Trigonometric function is a kind of transcendental function in elementary function in mathematics. Their essence is the mapping between the set of arbitrary angles and a set of ratio variables. The usual trigonometric function is defined in the plane rectangular coordinate system, and its domain is the whole real number domain. The other is defined in a right triangle, but it is incomplete. Modern mathematics describes them as the limit of infinite sequence and the solution of differential equation, and extends their definitions to complex system.

Because of the periodicity of trigonometric function, it does not have the inverse function in the sense of single-valued function.

Trigonometric functions have important applications in complex numbers. Trigonometric function is also a common tool in physics.

basic content

It has six basic functions (elementary basic representation):

function name

sine

cosine

tangent

cotangent

secant

cosecant

sine function

sinθ=y/r

cosine function

cosθ=x/r

Tangent function

tanθ=y/x

Cotangent function

cotθ=x/y

secant

secθ=r/x

Csc function

csθ= r/y

And two functions that are not commonly used and easily eliminated:

Sine and sine

Version θ

= 1-cosθ

anticosine

Cosine θ

= 1-sinθ

The basic relationship between trigonometric functions with the same angle;

Square relation:

sin^2(α)+cos^2(α)= 1

tan^2(α)+ 1=sec^2(α)

cot^2(α)+ 1=csc^2(α)

Relationship between products:

sinα=tanα*cosα

cosα=cotα*sinα

tanα=sinα*secα

cotα=cosα*cscα

secα=tanα*cscα

csα= secα* cotα

Reciprocal relationship:

tanα cotα= 1

sinα cscα= 1

cosα secα= 1

Constant deformation formula of trigonometric function;

Trigonometric function of sum and difference of two angles;

cos(α+β)=cosα cosβ-sinα sinβ

cos(α-β)=cosα cosβ+sinα sinβ

sin(α β)=sinα cosβ cosα sinβ

tan(α+β)=(tanα+tanβ)/( 1-tanαtanβ)

tan(α-β)=(tanα-tanβ)/( 1+tanαtanβ)

Auxiliary angle formula:

Asinα+bcosα = (A2+B2) (1/2) sin (α+t), where

sint=B/(A^2+B^2)^( 1/2)

cost=A/(A^2+B^2)^( 1/2)

Double angle formula:

sin(2α)=2sinα cosα

cos(2α)=cos^2(α)-sin^2(α)=2cos^2(α)- 1= 1-2sin^2(α)

tan(2α)=2tanα/[ 1-tan^2(α)]

Triple angle formula:

sin3α=3sinα-4sin^3(α)

cos3α=4cos^3(α)-3cosα

Half-angle formula:

sin^2(α/2)=( 1-cosα)/2

cos^2(α/2)=( 1+cosα)/2

tan^2(α/2)=( 1-cosα)/( 1+cosα)

tan(α/2)= sinα/( 1+cosα)=( 1-cosα)/sinα

General formula:

sinα=2tan(α/2)/[ 1+tan^2(α/2)]

cosα=[ 1-tan^2(α/2)]/[ 1+tan^2(α/2)]

tanα=2tan(α/2)/[ 1-tan^2(α/2)]

Product sum and difference formula:

sinαcosβ=( 1/2)[sin(α+β)+sin(α-β)]

cosαsinβ=( 1/2)[sin(α+β)-sin(α-β)]

cosαcosβ=( 1/2)[cos(α+β)+cos(α-β)]

sinαsinβ=-( 1/2)[cos(α+β)-cos(α-β)]

Sum-difference product formula:

sinα+sinβ= 2 sin[(α+β)/2]cos[(α-β)/2]

sinα-sinβ= 2cos[(α+β)/2]sin[(α-β)/2]

cosα+cosβ= 2cos[(α+β)/2]cos[(α-β)/2]

cosα-cosβ=-2 sin[(α+β)/2]sin[(α-β)/2]

* Others:

sinα+sin(α+2π/n)+sin(α+2π* 2/n)+sin(α+2π* 3/n)+……+sin[α+2π*(n- 1)/n]= 0

cosα+cos(α+2π/n)+cos(α+2π* 2/n)+cos(α+2π* 3/n)+……+cos[α+2π*(n- 1)/n]= 0

and

sin^2(α)+sin^2(α-2π/3)+sin^2(α+2π/3)=3/2

tanAtanBtan(A+B)+tanA+tan B- tan(A+B)= 0