According to sine theorem
a/sinA=b/sinB=c/sinC=2R
therefore
a=2R*sinA
b=2R*sinB
c=2R*sinC
Add up a+b+c=2R*(sinA+sinB+sinC) and bring it in.
(a+b+c)/(sinA+sin b+ sinC)= 2R *(sinA+sin b+ sinC)/(sinA+sin b+ sinC)= 2R
Two-angle sum formula
sin(A+B)=sinAcosB+cosAsinB
sin(A-B)=sinAcosB-cosAsinB
cos(A+B)=cosAcosB-sinAsinB
cos(A-B)=cosAcosB+sinAsinB
tan(A+B)=(tanA+tanB)/( 1-tanA tanB)
tan(A-B)=(tanA-tanB)/( 1+tanA tanB)
cot(A+B)=(cotA cotB- 1)/(cot B+cotA)
cot(A-B)=(cotA cotB+ 1)/(cot b-cotA)
Double angle formula
Sin2A=2SinA? Kosa
Properties and Derivation of Logarithm
Use 0 to represent the power, and log(a)(b) to represent the logarithm of b with a as the base.
* means multiplication symbol,/means division symbol.
Define formula:
If a n = b(a >;; 0 and a ≠ 1)
Then n=log(a)(b)
Basic nature:
1.a^(log(a)(b))=b
2 . log(a)(MN)= log(a)(M)+log(a)(N);
3 . log(a)(M/N)= log(a)(M)-log(a)(N);
4.log(a)(M^n)=nlog(a)(M)
infer
1. This need not be pushed, but can be obtained directly from the definition (bring [n=log(a)(b)] in the definition into a n = b).
2.
MN=M*N
By the basic properties of 1 (replacing m and n)
a^[log(a)(mn)]=a^[log(a)(m)]*a^[log(a)(n)]
According to the nature of the index
a^[log(a)(mn)]=a^{[log(a)(m)]+[log(a)(n)]}
And because exponential function is monotone function, so
log(a)(MN)=log(a)(M)+log(a)(N)
3. Similar to 2.
MN=M/N
By the basic properties of 1 (replacing m and n)
a^[log(a)(m/n)]=a^[log(a)(m)]/a^[log(a)(n)]
According to the nature of the index
a^[log(a)(m/n)]=a^{[log(a)(m)]-[log(a)(n)]}
And because exponential function is monotone function, so
Logarithm (a)(M/N)= Logarithm (a)(M)- Logarithm (a)(N)
4. Similar to 2.
M^n=M^n
From the basic attribute 1 (replace m)
a^[log(a)(m^n)]={a^[log(a)(m)]}^n
According to the nature of the index
a^[log(a)(m^n)]=a^{[log(a)(m)]*n}
And because exponential function is monotone function, so
log(a)(M^n)=nlog(a)(M)
Other attributes:
Attribute 1: bottoming formula
log(a)(N)=log(b)(N)/log(b)(a)
Derived as follows
N=a^[log(a)(N)]
a=b^[log(b)(a)]
By combining the two formulas, it can be concluded that.
n={b^[log(b)(a)]}^[log(a)(n)]=b^{[log(a)(n)]*[log(b)(a)]}
And because n = b [log (b) (n)]
therefore
b^[log(b)(n)]=b^{[log(a)(n)]*[log(b)(a)]}
therefore
log(b)(n)=[log(a)(n)]*[log(b)(a)]
So log(a)(N)=log(b)(N)/log(b)(a)
Nature 2: (I don't know what it's called)
log(a^n)(b^m)=m/n*[log(a)(b)]
Derived as follows
Through the formula [lnx is log (e) (x), and e is called the base of natural logarithm]
log(a^n)(b^m)=ln(a^n)/ln(b^n)
It can be obtained from basic attribute 4.
log(a^n)(b^m)=[n*ln(a)]/[m*ln(b)]=(m/n)*{[ln(a)]/[ln(b)]}
Then according to the bottom changing formula
log(a^n)(b^m)=m/n*[log(a)(b)]
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Formula 3:
log(a)(b)= 1/log(b)(a)
Proved as follows:
Log(a)(b)= log(b)(b)/log(b)(a)- Logarithm based on b, log(b)(b)= 1.
= 1/log(b)(a)
Also deformable:
log(a)(b)*log(b)(a)= 1
Square relation:
sin^2(α)+cos^2(α)= 1
tan^2(α)+ 1=sec^2(α)
cot^2(α)+ 1=csc^2(α)
Relationship between businesses:
tanα=sinα/cosαcotα=cosα/sinα
Reciprocal relationship:
tanα? cotα= 1
sinα? cscα= 1
cosα? secα= 1
General formula:
sinα=2tan(α/2)/[ 1+tan^2(α/2)]
cosα=[ 1-tan^2(α/2)]/[ 1+tan^2(α/2)]
tanα=2tan(α/2)/[ 1-tan^2(α/2)]
Commonly used inductive formulas have the following groups:
Formula 1:
Let α be an arbitrary angle, and the values of the same trigonometric function with the same angle of the terminal edge are equal:
sin(2kπ+α)=sinα
cos(2kπ+α)=cosα
tan(2kπ+α)=tanα
cot(2kπ+α)=cotα
Equation 2:
Let α be an arbitrary angle, and the relationship between the trigonometric function value of π+α and the trigonometric function value of α;
Sine (π+α) =-Sine α
cos(π+α)=-cosα
tan(π+α)=tanα
cot(π+α)=cotα
Formula 3:
The relationship between arbitrary angle α and the value of-α trigonometric function;
Sine (-α) =-Sine α
cos(-α)=cosα
tan(-α)=-tanα
Kurt (-α) =-Kurt α
Equation 4:
The relationship between π-α and the trigonometric function value of α can be obtained by Formula 2 and Formula 3:
Sine (π-α) = Sine α
cos(π-α)=-cosα
tan(π-α)=-tanα
cot(π-α)=-coα
Formula 5:
The relationship between 2π-α and the trigonometric function value of α can be obtained by formula 1 and formula 3:
Sine (2π-α)=- Sine α
cos(2π-α)=cosα
tan(2π-α)=-tanα
Kurt (2π-α)=- Kurt α
Equation 6:
The relationship between π/2 α and 3 π/2 α and the trigonometric function value of α;
sin(π/2+α)=cosα
cos(π/2+α)=-sinα
tan(π/2+α)=-cotα
cot(π/2+α)=-tanα
sin(π/2-α)=cosα
cos(π/2-α)=sinα
tan(π/2-α)=cotα
cot(π/2-α)=tanα
sin(3π/2+α)=-cosα
cos(3π/2+α)=sinα
tan(3π/2+α)=-cotα
cot(3π/2+α)=-tanα
sin(3π/2-α)=-cosα
cos(3π/2-α)=-sinα
tan(3π/2-α)=cotα
cot(3π/2-α)=tanα
(higher than k∈Z)
The most commonly used formulas are:
Sin(A+B)=SinA*CosB+SinB*CosA
Sin(A-B)=SinA*CosB-SinB*CosA
Cos(A+B)=CosA*CosB-SinA*SinB
Cos(A-B)=CosA*CosB+SinA*SinB
tan(A+B)=(TanA+TanB)/( 1-TanA * TanB)
tan(A-B)=(TanA-TanB)/( 1+TanA * TanB)
Square relation:
sin^2(α)+cos^2(α)= 1
tan^2(α)+ 1=sec^2(α)
cot^2(α)+ 1=csc^2(α)
Product relationship:
sinα=tanα*cosα
cosα=cotα*sinα
tanα=sinα*secα
cotα=cosα*cscα
secα=tanα*cscα
csα= secα* cotα
Reciprocal relationship:
tanα? cotα= 1
sinα? cscα= 1
cosα? secα= 1
In the right triangle ABC,
The sine value of angle a is equal to the ratio of the opposite side to the hypotenuse of angle a,
Cosine is equal to the adjacent side of angle a than the hypotenuse.
The tangent is equal to the opposite side of the adjacent side,
Constant deformation formula of trigonometric function
Trigonometric function of sum and difference of two angles;
cos(α+β)=cosα? cosβ-sinα? sinβ
cos(α-β)=cosα? cosβ+sinα? sinβ
sin(α β)=sinα? cosβ cosα? sinβ
tan(α+β)=(tanα+tanβ)/( 1-tanα? tanβ)
tan(α-β)=(tanα-tanβ)/( 1+tanα? tanβ)
Auxiliary angle formula:
Asinα+bcosα = (A2+B2) (1/2) sin (α+t), where
sint=B/(A^2+B^2)^( 1/2)
cost=A/(A^2+B^2)^( 1/2)
Double angle formula:
sin(2α)=2sinα? cosα=2/(tanα+cotα)
cos(2α)=cos^2(α)-sin^2(α)=2cos^2(α)- 1= 1-2sin^2(α)
tan(2α)=2tanα/[ 1-tan^2(α)]
Triple angle formula:
sin(3α)=3sinα-4sin^3(α)
cos(3α)=4cos^3(α)-3cosα
Half-angle formula:
sin(α/2)= √(( 1-cosα)/2)
cos(α/2)= √(( 1+cosα)/2)
tan(α/2)=√(( 1-cosα)/( 1+cosα))= sinα/( 1+cosα)=( 1-cosα)/sinα
Reduced power formula
sin^2(α)=( 1-cos(2α))/2=versin(2α)/2
cos^2(α)=( 1+cos(2α))/2=vercos(2α)/2
tan^2(α)=( 1-cos(2α))/( 1+cos(2α))
General formula:
sinα=2tan(α/2)/[ 1+tan^2(α/2)]
cosα=[ 1-tan^2(α/2)]/[ 1+tan^2(α/2)]
tanα=2tan(α/2)/[ 1-tan^2(α/2)]
Product sum and difference formula:
sinα? cosβ=( 1/2)[sin(α+β)+sin(α-β)]
cosα? sinβ=( 1/2)[sin(α+β)-sin(α-β)]
cosα? cosβ=( 1/2)[cos(α+β)+cos(α-β)]
sinα? sinβ=-( 1/2)[cos(α+β)-cos(α-β)]
Sum-difference product formula:
sinα+sinβ= 2 sin[(α+β)/2]cos[(α-β)/2]
sinα-sinβ= 2cos[(α+β)/2]sin[(α-β)/2]
cosα+cosβ= 2cos[(α+β)/2]cos[(α-β)/2]
cosα-cosβ=-2 sin[(α+β)/2]sin[(α-β)/2]
Others:
sinα+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+2π * (n-1)/n] = 0 and
sin^2(α)+sin^2(α-2π/3)+sin^2(α+2π/3)=3/2
tanAtanBtan(A+B)+tanA+tan B- tan(A+B)= 0