formulas of trigonometric functions

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Trigonometric function is a kind of transcendental function in elementary function in mathematics. Their essence is the mapping between any set of angles and a set of ratio variables. The usual trigonometric function is defined in a plane rectangular coordinate system. It defines a city as the whole real city. The other is defined in a right triangle, but it is incomplete. Modern mathematics describes them as the limits of infinite sequences and the solutions of differential equations, and extends their definitions to complex systems.

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Formula classification

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Formula classification

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)

Two commonly used formulas under different conditions

Sin? α+cos？ α= 1

tan α *cot α= 1

Acute angle formula of trigonometric function

Sine: the opposite side of sin α = the hypotenuse of ∠ α/∠α.

Cosine: the adjacent side of cos α = the hypotenuse of ∠ α/∠α.

Tangent: the opposite side of tan α = the adjacent side of ∠ α/∠α.

Cotangent: cotα = the adjacent side of ∠ α/the opposite side of ∠α.

Double angle formula

sine

sin2A=2sinA cosA

cosine

1.cos2a=cos^2(a)-sin^2(a)=2cos^2(a)- 1 = 1-2sin^2(a)

2.Cos2a= 1-2Sin^2(a)

3.Cos2a=2Cos^2(a)- 1

tangent

tan2A=(2tanA)/( 1-tan？ answer

Triple angle formula

sin3α=4sinα sin(π/3+α)sin(π/3-α)

cos3α=4cosα cos(π/3+α)cos(π/3-α)

tan3a = tan a tan(π/3+a) tan(π/3-a)

Derivation of triple angle formula

Sin (3a)

=sin(a+2a)

=sin2acosa+cos2asina

=2sina( 1-sin？ a)+( 1-2sin？ A) Sina

=3sina-4sin^3a

cos3a

=cos(2a+a)

=cos2acosa-sin2asina

=(2cos？ a- 1)cosa-2( 1-cos^a)cosa

=4cos^3a-3cosa

sin3a=3sina-4sin^3a

=4sina(3/4-sin？ answer

=4sina[(√3/2)？ Sin? Answer]

=4sina (sin? 60- sin? answer

=4sina(sin60 +sina)(sin60 -sina)

= 4 Sina * 2 sin[(60+a)/2]cos[(60-a)/2]* 2 sin[(60-a)/2]cos[(60-a)/2]

=4sinasin(60 +a)sin(60 -a)

cos3a=4cos^3a-3cosa

=4cosa(cos？ a-3/4)

= 4c OSA【cos？ a-(√3/2)^2]

=4cosa(cos？ a-cos？ 30 )

=4cosa(cosa+cos30 )(cosa-cos30)

= 4 cosa * 2cos[(a+30)/2]cos[(a-30)/2]* {-2 sin[(a+30)/2]sin[(a-30)/2]}

=-4 eicosapentaenoic acid (a+30) octyl (a-30)

=-4 Coxsacin [90-(60-a)] Xin [-90 +(60 +a)]

=-4 cos(60-a)[-cos(60+a)]

= 4 cos(60-a)cos(60+a)

Comparing the above two formulas, we can get

tan3a=tanatan(60 -a)tan(60 +a)

half-angle formula

tan(A/2)=( 1-cosA)/sinA = sinA/( 1+cosA)；

cot(A/2)= sinA/( 1-cosA)=( 1+cosA)/sinA。

sin^2(a/2)=( 1-cos(a))/2

cos^2(a/2)=( 1+cos(a))/2

tan(a/2)=( 1-cos(a))/sin(a)= sin(a)/( 1+cos(a))

Sum difference product

sinθ+sinφ= 2 sin[(θ+φ)/2]cos[(θ-φ)/2]

sinθ-sinφ= 2 cos[(θ+φ)/2]sin[(θ-φ)/2]

cosθ+cosφ= 2 cos[(θ+φ)/2]cos[(θ-φ)/2]

cosθ-cosφ=-2 sin[(θ+φ)/2]sin[(θ-φ)/2]

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

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

Sum difference product

cos(α+β)=cosαcosβ-sinαsinβ

cos(α-β)=cosαcosβ+sinαsinβ

sin(α+β)=sinαcosβ+cosαsinβ

sin(α-β)=sinαcosβ -cosαsinβ

Sum and difference of products

sinαsinβ=-[cos(α-β)-cos(α+β)]/2

cosαcosβ = [cos(α+β)+cos(α-β)]/2

sinαcosβ = [sin(α+β)+sin(α-β)]/2

cosαsinβ = [sin(α+β)-sin(α-β)]/2

Hyperbolic function

sinh(a) = [e^a-e^(-a)]/2

cosh(a) = [e^a+e^(-a)]/2

tanh(a) = sin h(a)/cos h(a)

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 the trigonometric function values of 2π-α and α can be obtained by Formula-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)

a sin(ωt+θ)+B sin(ωt+φ)= 1

√{(A？ +B？ +2 abcos(θ-φ)} sin {ωt+arcsin[(a sinθ+b sinφ)/√{a^2 +b^2； +2ABcos(θ-φ)} }

√ indicates the root number, including the contents in {...}.

Inductive formula

Sine (-α) =-Sine α

cos(-α) = cosα

tan (-α)=-tanα

sin(π/2-α) = cosα

cos(π/2-α) = sinα

sin(π/2+α) = cosα

cos(π/2+α) = -sinα

Sine (π-α) = Sine α

cos(π-α) = -cosα

Sine (π+α) =-Sine α

cos(π+α) = -cosα

tanA= sinA/cosA

tan(π/2+α)=-cotα

tan(π/2-α)=cotα

tan(π-α)=-tanα

tan(π+α)=tanα

Inductive formula memory skills: odd variables are unchanged, and symbols look at quadrants.

General formula of trigonometric function

sinα= 2tan(α/2)/[ 1+(tan(α/2))？ ]

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

tanα= 2tan(α/2)/[ 1-(tan(α/2))？ ]

Other formulas

( 1) (sinα)？ +(cosα)？ = 1

(2) 1+(tanα)？ =(secα)？

(3) 1+(cotα)？ =(cscα)？

To prove the following two formulas, just divide one formula by the left and right (sinα)? , seconds divided by (cosα)? Just do it.

(4) For any non-right triangle, there is always

tanA+tanB+tanC=tanAtanBtanC

Certificate:

A+B=π-C

tan(A+B)=tan(π-C)

(tanA+tanB)/( 1-tanA tanB)=(tanπ-tanC)/( 1+tanπtanC)

Surface treatment can be carried out.

tanA+tanB+tanC=tanAtanBtanC

Obtain a certificate

It can also be proved that this relationship holds when x+y+z=nπ(n∈Z).

The following conclusions can be drawn from tana+tanbtana+tanb+tanc = tanatanbtanc.

(5)cotAcotB+cotAcotC+cotbctc = 1

(6) Cost (A/2)+ Cost (B/2)+ Cost (C/2)= Cost (A/2) Cost (B/2)

(7)(cosA)？ +(cosB)？ +(cosC)？ = 1-2 osacosbcosc

(8) (Sina)? +(sinB)？ +(sinC)？ = 2+2 coscosbcosc

Other non-critical trigonometric functions

csc(a) = 1/sin(a)

Seconds (a)= 1/ cosine (a)

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Content law

Trigonometric functions seem to be many and complicated, but as long as we master the essence and internal laws of trigonometric functions, we will find that there is a strong connection between the formulas of trigonometric functions. And mastering the inherent law and essence of trigonometric function is also the key to learn trigonometric function well.

1, the essence of trigonometric function:

According to the picture on the right, there are

sinθ= y/r； cosθ= x/r； tanθ= y/x； cotθ=x/y .

Have a deep understanding of this point, all the following trigonometric formulas can be derived from this point, for example.

Sin(A+B) = sinAcosB+cosAsinB For example:

Deduction:

Draw a unit circle with the X axis at C and D, and there are any points A and B on the unit circle. The angle AOD is α and BOD is β. Rotating AOB makes OB and OD overlap to form a new A'OD.

A(cosα，sinα)，B(cosβ，sinβ)，A'(cos(α-β)，sin(α-β))

OA'=OA=OB=OD= 1，D( 1，0)

∴[cos(α-β)- 1]^2+[sin(α-β)]^2=(cosα-cosβ)^2+(sinα-sinβ)^2

The sum-difference product and product-difference reduction method can be derived by combining the above formulas (for (a+b)/2 and (a-b)/2).

Definition of unit circle

unit circle

You can also define six trigonometric functions according to the unit circle with a radius centered on a center. The definition of unit circle is of little value in practical calculation. In fact, for most angles, it depends on the right triangle. But the definition of the unit circle does allow trigonometric functions to define all positive and negative radians, not just the angle between 0 and π/2 radians. It also provides images containing all the important trigonometric functions. According to Pythagorean theorem, the equation of unit circle is:

Some angles expressed in radians are given in the figure. The counterclockwise measurement is a positive angle, while the clockwise measurement is a negative angle. Let a straight line passing through the origin make an angle θ with the positive half of the X axis and intersect the unit circle. The x and y coordinates of this intersection point are equal to cos θ and sin θ respectively. The triangle in the image ensures this formula; The radius is equal to the hypotenuse and the length is 1, so there are sin θ = y/ 1 and cos θ = x/ 1. The unit circle can be regarded as a way to view an infinite number of triangles by changing the lengths of adjacent sides and opposite sides, but keeping the hypotenuse equal to 1.

Two-angle sum formula

sin(A+B) = sinAcosB+cosAsinB

sin(A-B) = sinAcosB-cosAsinB

cos(A+B) = cosAcosB-sinAsinB

cos(A-B) = cosAcosB+sinAsinB

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

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

cot(A+B)=(cotA cotB- 1)/(cot B+cotA)

cot(A-B)=(cotA cotB+ 1)/(cot b-cotA)