tangent

[zhèng qiē]

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Tangent, mathematical term, in Rt △ ABC（ right triangle ）In, ∠ C=90 °, AB is ∠ C Opposite side c. BC is the opposite side of ∠ A, AC is the opposite side of ∠ B, Tangent function namely tan B=b/a， That is, tanB=AC/BC.^[1]

Chinese name: tangent
Foreign name: Tan gent (short for tan, formerly tg)
Research discipline: mathematics

Value range: Entire set of real numbers
Define Fields: {x|x≠(π/2)+kπ,k∈Z}
Cycle: kπ,k∈Z

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three Properties of Tangent Function Image
four Special angle
five Tangent theorem

trigonometric function

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Figure 1 (1 sheet)

trigonometric function It belongs to Elementary function In Transcendental function Class of function 。^[1] Their essence is Arbitrary angle The mapping between the variables of a set of and a set of ratios. The usual trigonometric function is Rectangular coordinate system The definition field of is the whole real number field. Another definition is right triangle Medium, but not complete. Modern mathematics describes them as infinite sequence Of limit And the solution of differential equation, and extend its definition to complex Department. As shown in Figure 1.

because trigonometric function The periodicity of Single valued function Meaningful Inverse function 。

Trigonometric functions have important applications in complex numbers. stay physics Trigonometric functions are also commonly used tools.

Diagram of trigonometric function

In Rt △ ABC, if acute angle A is determined, then the ratio of the opposite side to the adjacent side of angle A will be determined. This ratio is called the tangent of angle A and recorded as tanA.

Namely: tanA=opposite side of ∠ A/adjacent side of ∠ A 。

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Six basic functions

Function name	formula
Sine function	sinθ=y/r
cosine function	cosθ=x/r
Tangent function	tanθ=y/x
Cotangent function	cotθ=x/y
Secant function	secθ=r/x
Cosecant function	cscθ=r/y

Homoangular trigonometric function

type	formula
square relationship	sin^2(α)+cos^2(α)=1
	tan^2(α)+1=sec^2(α)
	cot^2(α)+1=csc^2(α)
product Relationship of	sinα=tanαcosα cosα=cotαsinα
	tanα=sinαsecα cotα=cosαcscα
	secα=tanαcscα cscα=secαcotα
reciprocal relationship	tanα·cotα=1
	sinα·cscα=1
	cosα·secα=1

Identical deformation formula

Trigonometric function of sum and difference of two angles	cos(α+β)=cosα·cosβ-sinα·sinβ
	cos(α-β)=cosα·cosβ+sinα·sinβ
	sin(α±β)=sinα·cosβ±cosα·sinβ
	tan(α+β)=(tanα+tanβ)/(1-tanα·tanβ)
	tan(α-β)=(tanα-tanβ)/(1+tanα·tanβ)

Double angle formula

sin(2α)=2sinα·cosα

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

tan(2α)=2tanα/[1-tan^2(α)]^[2]

Triple angle formula

sin3α=3sinα-4sin^3(α)

cos3α=4cos^3(α)-3cosα

Half angle formula

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)=sinα/(1+cosα)=(1-cosα)/sinα

Power reducing formula

sin^2(α)=(1-cos(2α))/2

cos^2(α)=(1+cos(2α))/2

tan^2(α)=(1-cos(2α))/(1+cos(2α))

Universal formula

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)]

Integral sum difference formula

sinα·cosβ=(1/2)[sin(α+β)+sin(α-β)]

cosα·sinβ=(1/2)[sin(α+β)-sin(α-β)]

cosα·cosβ=(1/2)[cos(α+β)+cos(α-β)]

sinα·sinβ=-(1/2)[cos(α+β)-cos(α-β)]

Sum difference product formula

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]

other

tanA·tanB·tan(A+B)+tanA+tanB-tan(A+B)=0

In Advanced Algebra trigonometric function Exponential representation of (by taylor series Easy to get):

sinx=[e^(ix)-e^(-ix)]/(2i)

cosx=[e^(ix)+e^(-ix)]/2

tanx=[e^(ix)-e^(-ix)]/[ie^(ix)+ie^(-ix)]

tanA·tanB=1

Properties of Tangent Function Image

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Definition field:{ x|x ≠(π/2)+ k π, k ∈Z}

Value field: R

Parity: Yes, for Odd function

Periodicity: Yes

Minimum positive period: π

Monotonicity: Yes

Monotonic increasing interval: (- π/2+k π,+π/2+k π), k ∈ Z

Monotone decreasing interval: none

Special angle

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tan15°	2-√3
tan30°	√3/3
tan45°	one
tan60°	√3
tan75°	2+√3

Tangent theorem

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In a plane triangle, Tangent theorem Explain that the quotient obtained by dividing the sum of any two edges by the difference between the first edge and the second edge is equal to the quotient obtained by the positive resection of half of the diagonal sum of the two edges and the tangent of half of the difference between the diagonal of the first edge and the diagonal of the second edge.

Francis Veda (Franç ois Vi è te) proposed the tangent theorem in his first book on the study of trigonometry, Mathematical Rules Applied to Triangles. Modern middle school textbooks have been rarely mentioned. For example, since the People's Republic of China once criticized the former Soviet Union and its pedagogy, the tangent theorem was deleted from middle schools between 1966 and 1977 mathematics Textbooks. However, this theorem can be compared with Cosine theorem Easier to use logarithm To calculate projection and other problems.

Tangent theorem ： (a + b) / (a - b) = tan((α+β)/2) / tan((α-β)/2)

prove Start with the following formula:

from Sine theorem obtain

(See Trigonometric identity )

Tangent function yes right triangle Medium, Opposite side Ratio to adjacent edge. Put on Rectangular coordinate system Middle (as shown in the Definition Diagram), that is, tan θ=y/x

Definition Diagram

It is also expressed as tg θ=y/x, but tan θ=y/x is commonly used. It used to be abbreviated as tg, and now it is no longer used. It is only used in books published before the 1990s.^[3]

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