The so-called trigonometric function induction formula is to transform the trigonometric function of angle n (π/2) α into the trigonometric function of angle α.

Common inductive formulas

Formula 1: Let α be an arbitrary angle, and the values of the same trigonometric functions with the same terminal angles 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, the relationship between the trigonometric function value of π+α and the trigonometric function value of α:

sin(π+α)=-sinα k∈z

cos(π+α)=-cosα k∈z

tan(π+α)=tanα k∈z

cot(π+α)=cotα k∈z

Equation 3: Relationship between trigonometric function values of arbitrary angles α and-α:

Sine (-α) =-Sine α

cos(-α)=cosα

tan(-α)=-tanα

Kurt (-α) =-Kurt α

Equation 4: Using Equation 2 and Equation 3, we can get the relationship between 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)=-coα

Equation 5: Using Equation 1 and Equation 3, we can get the relationship between the trigonometric function values of 2π-α and α:

Sine (2π-α)=- Sine α

cos(2π-α)=cosα

tan(2π-α)=-tanα

Kurt (2π-α)=- Kurt α

Equation 6: Relationship between π/2α and 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

Inductive formula memory formula: "parity is constant, symbols look at quadrants."

"Odd and even" refers to the parity of a multiple of π/2, and "change and unchanging" refers to the change of the name of a trigonometric function:

"Change" refers to sine changing into cosine and tangent changing into cotangent. (and vice versa) The meaning of "symbols look at quadrants" is:

Take the angle α as an acute angle, regardless of the quadrant where the angle α is located, and see what the quadrant angle N (π/2) α is, so we can get and so on.

Whether the right side of the formula is positive or negative.

Symbolic judgment formula: "a full pair; Two sinusoids; Cut in twos and threes; Four cosines ". The meaning of twelve-character formula

Thinking means that the four trigonometric functions at any angle in the first quadrant are "+"; The second quadrant is only sine.

Is "+",the rest are "-"; In the third quadrant, only the tangent and cotangent are "+",and the rest are "-";

In the fourth quadrant, only cosine is "+",others are "-". "ASCT" is the antonym of z, which means "all".

"sin", "cos" and "tan" are positive values according to the trigonometric function corresponding to the quadrant occupied by the letter Z.

Other trigonometric function knowledge

Basic relations of trigonometric functions with the same angle

Reciprocal relationship

tanα cotα= 1

sinα cscα= 1

cosα secα= 1

Relationship of quotient

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

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

The two functions on the diagonal of reciprocal relation are reciprocal;

The function value of any vertex of the quotient relation hexagon is equal to the product of the function values of 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.

Square relation In a triangle with hatched lines, the sum of squares of trigonometric function values of the top two vertices is equal to the following.

Square of trigonometric function value on the vertex of a face.

Two-angle sum and difference formula

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)

Sine, cosine and tangent formulas of double angles

sin2α=2sinαcosα

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

tan2α=2tanα/( 1-tan^2(α))

Sine, cosine and tangent formulas of half angle

sin^2(α/2)=( 1-cosα)/2

cos^2(α/2)=( 1+cosα)/2

tan^2(α/2)=( 1-cosα)/( 1+cosα)

tan(α/2)=( 1—cosα)/sinα= sinα/ 1+cosα

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))

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))

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)

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)]

Edit this paragraph formula derivation process.

Derivation of universal formula

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.

Derivation of triple angle formula

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

Derivation of sum-difference product formula

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)