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
Other commonly used trigonometric formulas
(sinα)^2-(sinβ)^2=(cosβ)^2-(cosα)^2=sin(α+β)sin(α-β)
(cosα)^2-(sinβ)^2=(cosβ)^2-(sinα)^2=cos(α+β)cos(α-β)
In triangle ABC,
cotAcotB+cotBcotC+cotAcotC = tan(A/2)tan(B/2)+tan(B/2)tan(C/2)+tan(C/2)tan(A/2)= 1
Solid geometry, Steiner theorem: in tetrahedron ABCD, let the angle formed by a straight line AB and CD be (AB, CD) and the distance be d(AB, CD), then
cos(ab,cd)=|[(ac^2+bd^2)-(ad^2+bc^2)]/2ab*cd|
d(AB,CD)=6V/[AB*CD*sin(AB,CD)]
De Morgan formula in set
Cu(A∪B)=(CuA)∩(CuB)
Cu(A∩B)=(CuA)∩(CuB)
Common formulas of complex numbers
Z = r(cosθ+is innθ), then z n = r n (cos θ+isinnθ).
Other commonly used formulas, such as plane geometry, number theory and combination, are too biased towards competition, so I won't talk about it.
This is all done by hand. . . I'm exhausted. Give me points ~ ~