math_ Angular function & inverse trigonometric function

Trigonometric functions & Anti trigonometric function

Triangle theory reference

• Trigonometric functions - Wikipedia
• List of trigonometric identities - Wikipedia

wikipedia Simplified Chinese version

• 1 history
• 2 Geometric definition
  • 2.1 Definition in right triangle
  • 2.2 Definition in rectangular coordinate system
  • 2.3 Unit circle definition
• 3 Basic properties
  • 3.1 Trigonometric identity
  • 3.2 Differential and integral calculus
• 4 Analytical definition
  • 4.1 Series definition
    • 4.1.1 Relation with exponential function and complex number
  • 4.2 Less common trigonometric functions
  • 4.3 Definition of differential equation
    • 4.3.1 The importance of radian
  • 4.4 Use function equation to define trigonometric function
• 5 Calculation
  • 5.1 Special values of trigonometric functions
• 6 Anti trigonometric function
• 7 Related theorems
  • 7.1 Sine theorem
  • 7.2 Cosine theorem
  • 7.3 Tangent theorem
  • 7.4 Cotangent theorem
  • 7.5 Periodic function

Images : Six basic trigonometric function images

• The first three are high school content
• The last three are self-study contents

The basic connotation of trigonometric function

• Trigonometric functions （ English ：Trigonometric functions) yes mathematics A common class of information about angle Of function .

• Trigonometric functions will right triangle Between the inner corner of a and its two sides The ratio of Related to , It can also be used equivalently with Unit circle About the length of various line segments .

• Trigonometric functions play an important role in studying the properties of geometric shapes such as triangles and circles , It also studies vibration 、 wave 、 Celestial motion and all kinds of Periodic phenomena Basic mathematical tools

• stay Mathematical analysis in , Trigonometric functions are also defined as Infinite series Or specific Differential equations Solution , Allow their values to be extended to any real value , Even The plural value .

• The relationship between different trigonometric functions can be obtained through geometric intuition or calculation , be called Trigonometric identity .

• Trigonometric functions are generally used to calculate triangle in Edges of unknown length and angles , In navigation 、 Engineering and physics have a wide range of uses .

• in addition , Take trigonometric function as template , Sure Define a class of similar functions , be called Hyperbolic function [2].

  • The common hyperbolic function is also called Hyperbolic sine function 、 Hyperbolic cosine Function, etc. .

Definition in right triangle

Definition in rectangular coordinate system

Unit circle definition ( Six basic trigonometric functions )& Geometric meaning

Trigonometric function value table of special angle

Induction formula between trigonometric functions

• A combination of numbers and shapes , Use symmetry to understand $sin(\theta ),sin(\pi \pm \theta ),sin(2\pi -\theta )$ 4 The relationship between values
• A similar conclusion can be drawn $cos\theta$ And its variants
  

• among (1,6);(2,5);(3,4) The product of each pair is 1( For the same θ horn )
  • sine (sine)* Cosecant (co-secant)=1
  • Secant (secant)* cosine (co-sine)=1
  • tangent (tangent)* Cotangent (co-tangent)=1

tan·gent	co·tan·gent	se·cant	co·se·cant
/ˈtanjənt/	/kōˈtanjənt/	/ˈsēˌkant,ˈsēˌkənt/	/kōˈsēkənt/
tangent	Cotangent	Secant	Cosecant

more (Reflections, shifts, and periodicity)

• $cos\alpha =cos(\frac{\pi }{2}-\theta )=sin\theta$

• A more general , When $\alpha +\beta =\frac{\pi }{2}$ when , Yes

  • $$\begin{gathered} cos\alpha =sin\beta \\ sin\alpha =cos\beta \end{gathered}$$

Reflections

Shifts and periodicity

Trigonometric function related formula theorem

Parity( Parity )

( Only cos&sec It's even function , The rest are odd functions )

• The cosine and the secant are even functions;
• the other trigonometric functions are odd functions. That is:

• See also:Proofs of trigonometric identities § Angle sum identities, andSmall-angle approximation § Angle sum and difference

The sum difference formula of two angles Angle sum and difference identities

• These are also known as the angle addition and subtraction theorems (or formulae).

• 

Geometric meaning

• For the convenience of description , We use vertex letters to describe line segments
• The picture above is in a rectangle ABCD, It has the following characteristics
  • AEFD It's a diameter of 1 The inscribed quadrilateral of the circle of (DE=1,DE It's a diameter )
  • Make separately :
    • AE Extension cord for ( Record as a straight line L1)
    • Passing point D And with the AE Parallel lines ( Record as a straight line L2)
    • after F Point and with AD Parallel lines ( Record as a straight line L3)
  • L1&L2 and L3 Decibels intersect at B,C At two o 'clock
  • such , So we have a rectangle ABCD Four points of , A rectangle with certain characteristics is determined ( It can be used to demonstrate the derivation of angle doubling formula )
• It was said that , A line segment in a rectangle DE The length of is 1, It's important ,( It is equivalent to using the unit circle to describe the basic x=cosx,y=sinx)
• Based on this rectangle ( Without losing the generality ) And the inner edges ( It is mainly the 4 individual RT triangle ), You can find the length of each line segment :
  • $△DEF$ in , because DE=1,$\angle EDF=\beta ,EF=sin\beta ;DF=cos\beta$,
  • remember $\angle CDF=\alpha$, be $sin\alpha =\frac{CF}{DF}=\frac{CF}{cos\beta }$
    • so ,$CF=\sin \alpha \cos \beta$
  • Allied , utilize
    • The offset angles of parallel lines are equal and
    • The sum of the interior angles of a triangle is equal (180 degree )
    • The opposite sides of parallelogram are equal in length
  • You can deduce each edge about $\alpha ,\beta ,\alpha +\beta$ Between ,cos&sin The relationship between values
    • for example , Make use of opposite side equality CD=AB=AE+EB, Corresponding $\cos \alpha \cos \beta =\cos (\alpha +\beta )+\sin \alpha \sin \beta$,
      • After moving to , It can be written in this form ( Formula form ):$\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$
  • take β Value -β, Bring in the sum angle formula , Get the formula of two angle difference

Double angle formula

When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.

$$\tan 2x=\frac{2sinx\cos x}{co{s}^{2}x-si{n}^{2}x}=\frac{\frac{2sinxcosx}{co{s}^{2}x}}{\frac{co{s}^{2}x}{co{s}^{2}x}-\frac{si{n}^{2}x}{co{s}^{2}x}}=\frac{2tanx}{1-ta{n}^{2}x}$$

Geometric diagram of angle doubling formula

Trigonometric function integral

• These identities can be used to derive the product-to-sum identities.
  

• The derivatives of trigonometric functions result from those of sine and cosine by applying quotient rule. ( Fraction ( Division ) Derivation rule )

• The values given for the antiderivatives in the following table can be verified by differentiating them.

• The number C is a constant of integration.

• this is the tangent half-angle substitution, which reduces the computation of integrals ( [ˈɪntɪɡrəl] integral )and antiderivatives ( Indefinite integral )of trigonometric functions to that of rational fractions.

Anti trigonometric function

Anti trigonometric function (wikipedia.org)

Inverse trigonometric function image

|

Definition domain of inverse trigonometric function & range

sum-to-product( And differential product )

In the diagram below , Yes
$$\begin{gathered} cos\theta +cos\phi =2cos(p)cos(q)=2EF \\ △AFG\cong △FCE \\ AG=FE \\ cosp=\frac{AG}{AF}=\frac{AG}{cosq} \\ \Rightarrow AG=cosp\times cosq \end{gathered}$$

and ( Bad ) Chemical product diagram

• Diagram illustrating sum-to-product identities for sine and cosine.

• The blue right-angled triangle has angle and the red right-angled triangle has angle .

• Both have a hypotenuse of length 1.

  • Auxiliary angles, here called and , are constructed such that and .
  • Therefore, and .
  • This allows the two congruent( Coincident consistency ) purple-outline triangles and to be constructed, each with hypotenuse and angle at their base.
  • The sum of the heights of the red and blue triangles is , and this is equal to twice the height of one purple triangle,
    • i.e. . Writing and in that equation in terms of and yields the sum-to-product identity for sine.
    • Similarly, the sum of the widths of the red and blue triangles yields the corresponding identity for cosine.

• $\phi =p-q;\theta =p+q;p>q$

• $$\begin{gathered} p=\frac{\theta +\phi }{2} \\ q=\frac{\theta -\phi }{2} \end{gathered}$$

product-to-sum ( Integrable sum difference )

refencen

glossary

formula&formulae

/ˈfɔːmjʊlə/

noun

plural noun: formulae

identity( Identity )

• MATHEMATICS

  • a transformation that leaves an object unchanged.

  • an element of a set which, if combined with another element by a specified binary operation, leaves that element unchanged.

    noun: identity element; plural noun: identity elements

• MATHEMATICS

  • the equality of two expressions for all values of the quantities expressed by letters, or an equation expressing this, e.g. ( x + 1)2 = x 2 + 2 x + 1.

Trigonometric function

Contents

• 1Notation

• 2Right-angled triangle definitions

• 3Radians versus degrees

• 4Unit-circle definitions

• 5Algebraic values

  • 5.1Simple algebraic values

• 6In calculus

  • 6.1Definition by differential equations
  • 6.2Power series expansion
  • 6.3Continued fraction expansion
  • 6.4Partial fraction expansion
  • 6.5Infinite product expansion
  • 6.6Relationship to exponential function (Euler’s formula)
  • 6.7Definitions using functional equations
  • 6.8In the complex plane

• 7Basic identities

  • 7.1Parity
  • 7.2Periods
  • 7.3Pythagorean identity
  • 7.4Sum and difference formulas
  • 7.5Derivatives and antiderivatives

• 8Inverse functions

• 9Applications

  • 9.1Angles and sides of a triangle
    • 9.1.1Law of sines
    • 9.1.2Law of cosines
    • 9.1.3Law of tangents
    • 9.1.4Law of cotangents
  • 9.2Periodic functions

• 10History

• 11Etymology