Square relation:

sin^2(α)+cos^2(α)= 1 cos^2a=( 1+cos2a)/2

tan^2(α)+ 1=sec^2(α)sin^2a=( 1-cos2a)/2

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

In the right triangle ABC,

The sine value of angle a is equal to the ratio of the opposite side to the hypotenuse of angle a,

Cosine is equal to the adjacent side of angle a than the hypotenuse.

The tangent is equal to the opposite side of the adjacent side,

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)

Trigonometric function of trigonometric sum:

sin(α+β+γ)= sinαcosβcosγ+cosαsinβcosγ+cosαcosβsinγ-sinαsinβsinγ

cos(α+β+γ)= cosαcosβcosγ-cosαsinβsinγ-sinαcosβsinγ-sinαsinαsinβcosγ-sinαsinβcosγ

tan(α+β+γ)=(tanα+tanβ+tanγ-tanαtanβtanγ)/( 1-tanαtanβ-tanβtanγ-tanγtanα)

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)

tant=B/A

asinα+bcosα=(a^2+b^2)^( 1/2)cos(α-t),tant=a/b

Double angle formula:

sin(2α)=2sinα cosα=2/(tanα+cotα)

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:

sin(3α)=3sinα-4sin^3(α)

cos(3α)=4cos^3(α)-3cosα

Half-angle formula:

sin(α/2)= √(( 1-cosα)/2)

cos(α/2)= √(( 1+cosα)/2)

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

Power reduction formula

sin^2(α)=( 1-cos(2α))/2=versin(2α)/2

cos^2(α)=( 1+cos(2α))/2=covers(2α)/2

tan^2(α)=( 1-cos(2α))/( 1+cos(2α))

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]

Derived formula

tanα+cotα=2/sin2α

tanα-cotα=-2cot2α

1+cos2α=2cos^2α

1-cos2α=2sin^2α

1+sinα=(sinα/2+cosα/2)^2

* Others:

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

Cos α+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

cosx+cos2x+...+cosnx =[sin(n+ 1)x+sinnx-sinx]/2 sinx

Prove:

Left = 2sinx (cosx+cos2x+...+cosnx)/2sinx

= [sin2x-0+sin3x-sinx+sin4x-sin2x+...+sinnx-sin (n-2) x+sin (n+1) x-sin (n-1) x]/2sinx (sum and difference of products)

=[sin(n+ 1)x+sinnx-sinx]/2 sinx = right。

Proof of equality

sinx+sin2x+...+sinnx =-[cos(n+ 1)x+cosnx-cosx- 1]/2 sinx

Prove:

Left =-2sinx [sinx+sin2x+...+sinnx]/(-2sinx)

=[cos2x-cos0+cos3x-cosx+...+cos NX-cos(n-2)x+cos(n+ 1)x-cos(n- 1)x]/(-2 sinx)

=-[cos(n+ 1)x+cosnx-cosx- 1]/2 sinx = right。

Proof of equality