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
Kurt (π-α) =-Kurt α
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)
Relationship between products:
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-tanαtanβ)
tan(α-β)=(tanα-tanβ)/( 1+tanαtanβ)
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
Power reduction 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:
sinαcosβ=( 1/2)[sin(α+β)+sin(α-β)]
cosαsinβ=( 1/2)[sin(α+β)-sin(α-β)]
cosαcosβ=( 1/2)[cos(α+β)+cos(α-β)]
sinαsinβ=-( 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
Some advanced content
Exponential representation of trigonometric functions in higher algebra (easily obtained 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 domain of trigonometric function has been extended to the whole complex set.
Trigonometric function as the solution of differential equation;
For the differential equation y =-y ""; Y=y'', there is a general solution q, which can be proved.
Q=Asinx+Bcosx, so trigonometric functions can also be defined from this angle.
Supplement: Represented by the corresponding exponent, a similar function-hyperbolic function can be defined, which has many similar properties with trigonometric function, and both are very interesting.
Special trigonometric function value
A0 ' 30 ' 45 ' 60 ' 90 '
Sina 0 1/2ì2/2ì3/2 1
cosa 1√3/2√2/2 1/20
Tana 0√3/3 1√3 None.
Ketone √3 1√3/30
Calculation of trigonometric function
power series
c0+c 1x+c2x2+...+cnxn+...=∑cnxn(n=0..∞)
c0+c 1(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, where c0, c 1, c2, ... communication network (abbreviation of Communicating Net) ... and A are constants, and 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 | & lt 1)
sinx=x-x3/3！ +x5/5！ -...(- 1)k- 1 * x2k- 1/(2k- 1)！ +...(-∞& lt； x & lt∞)
cosx= 1-x2/2！ +x4/4！ -...(- 1)k*x2k/(2k)！ +...(-∞& lt； x & lt∞)
arcsinx = x+ 1/2 * x3/3+ 1 * 3/(2 * 4)* X5/5+...(| x | & lt 1)
arc cosx =π-(x+ 1/2 * x3/3+ 1 * 3/(2 * 4)* X5/5+...)(| x | & lt 1)
arctanx=x-x^3/3+x^5/5-...(x≤ 1)
sinhx=x+x3/3！ +x5/5！ +...(- 1)k- 1 * x2k- 1/(2k- 1)！ +...(-∞& lt； x & lt∞)
coshx= 1+x2/2！ +x4/4！ +...(- 1)k*x2k/(2k)！ +...(-∞& lt； x & lt∞)
arcsinhx = x- 1/2 * x3/3+ 1 * 3/(2 * 4)* X5/5-...(| x | & lt 1)
arctanhx=x+x^3/3+x^5/5+...(| x | & lt 1)
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Fourier series (trigonometric series)
f(x)=a0/2+∑(n=0..∞)(ancos NX+bns inx)
a0= 1/π∫(π..-π)(f(x))dx
an= 1/π∫(π..-π)(f(x)cosnx)dx
bn= 1/π∫(π..-π)(f(x)sinnx)dx
Note: Tangent can also be expressed as "Tg", such as TanA=TgA.
Sin2a=2SinaCosa
Cos2a=Cosa^2-Sina^2
= 1-2Sina^2
=2Cosa^2- 1
Tan2a=2Tana/ 1-Tana^2