Basic formula of high school mathematics

Summary of all formulas in senior high school mathematics

1 formulas of trigonometric functions table

Basic relations of trigonometric functions with the same angle

Reciprocal relation: quotient relation: square relation:

tanα cotα= 1？

sinα cscα= 1？

cosαsecα= 1 sinα/cosα= tanα= secα/CSCα？

cosα/sinα= cotα= CSCα/secαsin 2α+cos 2α= 1？

1+tan2α=sec2α？

1+cot2α=csc2α

(Hexagon mnemonic method: the graphic structure is "upper chord cut, Zuo Zheng middle cut,1"; The product of two functions on the diagonal is1; The sum of squares of trigonometric function values of two vertices on the shadow triangle is equal to the square of trigonometric function value of the next vertex; The trigonometric function value of any vertex is equal to the product of the trigonometric function values of two adjacent vertices. " )

Inductive formula (formula: odd variable couple, sign according to quadrant. )

sin(-α)=-sinα？

cos(-α)=cosα tan(-α)=-tanα？

Kurt (-α) =-Kurt 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(π-α)=sinα？

cos(π-α)=-cosα？

tan(π-α)=-tanα？

Kurt (π-α) =-Kurt α

sin(π+α)=-sinα？

cos(π+α)=-cosα？

tan(π+α)=tanα？

cot(π+α)=cotα

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

sin(2π-α)=-sinα？

cos(2π-α)=cosα？

tan(2π-α)=-tanα？

Kurt (2π-α)=- Kurt α

sin(2kπ+α)=sinα？

cos(2kπ+α)=cosα？

tan(2kπ+α)=tanα？

cot(2kπ+α)=cotα？

(where k∈Z)

General formula for sum and difference of formulas of trigonometric functions's two angles

sin(α+β)=sinαcosβ+cosαsinβ？

sin(α-β)=sinαcosβ-cosαsinβ？

cos(α+β)=cosαcosβ-sinαsinβ？

cos(α-β)=cosαcosβ+sinαsinβ

tanα+tanβ？

tan(α+β)=———?

1-tanα tanβ

tanα-tanβ？

tan(α-β)=———?

1+tanα tanβ

2tan(α/2)？

sinα=—————?

1+tan2(α/2)

1-tan2(α/2)？

cosα=—————?

1+tan2(α/2)

2tan(α/2)？

tanα=—————?

1-tan2(α/2)

Sine, cosine and tangent formulas of half angle; Power reduction formula of trigonometric function

Sine, cosine and tangent formulas of double angles Sine, cosine and tangent formulas of triangle

sin2α=2sinαcosα

cos 2α= cos 2α-sin 2α= 2 cos 2α- 1 = 1-2 sin 2α

2tanα？

tan2α=————？

1-tan2α

sin3α=3sinα-4sin3α

cos3α=4cos3α-3cosα

3tanα-tan3α？

tan3α=—————?

1-3tan2α

Sum and difference product formula of trigonometric function

α+β α-β ?

sinα+sinβ= 2 sin————cos——？

2 2 ?

α+β α-β ?

sinα-sinβ= 2cos————sin——？

2 2 ?

α+β α-β ?

cosα+cosβ= 2cos————cos——？

2 2 ?

α+β α-β ?

cosα-cosβ=-2 sin——sin——？

2 2 1 ?

sinα cosβ=-[sin(α+β)+sin(α-β)]？

2 ?

1 ?

cosα sinβ=-[sin(α+β)-sin(α-β)]？

2 ?

1 ?

cosα cosβ=-[cos(α+β)+cos(α-β)]？

2 ?

1 ?

sinαsinβ=--[cos(α+β)-cos(α-β)]？

Convert asinα bcosα into trigonometric function form of angle (formulas of trigonometric functions of auxiliary angle

Set, function

Set simple logic?

Any x∈A x∈B, recorded as A B?

A B，B A A=B？

A b = {x | x ∈ a, and x∈B}?

A b = {x | x ∈ a, or x∈B}

Card (A B)= Card (A)+ Card (B)- Card (A B)?

(1) proposition?

If the original proposition is p, then q?

Inverse proposition if q is p?

P is q and there is no proposition?

If the negative proposition is q, then p?

(2) What is the relationship between the four propositions?

(3)A B, a sufficient condition for b?

B A, a is a necessary condition for b to be established?

A B, a is the necessary and sufficient condition of b.

Property index and logarithm of function?

(1) Definition domain, value domain and corresponding rules?

(2) Monotonicity?

For any x 1, x2∈D?

If X 1 < X2F (X 1) < F (X2), is F (X) a increasing function on D?

If x 1 < x2 f(x 1) > f (x2), is f(x) a decreasing function on d?

(3) parity?

If f (-x) = f(x), is f(x) an even function for any x in the domain of function f(x)?

If f (-x) =-f(x), is f(x) odd function?

(4) periodicity?

For any x in the definition domain of function f(x), if there is a constant t that makes f(x+t) = f (x), then f(x) is said to be a fractional exponential power of a periodic function (1)?

The significance of positive fractional exponential power is

The significance of negative fractional exponential power is

(2) The nature and algorithm of logarithm.

loga(MN)=logaM+logaN

logaMn=nlogaM(n∈R)

Exponential function logarithmic function?

(1) y = ax (a > 0, a≠ 1) is an exponential function?

(2)x∈R，y>0？

Image pass (0, 1)?

When a > 1, x > 0, y > 1; When x 1, y = ax is increasing function?

When 0 < a < 1, y = ax is a decreasing function (1), and y = logax (a > 0, a≠ 1) is called a logarithmic function?

(2)x>0，y∈R？

Image pass (1, 0)?

When a > 1, x > 1, y > 0; 0 1, y = logax is increasing function?

0 < a < 1, y = logax is a decreasing function?

Exponential equation and logarithmic equation?

Basic type?

logaf(x)=b f(x)=ab(a>0，a≠ 1)？

Same bottom type

logaf(x)= logag(x)f(x)= g(x)> 0(a > 0，a≠ 1)？

Substitution type f (ax) = 0 or f (logax) = 0.

Ordered sequence

What is the basic concept of arithmetic progression of sequence?

The general formula of (1) series an = f (n)?

(2) Recursive formula of series?

(3) The relationship between the general term formula of the sequence and the sum of the first n terms.

an+ 1-an=d？

an=a 1+(n- 1)d？

A, a and b are equal. 2a = a+b？

m+n=k+l am+an=ak+al

Common summation formula of geometric series?

an=a 1qn_ 1？

A, G, B equal proportion G2 = AB?

M+n=k+l Oman = Aka

inequality

Basic Properties of Inequalities Important Inequalities?

a>b bb，b>c a>c？

a>b a+c>b+c？

a+b>c a>c-b？

a>b，c>d a+c>b+d？

a>b，c>0 ac>bc？

a>b，c0，c>d>0 acb>0 dn>bn(n∈Z，n> 1)？

a>b>0 > (n∈Z，n> 1)？

(a-b)2≥0？

a，b∈R a2+b2≥2ab

|a|-|b|≤|a b|≤|a|+|b|？

Basic method of inequality proof?

Comparative method?

(1) Prove the inequality A > B (or A < B), just prove?

A-b > 0 (or a-b < 0 =?

(2) If B > 0, to prove A > B, just prove,?

To prove a < b, prove it?

Synthesis method is a method to deduce the inequality to be proved (from cause to effect) from the known or proved inequality according to the nature of inequality. ?

Analytical method is to seek the sufficient conditions for the conclusion to be established, and gradually seek the sufficient conditions for the required conditions to be established until the required conditions are known to be correct, which is obviously manifested as "holding the fruit"

plural

Algebraic form, triangular form?

A+bi =c+ Adi = c and B = D.

(a+bi)+(c+di)=(a+c)+(b+d)i？

(a+bi)-(c+di)=(a-c)+(b-d)i？

(a+bi)(c+di )=(ac-bd)+(bc+ad)i

a+bi = r(cosθ+isθ)？

r 1 =(cosθ 1+isθ 1)？ R2(cosθ2+isθ2)？

=r 1？ R2〔cos(θ 1+θ2)+isin(θ 1+θ2)〕？

〔r(cosθ+sinθ)〕n = rn(cosnθ+isinnθ)

k=0， 1，…，n- 1

Analytic geometry

1, straight line

Linear equation of the distance between two points and the bisector?

|AB|=| |？

|P 1P2|=

y-y 1=k(x-x 1)？

y=kx+b

The positional relationship between two straight lines, including angle and distance.

Or k 1 = k2, and b 1≠b2?

L 1 coincides with l2?

Or k 1 = k2 and b 1 = B2?

L 1 intersects with l2?

Or k 1≠k2?

l2⊥l2？

Or k 1k2 =- 1L 1 to l2.

The angle between l 1 and l2.

Distance from point to straight line

2. Conic curve?

Elliptical circle?

The standard equation (X-A) 2+(Y-B) 2 = R2?

The center of the circle is (a, b) and the radius is r?

The general equation x2+y2+dx+ey+f = 0?

Where the center is (),?

Radius r?

(1) Do you use the distance d from the center of the circle to the straight line and the radius r of the circle to judge or use the discriminant to judge the positional relationship between the straight line and the circle?

(2) Use the sum and difference of center distance d and radius to judge the positional relationship between two circles.

Focus f 1 (-c, 0), F2(c, 0)?

(b2=a2-c2)？

Centrifugal rate?

Collinear equation?

Focal radius | mf 1 | = a+ex0, | mf2 | = a-ex0?

Hyperbolic parabola?

Hyperbolic?

Focus f 1 (-c, 0), F2(c, 0)?

(a，b>0，b2=c2-a2)？

Centrifugal rate?

Collinear equation?