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)]

- (

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