Find all the formula theorems of senior one mathematics!

Square relation:

sin^2α+cos^2α= 1

1+tan^2α=sec^2α

1+cot^2α=csc^2α

Relationship between products:

sinα=tanα×cosα

cosα=cotα×sinα

tanα=sinα×secα

cotα= cosα×csα

secα=tanα×cscα

cscα=secα×cotα

Reciprocal relationship:

tanα cotα= 1

sinα cscα= 1

cosα secα= 1

Relationship between businesses:

sinα/cosα=tanα=secα/cscα

cosα/sinα=cotα=cscα/secα

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,

[1] trigonometric function identity deformation formula

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=(A？ 0? 5+B？ 0? 5) (1/2) sin (α+t), where

sint=B/(A？ 0? 5+B？ 0? 5)^( 1/2)

Cost =A/(A? 0? 5+B？ 0? 5)^( 1/2)

tant=B/A

Asinα-Bcosα=(A？ 0? 5+B？ 0? 5)^( 1/2)cos(α-t),tant=a/b

Double angle formula:

sin(2α)=2sinα cosα=2/(tanα+cotα)

cos(2α)=cos？ 0? 5(α)-sin？ 0? 5(α)=2cos？ 0? 5(α)- 1= 1-2sin？ 0? 5(α)

tan(2α)=2tanα/[ 1-tan？ 0? 5(α)]

Triple angle formula:

sin(3α)=3sinα-4sin？ 0? 6(α)=4sinα sin(60+α)sin(60-α)

cos(3α)=4cos？ 0? 6(α)-3 cosα= 4 cosαcos(60+α)cos(60-α)

tan(3α)= tan a tan(π/3+a)tan(π/3-a)

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? 0? 5(α)=( 1-cos(2α))/2 = versin(2α)/2

Because? 0? 5(α)=( 1+cos(2α))/2 = covers(2α)/2

Tan? 0? 5(α)=( 1-cos(2α))/( 1+cos(2α))

General formula:

sinα=2tan(α/2)/[ 1+tan？ 0? 5(α/2)]

cosα=[ 1-tan？ 0? 5(α/2)]/[ 1+tan？ 0? 5(α/2)]

tanα=2tan(α/2)/[ 1-tan？ 0? 5(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？ 0? 5α

1-cos2α=2sin？ 0? 5α

1+sinα=(sinα/2+cosα/2)？ 0? five

* 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? 0? 5(α)+sin？ 0? 5(α-2π/3)+sin？ 0? 5(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

Inductive formula

Formula 1:

Let α be an arbitrary angle, and the values of the same trigonometric function with the same angle of the terminal edge are equal:

sin(2kπ+α)=sinα

cos(2kπ+α)=cosα

tan(2kπ+α)=tanα

cot(2kπ+α)=cotα

Equation 2:

Let α be an arbitrary angle, and the relationship between the trigonometric function value of π+α and the trigonometric function value of α;

Sine (π+α) =-Sine α

cos(π+α)=-cosα

tan(π+α)=tanα

cot(π+α)=cotα

Formula 3:

The relationship between arbitrary angle α and the value of-α trigonometric function;

Sine (-α) =-Sine α

cos(-α)=cosα

tan(-α)=-tanα

Kurt (-α) =-Kurt α

Equation 4:

The relationship between π-α and the trigonometric function value of α can be obtained by Formula 2 and Formula 3:

Sine (π-α) = Sine α

cos(π-α)=-cosα

tan(π-α)=-tanα

cot(π-α)=-coα

Formula 5:

The relationship between 2π-α and the trigonometric function value of α can be obtained by formula 1 and formula 3:

Sine (2π-α)=- Sine α

cos(2π-α)=cosα

tan(2π-α)=-tanα

Kurt (2π-α)=- Kurt α

Equation 6:

The relationship between π/2 α and 3 π/2 α and the trigonometric function value of α;

sin(π/2+α)=cosα

cos(π/2+α)=-sinα

tan(π/2+α)=-cotα

cot(π/2+α)=-tanα

sin(π/2-α)=cosα

cos(π/2-α)=sinα

tan(π/2-α)=cotα

cot(π/2-α)=tanα

sin(3π/2+α)=-cosα

cos(3π/2+α)=sinα

tan(3π/2+α)=-cotα

cot(3π/2+α)=-tanα

sin(3π/2-α)=-cosα

cos(3π/2-α)=-sinα

tan(3π/2-α)=cotα

cot(3π/2-α)=tanα

(higher than k∈Z)

Sine theorem means that in a triangle, the sine of each side is equal to the angle it faces, that is, a/sina = b/sinb = c/sinc = 2r (where r is the radius of the circumscribed circle).

Cosine theorem means that the square of any side in a triangle is equal to the sum of the squares of the other two sides MINUS twice the product of the cosine of the angle between these two sides, that is, A 2 = B 2+C 2-2BC COSA.

The ratio of the opposite side to the hypotenuse of Angle A is called sine of Angle A, and it is recorded as sinA, that is, sinA= the opposite side/hypotenuse of Angle A..

The angle α between the hypotenuse and the adjacent edge.

sin=y/r

Whether y>x or y≤x

No matter how big or small A is, it can be any size.

The maximum value of sine is 1 and the minimum value is-1.

Trigonometric identity

For any non-right triangle, such as triangle ABC, there is always tana+tan b+ tanc = tanatanbanc.

Prove:

It is known that (A+B)=(π-C)

So tan(A+B)=tan(π-C)

Then (tana+tanb)/(1-tanatanb) = (tanπ-tanc)/(1+tanπ tanc)

Surface treatment can be carried out.

tanA+tanB+tanC=tanAtanBtanC

Similarly, we can also prove that when α+β+γ=nπ(n∈Z), there is always tanα+tanβ+tanγ=tanαtanβtanγ.

Vector calculation

Let a=(x, y) and b=(x', y').

1, vector addition

The addition of vectors satisfies parallelogram rule and triangle rule.

AB+BC=AC .

a+b=(x+x '，y+y ').

a+0=0+a=a .

Algorithm of vector addition;

Exchange law: a+b = b+a;

Law of association: (a+b)+c=a+(b+c).

2. Vector subtraction

If a and b are mutually opposite vectors, then the reciprocal of a=-b, b=-a and a+b =0. 0 is 0.

AB-AC=CB。 That is, "* * * the starting point is the same, and the direction is reduced"

A=(x, y) b=(x', y') Then a-b=(x-x', y-y').

4. Multiply the number by the vector

The product of real number λ and vector A is a vector, and λa = ∣ λ ∣∣∣∣∣∣∣∣∣∣∣∣∣.

When λ > 0, λa and A are in the same direction;

When λ < 0, λa and A are in opposite directions;

When λ=0, λa=0, and the direction is arbitrary.

When a=0, there is λa=0 for any real number λ.

Note: By definition, if λa=0, then λ=0 or A = 0.

Real number λ is called the coefficient of vector A, and the geometric meaning of multiplier vector λa is to extend or compress the directed line segment representing vector A. ..

When ∣ λ ∣ > 1, the directed line segment representing vector A extends to ∣λ ∣ times in the original direction (λ > 0) or in the reverse direction (λ < 0);

When ∣ λ ∣ < 1, the directed line segment representing vector A is shortened to ∣ λ ∣ times in the original direction (λ > 0) or in the reverse direction (λ < 0).

The multiplication of numbers and vectors satisfies the following algorithm.

Law of association: (λ a) b = λ (a b) = (a λ b).

The distribution law of vector logarithm (first distribution law): (λ+μ)a=λa+μa 。

The distribution law of number pair vector (second distribution law): λ(a+b)=λa+λb 。

The elimination method of number multiplication vector: ① If the real number λ≠0 and λa=λb, then A = B. ② If a≠0 and λa=μa, then λ = μ.

3. Quantity product of vectors

Definition: the included angle between two nonzero vectors is < a, b > and < a, b > ∈ [0, π].

Definition: the product of two vectors (inner product, dot product) is a quantity, denoted as a B. If A and B are not * * * lines, a b = | a || b | cos < a, b >；; If a, b***, then a b =+-∣ a ∣ ∣ b ∣.

The coordinates of the product of vectors are expressed as: a b = x x'+y y'.

Arithmetic ratio of the product of vectors

A b = b a (exchange rate);

(a+b) c = a c+b c (distribution rate);

Properties of scalar product of vectors

A a = the square of a |.

a⊥b = a b = 0 .

|a b|≤|a| |b| .

The main difference between vector product and real number operation

1, the product of vectors does not satisfy the associative law, that is, (a b) c ≠ a (b c); For example: (a b) 2 ≠ a 2 b 2.

2. The product of vectors does not satisfy the law of elimination, that is, b=c cannot be deduced from A = A = C (A ≠ 0).

3、a | b |≦| a | | b |