Commonly used inductive formulas have the following groups:

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 (k∈Z)

cos(2kπ+α)=cosα (k∈Z)

tan(2kπ+α)=tanα (k∈Z)

cot(2kπ+α)=cotα (k∈Z)

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

Kurt (π-α) =-Kurt α

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)

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:

For the trigonometric function value of π/2 * k α (k ∈ z),

① When k is an even number, the function value of α with the same name is obtained, that is, the function name is unchanged;

② When k is an odd number, the cofunction value corresponding to α is obtained, that is, sin→cos；; cos→sin； Tan → Kurt, Kurt → Tan.

(Odd and even numbers remain the same)

Then when α is regarded as an acute angle, the sign of the original function value is added.

(Symbols look at quadrants)

For example:

Sin (2π-α) = sin (4 π/2-α), and k = 4 is an even number, so we take sinα.

When α is an acute angle, 2π-α ∈ (270,360), sin (2π-α) < 0, and the symbol is "-".

So sin (2 π-α) =-sin α.

The above memory formula is:

Odd couples, symbols look at quadrants.

The symbols on the right side of the formula are angles k 360+α (k ∈ z),-α, 180 α, and when α is regarded as an acute angle, it is 360-α.

The sign of the original trigonometric function value in the quadrant can be remembered.

The name of horizontal induction remains unchanged; Symbols look at quadrants.

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How to judge the symbols of various trigonometric functions in four quadrants, you can also remember the formula "a full pair; Two sine (cotangent); Cut in twos and threes; Four cosines (secant) ".

The meaning of this 12 formula is:

The four trigonometric functions at any angle in the first quadrant are "+";

In the second quadrant, only the sine is "+",and the rest are "-";

The tangent function of the third quadrant is+and the chord function is-.

In the fourth quadrant, only cosine is "+",others are "-".

The above memory formulas are all positive, sine, inscribed and cosine.

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There is another way to define positive and negative according to the function type:

Function Type First Quadrant Second Quadrant Third Quadrant Fourth Quadrant

Sine ...........+............+............-............- ........

Cosine ...........+............-............-............+ ........

Tangent ...........+............-............+............- ........

I cut ...........+............-............+............- ........

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

Square relation:

sin^2(α)+cos^2(α)= 1

1+tan^2(α)=sec^2(α)

1+cot^2(α)=csc^2(α)

Hexagon memory method of equilateral trigonometric function relationship

Hexagonal mnemonics: (see pictures or links to resources)

The structure is "winding, cutting and cutting; Zuo Zheng, the right remainder and the regular hexagon of the middle 1 "are models.

(1) Reciprocal relation: The two functions on the diagonal are reciprocal;

(2) Quotient relation: the function value at any vertex of a hexagon is equal to the product of the function values at two adjacent vertices.

(Mainly the product of trigonometric function values at both ends of two dotted lines). From this, the quotient relation can be obtained.

(3) Square relation: In a triangle with hatched lines, the sum of squares of trigonometric function values on the top two vertices is equal to the square of trigonometric function values on the bottom vertex.

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+tanαtanβ)

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

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.

Triple angle formula

Sine, cosine and tangent formulas of triple angle

sin3α=3sinα-4sin^3(α)

cos3α=4cos^3(α)-3cosα

tan3α=[3tanα-tan^3(α)]/[ 1-3tan^2(α)]

Derivation of triple angle formula

Additional derivation:

tan3α=sin3α/cos3α

=(sin 2αcosα+cos 2αsinα)/(cos 2αcosα-sin 2αsinα)

=(2sinαcos^2(α)+cos^2(α)sinα-sin^3(α))/(cos^3(α)-cosαsin^2(α)-2sin^2(α)cosα)

Divided by COS 3 (α), we get:

tan3α=(3tanα-tan^3(α))/( 1-3tan^2(α))

sin 3α= sin(2α+α)= sin 2αcosα+cos 2αsinα

=2sinαcos^2(α)+( 1-2sin^2(α))sinα

=2sinα-2sin^3(α)+sinα-2sin^3(α)

=3sinα-4sin^3(α)

cos 3α= cos(2α+α)= cos 2αcosα-sin 2αsinα

=(2cos^2(α)- 1)cosα-2cosαsin^2(α)

=2cos^3(α)-cosα+(2cosα-2cos^3(α))

=4cos^3(α)-3cosα

that is

sin3α=3sinα-4sin^3(α)

cos3α=4cos^3(α)-3cosα

Associative memory of triangle formula

★ Memory method: homophonic association.

Sine Triangle: 3 yuan minus 4 yuan Triangle (debt (minus negative number), so "making money" (sounds like "sine").

Cosine triple angle: 4 yuan minus 3 yuan (there is a "remainder" after subtraction).

☆☆ Pay attention to the function name, that is, the triple angle of sine is expressed by sine, and the triple angle of cosine is expressed by cosine.

★ Another memory method:

Sine triangle: the mountain has no commander (the homonym is three without four stands), three refers to "triple" sinα, no negative sign, four refers to "quadruple", and three refers to sinα cube.

Cosine triple angle: commander without mountain is the same as above.

Sum-difference product formula

Sum and difference product formula of trigonometric function

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]

Product sum and difference formula

Formula of product and difference of trigonometric function

sinαcosβ= 0.5[sin(α+β)+sin(α-β)]

cosαsinβ= 0.5[sin(α+β)-sin(α-β)]

cosαcosβ= 0.5[cos(α+β)+cos(α-β)]

sinαsinβ=-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)