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)