A right triangle with sides A, B and C, in which an included angle is θ. Its six trigonometric functions are sine, cosine, tangent, cotangent, secant and cotangent.
sinθ=b/c cosθ=a/c tanθ=b/a
csθ= c/b secθ= c/a cotθ= a/b
If the radius of a circle is 1, its sine and cosine are the height and bottom of a right triangle respectively.
a=cosθ b=sinθ
According to Pythagorean theorem, we know that A2+B2 = C2. Therefore, for any angle θ on the circle, we can get the following complete equation:
cos2θ+sin2θ= 1
Trigonometric identity
According to the definitions described in the previous pages, the following identities can be obtained:
tanθ=sinθ/cosθ,cotθ=cosθ/sinθ
secθ= 1/cosθ,csθ= 1/sinθ
Divide cos 2θ+sin 2θ = 1 by cos 2θ and sin 2θ, respectively, and you can get:
sec 2θ–tan 2θ= 1,cs C2θ–cot 2θ= 1。
For negative angles, the six trigonometric functions are:
sin(–θ)=–sinθCSC(–θ)=–csθ
cos(–θ)= cosθsec(–θ)= secθ
tan(–θ)=–tanθcot(–θ)=–cotθ
When two angles are added, the angle and formula are used:
sin(α+β)= sinαcosβ+cosαsinβ
cos(α+β)= cosαcosβ–sinαsinβ
tan(α+β)= tanα+tanβ/ 1–tanαtanβ
If you encounter a double angle or a triple angle, please use the double angle formula:
sin 2α= 2 sinαcosαsin 3α= 3 sinαcos 2α–sin 3α
cos 2α= cos 2α–sin 2αcos 3α= cos 3α–3 sin 2αcosα
tan 2α= 2 tanα/ 1–tan 2α
tan 3α= 3 tanα–tan 3α/ 1–3 tan 2α
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)