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(c)=ba.
Answer? Sin (a)+b? Cos(a)=a2+b2cos(a-c) where tan(c)=ab.
1+sin(a)=(sin(a2)+cos(a2))2
1-sin(a)=(sin(a2)-cos(a2))2
formulas of trigonometric functions
Two-angle summation formula sin (a+b) = sinacosb+cosasinbsin (a-b) = sinacosb-sinbcosa.
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)
ctg(A+B)=(ctgActgB- 1)/(ctg B+ctgA)ctg(A-B)=(ctgActgB+ 1)/(ctg B-ctgA)
The angle doubling formula tan2a = 2tana/(1-tan2a) ctg2a = (ctg2a-1)/2ctga.
cos2a = cos2a-sin2a = 2 cos2a- 1 = 1-2 sin2a
Half-angle formula sin (a/2) = √ ((kloc-0/-COSA)/2) sin (a/2) =-√ ((kloc-0/-COSA)/2).
cos(A/2)=√(( 1+cosA)/2)cos(A/2)=-√(( 1+cosA)/2)
tan(A/2)=√(( 1-cosA)/(( 1+cosA))tan(A/2)=-√(( 1-cosA)/(( 1+cosA))
ctg(A/2)=√(( 1+cosA)/(( 1-cosA))ctg(A/2)=-√(( 1+cosA)/(( 1-cosA))
Sum-difference product 2sina cosb = sin (a+b)+sin (a-b) 2cosasinb = sin (a+b)-sin (a-b)
2 cosa cosb = cos(A+B)-sin(A-B)-2 sinasinb = cos(A+B)-cos(A-B)
sinA+sinB = 2 sin((A+B)/2)cos((A-B)/2 cosA+cosB = 2 cos((A+B)/2)sin((A-B)/2)
tanA+tanB = sin(A+B)/cosa cosb tanA-tanB = sin(A-B)/cosa cosb
ctgA+ctgBsin(A+B)/Sina sinb-ctgA+ctgBsin(A+B)/Sina sinb
The sum of the first n terms in some sequences is1+2+3+4+5+6+7+8+9+…+n = n (n+1)/21+3+5+7+9+/kloc-0.
2+4+6+8+ 10+ 12+ 14+…+(2n)= n(n+ 1) 12+22+32+42+52+62+72+82+…+N2 = n(n+ 1)(2n+ 1)/6
13+23+33+43+53+63+…n3 = N2(n+ 1)2/4 1 * 2+2 * 3+3 * 4+4 * 5+5 * 6+6 * 7+…+n(n+ 1)= n(n+ 1)(n+2)/3
Sine theorem a/sinA=b/sinB=c/sinC=2R Note: where r represents the radius of the circumscribed circle of a triangle.
Cosine Theorem b2=a2+c2-2accosB Note: Angle B is the included angle between side A and side C..
The standard equation of a circle (x-a)2+(y-b)2=r2 Note: (A, B) is the center coordinate.
General equation of circle x2+y2+Dx+Ey+F=0 Note: D2+E2-4f > 0
Parabolic standard equation y2=2px y2=-2px x2=2py x2=-2py
Lateral area of a straight prism S=c*h lateral area of an oblique prism s = c' * h.
Lateral area of a regular pyramid S= 1/2c*h' lateral area of a regular prism S= 1/2(c+c')h'
The lateral area of the frustum of a cone S = 1/2(c+c')l = pi(R+R)l The surface area of the ball S=4pi*r2.
Lateral area of cylinder S=c*h=2pi*h lateral area of cone s =1/2 * c * l = pi * r * l.
The arc length formula l=a*r a is the radian number r > of the central angle; 0 sector area formula s= 1/2*l*r
Conical volume formula V= 1/3*S*H Conical volume formula V= 1/3*pi*r2h
Oblique prism volume V=S'L Note: where s' is the straight cross-sectional area and l is the side length.
Cylinder volume formula V=s*h cylinder
Two-dimensional graph
Here are some formulas for the perimeter and area of two-dimensional graphics.
Circle:
Radius = r diameter d = 2r
Perimeter = 2π r = π d
Area = π R2 (π = 3. 14 15926 ...)
Ellipse:
Area = π ab
A and b represent half of the minor axis and the major axis, respectively.
Rectangular:
Area = ab
Circumference = 2a+2b
Parallelogram (parallelogram):
Area = BH = absinα
Circumference = 2a+2b
Trapezoid:
Area = 1/2h (a+b)
Circumference = a+b+h (secα+secβ)
Regular n-polygon:
Area = 1/2nb2cot (180/n)
Perimeter = nb
Quadrilateral (I):
Area = 1/2absinα
Quadrilateral (2):
Area =1/2 (h1+H2) b+ah1+CH2.
3D graphics
The following is the formula of the volume and surface area (including the bottom surface) of a three-dimensional solid.
Sphere:
Volume = 4/3 π R3
Surface area = 4 π R2
Cube:
Volume = ABC
Surface area = 2 (AB+AC+BC)
Cylinder:
Volume = π r2h
Surface area = 2π RH+2π R2
Cone:
Volume = 1/3 π r2h
Surface area = π r √ R2+H2+π R2
Triangular cone:
If the bottom area is a,
Volume = 1/3ah
Truncated body:
Volume = 1/3 π h (A2+AB+B2)
Surface area = π (a+b) c+π a2+π b2
Ellipsoid:
Volume = 4/3 π ABC
Torus:
Volume =1/4π 2 (a+b) (b–a) 2
Surface area = π 2 (B2–A2)
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