Let A, B and C be three internal angles of a triangle.
tanA+tanB+tanC=tanAtanBtanC
cotAcotB+cotBcotC+cotCcotA = 1
(cosa)^2+(cosb)^2+(cosc)^2+2cosacosbcosc= 1
cosA+cosB+cosC = 1+4 sin(A/2)sin(B/2)sin(C/2)
tan(A/2)tan(B/2)+tan(B/2)tan(C/2)+tan(C/2)tan(A/2)= 1
sin2A+sin2B+sin2C = 4 sinasinbsinc
sinA+sin B+sinC = 4 cos(A/2)cos(B/2)cos(C/2)
Double angle formula
sin2A=2sinA? Kosa
cos2a=cos^2a-sin^2a= 1-2sin^2a=2cos^2a- 1
tan2A=(2tanA)/( 1-tan^2A)
Triple angle formula
sin3α=4sinα sin(π/3+α)sin(π/3-α)
cos3α=4cosα cos(π/3+α)cos(π/3-α)
tan3a = tan a tan(π/3+a) tan(π/3-a)
Derivation of triple angle formula
sin3a
=sin(2a+a)
=sin2acosa+cos2asina
=2sina( 1-sin^2a)+( 1-2sin^2a)sina
=3sina-4sin^3a
cos3a
=cos(2a+a)
=cos2acosa-sin2asina
=(2cos^2a- 1)cosa-2( 1-cos^a)cosa
=4cos^3a-3cosa
sin3a=3sina-4sin^3a
=4sina(3/4-sin^2a)
=4sina[(√3/2)^2-sin^2a]
=4sina(sin^260 -sin^2a)
=4sina(sin60 +sina)(sin60 -sina)
= 4 Sina * 2 sin[(60+a)/2]cos[(60-a)/2]* 2 sin[(60-a)/2]cos[(60-a)/2]
=4sinasin(60 +a)sin(60 -a)
cos3a=4cos^3a-3cosa
=4cosa(cos^2a-3/4)
=4cosa[cos^2a-(√3/2)^2]
=4cosa(cos^2a-cos^230)
=4cosa(cosa+cos30 )(cosa-cos30)
= 4 cosa * 2cos[(a+30)/2]cos[(a-30)/2]* {-2 sin[(a+30)/2]sin[(a-30)/2]}
=-4 eicosapentaenoic acid (a+30) octyl (a-30)
=-4 Coxsacin [90-(60-a)] Xin [-90 +(60 +a)]
=-4 cos(60-a)[-cos(60+a)]
= 4 cos(60-a)cos(60+a)
Comparing the above two formulas, we can get
tan3a=tanatan(60 -a)tan(60 +a)
half-angle formula
tan(A/2)=( 1-cosA)/sinA = sinA/( 1+cosA);
cot(A/2)= sinA/( 1-cosA)=( 1+cosA)/sinA。
sin^2(a/2)=( 1-cos(a))/2
cos^2(a/2)=( 1+cos(a))/2
tan(a/2)=( 1-cos(a))/sin(a)= sin(a)/( 1+cos(a))
Sum difference product
sinθ+sinφ= 2 sin[(θ+φ)/2]cos[(θ-φ)/2]
sinθ-sinφ= 2 cos[(θ+φ)/2]sin[(θ-φ)/2]
cosθ+cosφ= 2 cos[(θ+φ)/2]cos[(θ-φ)/2]
cosθ-cosφ=-2 sin[(θ+φ)/2]sin[(θ-φ)/2]
tanA+tanB = sin(A+B)/cosa cosb = tan(A+B)( 1-tanA tanB)
tanA-tanB = sin(A-B)/cosa cosb = tan(A-B)( 1+tanA tanB)
Sum and difference of products
sinαsinβ = [cos(α-β)-cos(α+β)] /2
cosαcosβ = [cos(α+β)+cos(α-β)]/2
sinαcosβ = [sin(α+β)+sin(α-β)]/2
cosαsinβ = [sin(α+β)-sin(α-β)]/2
Hyperbolic function
sinh(a) = [e^a-e^(-a)]/2
cosh(a) = [e^a+e^(-a)]/2
tanh(a) = sin h(a)/cos h(a)
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 the trigonometric function values of 2π-α and α can be obtained by Formula-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)
a sin(ωt+θ)+B sin(ωt+φ)= 1
√{(A^2 +B^2 +2ABcos(θ-φ)}? sin{ ωt + arcsin[ (A? sinθ+B? sinφ)/√{a^2 +b^2; +2ABcos(θ-φ)} }
√ indicates the root number, including the contents in {...}.
Inductive formula
Sine (-α) =-Sine α
cos(-α) = cosα
tan (-α)=-tanα
sin(π/2-α) = cosα
cos(π/2-α) = sinα
sin(π/2+α) = cosα
cos(π/2+α) = -sinα
Sine (π-α) = Sine α
cos(π-α) = -cosα
Sine (π+α) =-Sine α
cos(π+α) = -cosα
tanA= sinA/cosA
tan(π/2+α)=-cotα
tan(π/2-α)=cotα
tan(π-α)=-tanα
tan(π+α)=tanα
Inductive formula memory skills: odd variables are unchanged, and symbols look at quadrants.
Other formulas
( 1)(sinα)^2+(cosα)^2= 1
(2) 1+(tanα)^2=(secα)^2
(3) 1+(cotα)^2=(cscα)^2
To prove the following two formulas, just divide one formula by (sin α) 2 and the second formula by (cos α) 2.
(4) For any non-right triangle, there is always
tanA+tanB+tanC=tanAtanBtanC
Certificate:
A+B=π-C
tan(A+B)=tan(π-C)
(tanA+tanB)/( 1-tanA tanB)=(tanπ-tanC)/( 1+tanπtanC)
Surface treatment can be carried out.
tanA+tanB+tanC=tanAtanBtanC
Obtain a certificate
It can also be proved that this relationship holds when x+y+z=nπ(n∈Z).
The following conclusions can be drawn from tana+tanbtana+tanb+tanc = tanatanbtanc.
(5)cotAcotB+cotAcotC+cotbctc = 1
(6) Cost (A/2)+ Cost (B/2)+ Cost (C/2)= Cost (A/2) Cost (B/2)
(7)(cosa)^2+(cosb)^2+(cosc)^2= 1-2cosacosbcosc
(8)(sina)^2+(sinb)^2+(sinc)^2=2+2cosacosbcosc
Other non-critical trigonometric functions
csc(a) = 1/sin(a)
Seconds (a)= 1/ cosine (a)