α+β)- cos( α-β)]
·Sum difference product formula:
sin α+ sin β= 2sin[( α+β)/ 2]cos[( α-β)/ 2]
sin α- sin β= 2cos[( α+β)/ 2]sin[( α-β)/ 2]
cos α+ cos β= 2cos[( α+β)/ 2]cos[( α-β)/ 2]
cos α- cos β=- 2sin[( α+β)/ 2]sin[( α-β)/ 2]
·Derivation formula
tan α+ cot α= 2/sin2 α
tan α- cot α=- 2cot2 α
1+cos2 α= 2cos ²α
1-cos2 α= 2sin ²α
1+sin α= (sin α/ 2+cos α/ 2) ²
·Others:
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
cosx+cos2x+...+cosnx= [sin(n+1)x+sinnx-sinx]/2sinx
prove:
Left = 2sinx (cosx + cos2x +... + cosnx) / 2sinx
=[sin2x-0 + sin3x SiNx + sin4x-sin2x +... + sinnx sin (n-2) x + sin (n + 1) x-sin (n-1) x] / 2sinx (integration sum difference)
=[sin (n + 1) x + sinnx SiNx] / 2sinx = right
Proof of equation
sinx+sin2x+...+sinnx= - [cos(n+1)x+cosnx-cosx-1]/2sinx
prove:
Left = - 2sinx [SiNx + sin2x +... + sinnx] / (- 2sinx)
=[cos2x-cos0+cos3x-cosx+...+cosnx-cos(n-2)x+cos(n+1)x-cos(n-1)x]/(-2sinx)
=-[cos (n + 1) x + cosnx-cosx-1] / 2sinx = right
Proof of equation
[edit this paragraph] induction formula of trigonometric function
Formula 1:
set up α Is an arbitrary angle, and the values of the same trigonometric function of the same angle with the same end edge are equal:
sin（2kπ＋ α）＝ sin α
cos（2kπ＋ α）＝ cos α
tan（2kπ＋ α）＝ tan α
cot（2kπ＋ α）＝ cot α
Formula 2:
set up α Is any angle, π+ α Trigonometric function value and α Relationship between trigonometric function values of:
sin（π＋ α）＝－ sin α
cos（π＋ α）＝－ cos α
tan（π＋ α）＝ tan α
cot（π＋ α）＝ cot α
Formula 3:
Arbitrary angle α And- α Relationship between trigonometric function values of:
sin（－ α）＝－ sin α
cos（－ α）＝ cos α
tan（－ α）＝－ tan α
cot（－ α）＝－ cot α
Formula 4:
π can be obtained by formula 2 and formula 3- α And α Relationship between trigonometric function values of:
sin（π－ α）＝ sin α
cos（π－ α）＝－ cos α
tan（π－ α）＝－ tan α
cot（π－ α）＝－ cot α
Formula 5:
2 π can be obtained by Formula 1 and formula 3- α And α Relationship between trigonometric function values of:
sin（2π－ α）＝－ sin α
cos（2π－ α）＝ cos α
tan（2π－ α）＝－ tan α
cot（2π－ α）＝－ cot α
Formula 6:
π/2± α And 3 π / 2 ± α And α Relationship between trigonometric function values of:
sin（π/2＋ α）＝ cos α
cos（π/2＋ α）＝－ sin α
tan（π/2＋ α）＝－ cot α
cot（π/2＋ α）＝－ tan α
sin（π/2－ α）＝ cos α
cos（π/2－ α）＝ sin α
tan（π/2－ α）＝ cot α
cot（π/2－ α）＝ tan α
sin（3π/2＋ α）＝－ cos α
cos（3π/2＋ α）＝ sin α
tan（3π/2＋ α）＝－ cot α
cot（3π/2＋ α）＝－ tan α
sin（3π/2－ α）＝－ cos α
cos（3π/2－ α）＝－ sin α
tan（3π/2－ α）＝ cot α
cot（3π/2－ α）＝ tan α
(above K ∈ z)
[edit this paragraph] sine cosine theorem
Sine theorem means that in a triangle, the ratio of the sine of each side to the angle it faces is equal, that is, a / Sina = B / SINB = C / sinc = 2R
Cosine theorem means that the square of any side of a triangle is equal to the sum of the squares of the other sides minus twice the product of the cosine of the angle between these two sides and them, that is, a ^ 2 = B ^ 2 + C ^ 2-2bc cosa
The ratio of the opposite side of angle a to the oblique side is called the sine of angle A and is recorded as Sina, that is, Sina = the opposite side / oblique side of angle A
Angle between beveled edge and adjacent edge a
sin=y/r
Whether Y > x or Y ≤ x
No matter how small a is, it can be of any size
The maximum value of sine is 1 and the minimum value is-
[edit this paragraph] some advanced contents
·Exponential representation of trigonometric functions in Higher Algebra (easy to get from Taylor series):
sinx=[e^(ix)-e^(-ix)]/(2i)
cosx=[e^(ix)+e^(-ix)]/2
tanx=[e^(ix)-e^(-ix)]/[ie^(ix)+ie^(-ix)]
Taylor expansion has infinite series, e ^ z = exp (z) = 1 + Z / 1! + Z ^ 2 / 2! + Z ^ 3 / 3! + Z ^ 4 / 4! +... + Z ^ n / N! +
At this time, the definition domain of trigonometric function has been extended to the whole complex set
·Trigonometric functions as differential equations
For the system of differential equations y = - y ''; y = y '', there is a general solution Q, which can be proved
Q = asinx + bcosx, so you can also define trigonometric functions from here
Supplement: by the corresponding exponent, we can define a similar function - hyperbolic function, which has many properties similar to trigonometric function
Trigonometric function of special angle:
Angle a 0 ° 30 ° 45 ° 60 ° 90 ° 120 ° 180 °
1.sina 0 1/2 √2/2 √3/2 1 √3/2 0
2.cosa 1 √3/2 √2/2 1/2 0 -1/2 -1
3. Tana 0 √ 3 / 3 1 √ 3 infinite - √ 3 0
4.cota / √3 1 √3/3 0 -√3/3 /
[edit this paragraph] calculation of trigonometric function
power series
c0+c1x+c2x2+...+cnxn+...=∑cnxn (n=0..∞)
c0+c1(x-a)+c2(x-a)2+...+cn(x-a)n+...=∑cn(x-a)n (n=0..∞)
Their terms are power functions of positive integer powers, in which C0, C1, C2,... CN... And a are constants. This series is called power series
Taylor expansion (power series expansion method):
f(x)=f(a)+f'(a)/1!*(x-a)+f''(a)/2!*(x-a)2+...f(n)(a)/n!*(x-a)n+...
Practical power series:
ex = 1+x+x2/2!+x3/3!+...+xn/n!+...
ln(1+x)= x-x2/3+x3/3-...(-1)k-1*xk/k+... (|x|

Given a sector, the arc length of the center angle of the circle with radian number 2 is 6, calculate the radius and area of the sector?

Circumference: arc length = 2 π: 2
So:
Circumference = 2 π × 6÷2=9π
Sector radius = 9 π ÷ 2 π = 4.5
Area = π × four point five ² ÷（2π÷2）=10.125

How to calculate the radius of an arc with known chord length and arch height? Given the chord length and arch height, the formula for finding the radius? Given the arc length and arch height, the formula for calculating the chord length?