sin(A+B)= Sina cosb+cosa sinb sin(A-B)= Sina cosb-sinb cosa

cos(A+B)= cosa cosb-Sina sinb cos(A-B)= cosa cosb+Sina sinb

tan(A+B)=(tanA+tanB)/( 1-tanA tanB)tan(A-B)=(tanA-tanB)/( 1+tanA tanB)

ctg(A+B)=(ctgActgB- 1)/(ctg B+ctgA)ctg(A-B)=(ctgActgB+ 1)/(ctg B-ctgA)

Double angle formula

tan2A = 2 tana/( 1-tan2A)ctg2A =(ctg2A- 1)/2c TGA

cos2a = cos2a-sin2a = 2 cos2a- 1 = 1-2 sin2a

half-angle formula

sin(A/2)=√(( 1-cosA)/2)sin(A/2)=-√(( 1-cosA)/2)

cos(A/2)=√(( 1+cosA)/2)cos(A/2)=-√(( 1+cosA)/2)

tan(A/2)=√(( 1-cosA)/(( 1+cosA))tan(A/2)=-√(( 1-cosA)/(( 1+cosA))

ctg(A/2)=√(( 1+cosA)/(( 1-cosA))ctg(A/2)=-√(( 1+cosA)/(( 1-cosA))

Sum difference product

2 Sina cosb = sin(A+B)+sin(A-B)2 cosa sinb = sin(A+B)-sin(A-B)

2 cosa cosb = cos(A+B)-sin(A-B)-2 sinasinb = cos(A+B)-cos(A-B)

sinA+sinB = 2 sin((A+B)/2)cos((A-B)/2 cosA+cosB = 2 cos((A+B)/2)sin((A-B)/2)

tanA+tanB = sin(A+B)/cosa cosb tanA-tanB = sin(A-B)/cosa cosb

ctgA+ctgBsin(A+B)/Sina sinb-ctgA+ctgBsin(A+B)/Sina sinb

The sum of the first n terms of some series

1+2+3+4+5+6+7+8+9+…+n = n(n+ 1)/2 1+3+5+7+9+ 1 1+ 13+ 15+…+(2n- 1)= N2

2+4+6+8+ 10+ 12+ 14+…+(2n)= n(n+ 1) 12+22+32+42+52+62+72+82+…+N2 = n(n+ 1)(2n+ 1)/6

13+23+33+43+53+63+…n3 = N2(n+ 1)2/4 1 * 2+2 * 3+3 * 4+4 * 5+5 * 6+6 * 7+…+n(n+ 1)= n(n+ 1)(n+2)/3

Sine theorem a/sinA=b/sinB=c/sinC=2R Note: where r represents the radius of the circumscribed circle of a triangle.

Cosine Theorem b2=a2+c2-2accosB Note: Angle B is the included angle between side A and side C..

The arc length formula l=a*r a is the radian number r > of the central angle; 0 sector area formula s= 1/2*l*r

Multiplication and factorization a2-b2=(a+b)(a-b)

a3+b3=(a+b)(a2-ab+b2)

a3-b3=(a-b(a2+ab+b2)

Trigonometric inequality | A+B |≤| A |+B|

|a-b|≤|a|+|b|

| a |≤b & lt； = & gt-b≤a≤b

|a-b|≥|a|-|b| -|a|≤a≤|a|

The solution of the unary quadratic equation -b+√(b2-4ac)/2a -b-√(b2-4ac)/2a

The relationship between root and coefficient x1+x2 =-b/ax1* x2 = c/a Note: Vieta theorem.

discriminant

B2-4ac=0 Note: This equation has two equal real roots.

B2-4ac >0 Note: The equation has two unequal real roots.

B2-4ac & lt； Note: The equation has no real root, but a complex number of the yoke.

Reduced power formula

(sin^2)x= 1-cos2x/2

(cos^2)x=i=cos2x/2

General formula of trigonometric function

Let tan(a/2)=t

sina=2t/( 1+t^2)

cosa=( 1-t^2)/( 1+t^2)

tana=2t/( 1-t^2)

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

(higher than k∈Z)

Note: When doing the problem, it is best to regard A as an acute angle.

Inductive formula memory formula

Summary of the law. ※。

The above inductive formula can be summarized as follows:

Odd couples, symbols look at quadrants.

Basic relations of trigonometric functions with the same angle

Basic relations of trigonometric functions with the same angle

Reciprocal relationship:

tanα？ cotα= 1

sinα？ cscα= 1

cosα？ secα= 1

Relationship between businesses:

sinα/cosα=tanα=secα/cscα

cosα/sinα=cotα=cscα/secα

Two-angle sum and difference formula

Formulas of trigonometric functions of sum and difference of two angles.

sin(α+β)=sinαcosβ+cosαsinβ

sin(α-β)=sinαcosβ-cosαsinβ

cos(α+β)=cosαcosβ-sinαsinβ

cos(α-β)=cosαcosβ+sinαsinβ

tan(α+β)=(tanα+tanβ)/( 1-tanαtanβ)

tan(α-β)=(tanα-tanβ)/( 1+tanα？ tanβ)

Double angle formula

Sine, Cosine and Tangent Formulas of Double Angles (Ascending Power and Shrinking Angle Formula)

sin2α=2sinαcosα

cos2α=cos^2(α)-sin^2(α)=2cos^2(α)- 1= 1-2sin^2(α)

tan2α=2tanα/[ 1-tan^2(α)]

half-angle formula

Sine, cosine and tangent formulas of half angle (power decreasing and angle expanding formulas)

sin^2(α/2)=( 1-cosα)/2

cos^2(α/2)=( 1+cosα)/2

tan^2(α/2)=( 1-cosα)/( 1+cosα)

And tan (α/2) = (1-cos α)/sin α = sin α/(1+cos α).

General formula of trigonometric function

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)]

Derivation of universal formula

Additional derivation:

sin2α=2sinαcosα=2sinαcosα/(cos^2(α)+sin^2(α))......*,

(Because cos 2 (α)+sin 2 (α) = 1)

Divide the * fraction up and down by COS 2 (α) to get SIN 2 α = 2 tan α/( 1+tan 2 (α)).

Then replace α with α/2.

Similarly, the universal formula of cosine can be derived. By comparing sine and cosine, a general formula of tangent can be obtained.

Sum-difference product formula

Sum and difference product formula of trigonometric function

sinα+sinβ=2sin[(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=-2sin[(α+β)/2]？ sin[(α-β)/2]

Product sum and difference formula

Formula of product and difference of trigonometric function

sinα？ cosβ=0.5[sin(α+β)+sin(α-β)]

cosα？ sinβ=0.5[sin(α+β)-sin(α-β)]

cosα？ cosβ=0.5[cos(α+β)+cos(α-β)]

sinα？ sinβ=-0.5[cos(α+β)-cos(α-β)]

Derivation of sum-difference product formula

Additional derivation:

First of all, we know that SIN (a+b) = Sina * COSB+COSA * SINB, SIN (a-b) = Sina * COSB-COSA * SINB.

We add these two expressions to get sin(a+b)+sin(a-b)=2sina*cosb.

So sin a * cosb = (sin (a+b)+sin (a-b))/2.

Similarly, if you subtract the two expressions, you get COSA * SINB = (SIN (A+B)-SIN (A-B))/2.

Similarly, we also know that COS (a+b) = COSA * COSB-SINA * SINB, COS (a-b) = COSA * COSB+SINA * SINB.

Therefore, by adding the two expressions, we can get cos(a+b)+cos(a-b)=2cosa*cosb.

So we get, COSA * COSB = (COS (A+B)+COS (A-B))/2.

Similarly, by subtracting two expressions, Sina * sinb =-(cos (a+b)-cos (a-b))/2 can be obtained.

In this way, we get the formulas of the sum and difference of four products:

Sina * cosb =(sin(a+b)+sin(a-b))/2

cosa * sinb =(sin(a+b)-sin(a-b))/2

cosa * cosb =(cos(a+b)+cos(a-b))/2

Sina * sinb =-(cos(a+b)-cos(a-b))/2

Well, with four formulas of sum and difference, we can get four formulas of sum and difference product with only one deformation.

Let a+b be X and A-B be Y in the above four formulas, then A = (X+Y)/2 and B = (X-Y)/2.

If a and b are represented by x and y respectively, we can get four sum-difference product formulas:

sinx+siny = 2 sin((x+y)/2)* cos((x-y)/2)

sinx-siny = 2cos((x+y)/2)* sin((x-y)/2)

cosx+cosy = 2cos((x+y)/2)* cos((x-y)/2)

cosx-cosy =-2 sin((x+y)/2)* sin((x-y)/2)

0 degrees

sina=0，cosa= 1，tana=0

30 degrees

sina= 1/2，cosa=√3/2，tana=√3/3

45 degrees

sina=√2/2，cosa=√2/2，tana= 1

60 degrees

sina=√3/2，cosa= 1/2，tana=√3

90 degrees

Sina= 1, cosa=0, tana does not exist.

120 degrees

sina=√3/2，cosa=- 1/2，tana=-√3

150 degrees

sina= 1/2，cosa=-√3/2，tana=-√3/3

180 degrees

sina=0，cosa=- 1，tana=0

270 degrees

Sina=- 1, cosa=0, tana does not exist.

360 degrees

sina=0，cosa= 1，tana=0

Geometric series formula

If a series starts from the second term and the ratio of each term to the previous term is equal to the same constant, this series is called geometric series. This constant is called the common ratio of geometric series and is usually represented by the letter Q.

The general formula of (1) geometric series is: an = a 1× q (n- 1).

If the general formula is transformed into an = a 1/q * q n (n ∈ n *), when q > 0, an can be regarded as a function of the independent variable n, and the point (n, an) is a set of isolated points on the curve y = a1/q * q X.

(2) the relationship between any two am and an is an=am? q^(n-m)

(3) From the definition of geometric series, the general term formula, the first n terms and the formula, we can deduce: a 1? an=a2？ an- 1=a3？ An -2=…=ak? an-k+ 1，k∈{ 1，2，…，n}

(4) Equal proportion: aq? AP = Ar 2, where Ar is the middle term in the ratio of AP to aq.

Write πn=a 1? A2…an, then π2n- 1=(an)2n- 1, π 2n+1= (an+1) 2n+1.

In addition, each term is a geometric series with positive numbers, and the same base number is taken to form a arithmetic progression; On the other hand, taking any positive number c as the cardinal number and a arithmetic progression term as the exponent, a power energy is constructed, which is a geometric series. In this sense, we say that a positive geometric series and an arithmetic series are isomorphic.

Nature:

(1) if m, n, p, q∈N*, and m+n = p+q, am? an=ap？ AQ；

(2) In geometric series, every k term is added in turn and still becomes a geometric series.

G is the median term in the equal proportion of A and B, and G 2 = AB (G ≠ 0).

(5) The sum of the first n terms of geometric series Sn = a1(1-q n)/(1-q) or Sn = (a1-an * q)/(q)

In geometric series, the first term A 1 and the common ratio q are not zero.

Note: in the above formula, a n stands for the n power of a.

Geometric series are often used in life.

For example, banks have a way of paying interest-compound interest.

That is, the interest and principal of the previous period are added together to calculate the principal.

Then calculate the interest of the next period, which is what people usually call rolling interest.

The formula for calculating the sum of principal and interest according to compound interest: sum of principal and interest = principal *( 1+ interest rate) deposit period.

arithmetical series formula

The general formula of arithmetic progression is: an = a1+(n-1) d.

Or an=am+(n-m)d

The first n terms and formulas are: Sn=na 1+n(n- 1)d/2 or Sn=(a 1+an)n/2.

If m+n=p+q, then: am+an=ap+aq exists.

If m+n=2p, then: am+an=2ap.

All the above n are positive integers.

Text translation

Value of material n = first material+(material number-1)* tolerance.

Sum of the first n items = (first item+last item) * Number of items /2

Tolerance = Last Item-First Item

Symmetric sequence formula

General formula of symmetric sequence:

The total number of items in a symmetric sequence: represented by the letter S.

Symmetric sequence items: represented by the letter C.

Symmetric sequence tolerance: represented by the letter d.

Common ratio of equal ratio symmetric sequence: expressed by the letter Q.

Let k=(s+ 1)/2.

General term solution of general sequence

Generally speaking, there are:

an=Sn-Sn- 1 (n≥2)

Cumulative sum (an-an-1= ... an-1-an-2 = ... a2-a1= ... add the above items to get one).

Quotient-by-quotient total multiplication (for a series with unknown numbers in the quotient of the latter term and the previous term).

Reduction method (transforming a sequence so that the reciprocal of the original sequence or the sum with a constant is equal to the difference or geometric series).

Special:

In arithmetic progression, there are always Sn S2n-Sn S3n-S2n.

2(S2n-Sn)=(S3n-S2n)+Sn

That is, the three are arithmetic progression and geometric progression. Tri-geometric series.

Fixed point method (often used in general fractional recurrence relation)

How to write the general term of special sequence?

1,2,3,4,5,6,7,8.......- an=n

1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8......- an= 1/n

2,4,6,8, 10, 12, 14.......- an=2n

1,3,5,7,9, 1 1, 13, 15.....- an=2n- 1

- 1, 1,- 1, 1,- 1, 1,- 1, 1......an=(- 1)^n

1,- 1, 1,- 1, 1,- 1, 1,- 1, 1......——an=(- 1)^(n+ 1)

1,0, 1,0, 1,0, 1,0 1,0, 1,0, 1....an=[(- 1)^(n+ 1)+ 1]/2

1,0,- 1,0, 1,0,- 1,0, 1,0,- 1,0......- an=cos(n- 1)π/2=sinnπ/2

9,99,999,9999,99999,.........an=( 10^n)- 1

1, 1 1, 1 1 1, 1 1 1 1, 1 1 1 1 1.......an=[( 10^n)- 1]/9

1,4,9, 16,25,36,49,.......an=n^2

1,2,4,8, 16,32......——an=2^(n- 1)

Solution of the summation formula of the first n terms of the sequence

1。 Arithmetic series:

The general formula an=a 1+(n- 1)d, the first term a 1, the tolerance d, the nth term of an.

An=ak+(n-k)d ak is the k th term.

If a, a and b constitute arithmetic progression, then A=(a+b)/2.

2. The sum of the first n items in the arithmetic series:

Let the sum of the first n terms of arithmetic progression be Sn.

That is, Sn=a 1+a2+...+ An;

Then Sn=na 1+n(n- 1)d/2.

= dn 2 (that is, the second power of n) /2+(a 1-d/2)n

There are also the following summation methods: 1, incomplete induction 2, accumulation 3, and inverse addition.

(2) 1. Geometric series:

The general formula an = a 1 * q (n- 1) (that is, n- 1 power of q) is the first term, and an is the nth term.

an=a 1*q^(n- 1),am=a 1*q^(m- 1)

Then an/am = q (n-m)

( 1)an=am*q^(n-m)

(2) If A, G, and B constitute a neutral term with equal proportion, then g 2 = ab (a, B, and G are not equal to 0).

(3) if m+n=p+q, am×an=ap×aq.

2. The first n sums of geometric series

Let a 1, a2, a3 ... a geometric series form.

The sum of the first n terms Sn=a 1+a2+a3 ... one; one

sn = a 1+a 1 * q+a 1 * q 2+...a 1 * q(n-2)+a 1 * q(n- 1)(。

sn=a 1( 1-q^n)/( 1-q)=(a 1-an*q)/( 1-q)；

Note: Q is not equal to1;

Sn=na 1 note: q= 1.

There are generally five methods for summation: 1, complete induction (that is, mathematical induction), 2 multiplication, 3 dislocation subtraction, 4 reverse summation, and 5 split term elimination.