tanα cotα= 1
sinα cscα= 1
cosαsecα= 1 sinα/cosα= tanα= secα/CSCα
cosα/sinα= cotα= CSCα/secαsin 2α+cos 2α= 1
1+tan2α=sec2α
1+cot2α=csc2α
Inductive formula
Sine (-α) =-Sine α
cos(-α)=cosα tan(-α)=-tanα
Kurt (-α) =-Kurt 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
Sine (π-α) = Sine α
cos(π-α)=-cosα
tan(π-α)=-tanα
cot(π-α)=-coα
Sine (π+α) =-Sine α
cos(π+α)=-cosα
tan(π+α)=tanα
cot(π+α)=cotα
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
Sine (2π-α)=- Sine α
cos(2π-α)=cosα
tan(2π-α)=-tanα
Kurt (2π-α)=- Kurt α
sin(2kπ+α)=sinα
cos(2kπ+α)=cosα
tan(2kπ+α)=tanα
cot(2kπ+α)=cotα
(where k∈Z)
General formula for sum and difference of formulas of trigonometric functions's two angles
sin(α+β)=sinαcosβ+cosαsinβ
sin(α-β)=sinαcosβ-cosαsinβ
cos(α+β)=cosαcosβ-sinαsinβ
cos(α-β)=cosαcosβ+sinαsinβ
tanα+tanβ
tan(α+β)=———
1-tanα tanβ
tanα-tanβ
tan(α-β)=———
1+tanα tanβ
2 tons (α/2)
sinα=————
1+tan2(α/2)
1-tan2(α/2)
cosα=————
1+tan2(α/2)
2 tons (α/2)
tanα=————
1-tan2(α/2)
Sine, cosine and tangent formulas of double angles Sine, cosine and tangent formulas of triangle
sin2α=2sinαcosα
cos 2α= cos 2α-sin 2α= 2 cos 2α- 1 = 1-2 sin 2α
2tanα
tan2α=———
1-tan2α
sin3α=3sinα-4sin3α
cos3α=4cos3α-3cosα
3tanα-tan3α
tan3α=————
1-3tan2α
Sum and difference product formula of trigonometric function
α+β α-β
sinα+sinβ= 2 sin—-cos——
2 2
α+β α-β
sinα-sinβ= 2cos—-sin——
2 2
α+β α-β
cosα+cosβ= 2cos—-cos——
2 2
α+β α-β
cosα-cosβ=-2 sin—-sin——
2 2 1
sinα cosβ=-[sin(α+β)+sin(α-β)]
2
1
cosα sinβ=-[sin(α+β)-sin(α-β)]
2
1
cosα cosβ=-[cos(α+β)+cos(α-β)]
2
1
sinαsinβ=-[cos(α+β)-cos(α-β)]
2
Convert asinα bcosα into trigonometric function of angle (formulas of trigonometric functions of auxiliary angle).
Supplementary difference grading formula
(sinx)'=cosx (cosx)'=-sinx
(tanx)'=(secx)^2
(cotx)'=-(cscx)^2
(secx)'=secx*tanxtx
(cscx)'=-cscx*cotx
arcsinx)'=( 1-x^2)^(- 1/2)
arccosx)'=-( 1-x^2)^(- 1/2)
arctanx)'=( 1+^2)^(- 1)
artcotx0'=- 1/( 1+x^2)
PS。 X 2 means the square of X.
1. inductive formula
sin(-a)=-sin(a)
cos(-a)=cos(a)
sin(π2-a)=cos(a)
cos(π2-a)=sin(a)
sin(π2+a)=cos(a)
cos(π2+a)=-sin(a)
sin(π-a)=sin(a)
cos(π-a)=-cos(a)
sin(π+a)=-sin(a)
cos(π+a)=-cos(a)
2. The trigonometric function of the sum and difference of two angles
sin(a+b)= sin(a)cos(b)+cos(α)sin(b)
cos(a+b)= cos(a)cos(b)-sin(a)sin(b)
sin(a-b)= sin(a)cos(b)-cos(a)sin(b)
cos(a-b)= cos(a)cos(b)+sin(a)sin(b)
tan(a+b)= tan(a)+tan(b) 1-tan(a)tan(b)
tan(a-b)= tan(a)-tan(b) 1+tan(a)tan(b)
3. Sum-difference product formula
sin(a)+sin(b)= 2s in(a+B2)cos(a-B2)
Crime (1)? sin(b)=2cos(a+b2)
cos(a)+cos(b)= 2cos(a+B2)cos(a-B2)
cos(a)-cos(b)=-2s in(a+B2)sin(a-B2)
4. Double angle formula
sin(2a)=2sin(a)cos(b)
cos(2a)= cos 2(a)-sin 2(a)= 2cos 2(a)- 1 = 1-2 sin 2(a)
5. Half-angle formula
sin2(a2)= 1-cos(a)2
cos2(a2)= 1+cos(a)2
tan(a2)= 1-cos(a)sin(a)= Sina 1+cos(a)
6. General formula
sin(a)=2tan(a2) 1+tan2(a2)
cos(a)= 1-tan 2(a2) 1+tan 2(a2)
tan(a)=2tan(a2) 1-tan2(a2)
7. Other formulas (derived)
Answer? Sin (a)+b? Cos(a)=a2+b2sin(a+c) where tan? =ba
Answer? Sin (a)+b? Cos(a)=a2+b2cos(a-c) where tan? =ab
1+sin(a)=(sin(a2)+cos(a2))2
1-sin(a)=(sin(a2)-cos(a2))2
Trigonometric identity
sin 2θ+cos 2θ= 1； 1+tan 2θ= sec 2θ； 1+cot2θ=csc2θ
Complex angle formula
sin(A+B)= Sina cosb+cosa sinb； sin(A–B)= Sina cosb–cosa sinb
cos(A+B)= cosa cosb–Sina sinb； cos(A–B)= cosa cosb+Sina sinb
Double angle formula
sin2θ=2sinθcosθ
cos 2θ= cos 2θ–sin 2θ= 2 cos 2θ– 1 = 1–2 sin 2θ
Double square
sin 2θ= 1-cos 2θ2； cos2θ= 1+cos2θ 2
Sum and difference of products
2 Sina cosb = sin(A+B)+sin(A–B)
2 cosa sinb = sin(A+B)-sin(A–B)
2 sinas inb = cos(A–B)–cos(A+B)
2 cos acosb = cos(A–B)+cos(A+B)
Basic formula of trigonometric function
Sinθ= diagonal (sine),
Cosθ= hypotenuse of adjacent side (cosine),
Tanθ=sinθ cosθ (tangent)
Cotθ=cosθ sinθ (cotangent),
Secθ= 1 cosθ (secant),
Csθ= 1 sinθ (cotangent)
1. inductive formula
sin(-a)=-sin(a)
cos(-a)=cos(a)
sin(π2-a)=cos(a)
cos(π2-a)=sin(a)
sin(π2+a)=cos(a)
cos(π2+a)=-sin(a)
sin(π-a)=sin(a)
cos(π-a)=-cos(a)
sin(π+a)=-sin(a)
cos(π+a)=-cos(a)
2. The trigonometric function of the sum and difference of two angles
sin(a+b)= sin(a)cos(b)+cos(α)sin(b)
cos(a+b)= cos(a)cos(b)-sin(a)sin(b)
sin(a-b)= sin(a)cos(b)-cos(a)sin(b)
cos(a-b)= cos(a)cos(b)+sin(a)sin(b)
tan(a+b)= tan(a)+tan(b) 1-tan(a)tan(b)
tan(a-b)= tan(a)-tan(b) 1+tan(a)tan(b)
3. Sum-difference product formula
sin(a)+sin(b)= 2s in(a+B2)cos(a-B2)
Crime (1)? sin(b)=2cos(a+b2)
cos(a)+cos(b)= 2cos(a+B2)cos(a-B2)
cos(a)-cos(b)=-2s in(a+B2)sin(a-B2)
4. Double angle formula
sin(2a)=2sin(a)cos(b)
cos(2a)= cos 2(a)-sin 2(a)= 2cos 2(a)- 1 = 1-2 sin 2(a)
5. Half-angle formula
sin2(a2)= 1-cos(a)2
cos2(a2)= 1+cos(a)2
tan(a2)= 1-cos(a)sin(a)= Sina 1+cos(a)
6. General formula
sin(a)=2tan(a2) 1+tan2(a2)
cos(a)= 1-tan 2(a2) 1+tan 2(a2)
tan(a)=2tan(a2) 1-tan2(a2)
7. Other formulas (derived)
Answer? Sin (a)+b? Cos(a)=a2+b2sin(a+c) where tan? =ba
Answer? Sin (a)+b? Cos(a)=a2+b2cos(a-c) where tan? =ab
1+sin(a)=(sin(a2)+cos(a2))2
1-sin(a)=(sin(a2)-cos(a2))2