In the plane rectangular coordinate system xOy, draw a ray OP from point O, let the rotation angle be θ, let OP=r, and the coordinate of point P be (x, y).

Sinθ=y/r

Cosine function cosθ=x/r

Tangent function tanθ=y/x

Cotangent function cotθ=x/y

Secθ secθ=r/x

Cotangent function csθ= r/y

(The hypotenuse is R, the opposite side is Y, and the adjacent side is X ...)

And two functions that are not commonly used and easily eliminated:

Positive vector function version θ = 1-cosθ

Covector function coversθ = 1-sinθ.

Edit the basic relationship between trigonometric functions with the same angle in this paragraph:

Square relation:

sin^2(α)+cos^2(α)= 1 cos^2a=( 1+cos2a)/2

tan^2(α)+ 1=sec^2(α)sin^2a=( 1-cos2a)/2

cot^2(α)+ 1=csc^2(α)

Relationship between products:

sinα=tanα*cosα

cosα=cotα*sinα

tanα=sinα*secα

cotα=cosα*cscα

secα=tanα*cscα

csα= secα* cotα

Reciprocal relationship:

tanα cotα= 1

sinα cscα= 1

cosα secα= 1

In the right triangle ABC,

The sine value of angle a is equal to the ratio of the opposite side to the hypotenuse of angle a,

Cosine is equal to the adjacent side of angle a than the hypotenuse.

The tangent is equal to the opposite side of the adjacent side,

Constant deformation formula of trigonometric function

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-tanαtanβ)

tan(α-β)=(tanα-tanβ)/( 1+tanαtanβ)

Trigonometric function of trigonometric sum:

sin(α+β+γ)= sinαcosβcosγ+cosαsinβcosγ+cosαcosβsinγ-sinαsinβsinγ

cos(α+β+γ)= cosαcosβcosγ-cosαsinβsinγ-sinαcosβsinγ-sinαsinαsinβcosγ-sinαsinβcosγ

tan(α+β+γ)=(tanα+tanβ+tanγ-tanαtanβtanγ)/( 1-tanαtanβ-tanβtanγ-tanγtanα)

Auxiliary angle formula:

Asinα+bcosα = (A2+B2) (1/2) sin (α+t), where

sint=B/(A^2+B^2)^( 1/2)

cost=A/(A^2+B^2)^( 1/2)

tant=B/A

asinα+bcosα=(a^2+b^2)^( 1/2)cos(α-t),tant=a/b

Double angle formula:

sin(2α)=2sinα cosα=2/(tanα+cotα)

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

tan(2α)=2tanα/[ 1-tan^2(α)]

Triple angle formula:

sin(3α)=3sinα-4sin^3(α)

cos(3α)=4cos^3(α)-3cosα

Half-angle formula:

sin(α/2)= √(( 1-cosα)/2)

cos(α/2)= √(( 1+cosα)/2)

tan(α/2)=√(( 1-cosα)/( 1+cosα))= sinα/( 1+cosα)=( 1-cosα)/sinα

Power reduction formula

sin^2(α)=( 1-cos(2α))/2=versin(2α)/2

cos^2(α)=( 1+cos(2α))/2=covers(2α)/2

tan^2(α)=( 1-cos(2α))/( 1+cos(2α))

General 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)]

Product sum and difference formula:

sinαcosβ=( 1/2)[sin(α+β)+sin(α-β)]

cosαsinβ=( 1/2)[sin(α+β)-sin(α-β)]

cosαcosβ=( 1/2)[cos(α+β)+cos(α-β)]

sinαsinβ=-( 1/2)[cos(α+β)-cos(α-β)]

Sum-difference product formula:

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]

Derived formula

tanα+cotα=2/sin2α

tanα-cotα=-2cot2α

1+cos2α=2cos^2α

1-cos2α=2sin^2α

1+sinα=(sinα/2+cosα/2)^2

* Others:

sinα+sin(α+2π/n)+sin(α+2π* 2/n)+sin(α+2π* 3/n)+……+sin[α+2π*(n- 1)/n]= 0

Cos α+cos (α+2π/n)+cos (α+2π * 2/n)+cos (α+2π * 3/n)+...+cos [α+2π * (n-1)/n] = 0 and

sin^2(α)+sin^2(α-2π/3)+sin^2(α+2π/3)=3/2

tanAtanBtan(A+B)+tanA+tan B- tan(A+B)= 0

cosx+cos2x+...+cosnx =[sin(n+ 1)x+sinnx-sinx]/2 sinx

Prove:

Left = 2sinx (cosx+cos2x+...+cosnx)/2sinx

= [sin2x-0+sin3x-sinx+sin4x-sin2x+...+sinnx-sin (n-2) x+sin (n+1) x-sin (n-1) x]/2sinx (sum and difference of products)

=[sin(n+ 1)x+sinnx-sinx]/2 sinx = right。

Proof of equality

sinx+sin2x+...+sinnx =-[cos(n+ 1)x+cosnx-cosx- 1]/2 sinx

Prove:

Left =-2sinx [sinx+sin2x+...+sinnx]/(-2sinx)

=[cos2x-cos0+cos3x-cosx+...+cos NX-cos(n-2)x+cos(n+ 1)x-cos(n- 1)x]/(-2 sinx)

=-[cos(n+ 1)x+cosnx-cosx- 1]/2 sinx = right。

Proof of equality

Edit the angle conversion of trigonometric function in this paragraph.

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 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)

Edit this sine and cosine theorem.

Sine theorem means that in a triangle, the ratio of sine of each side to its diagonal is equal, that is, a/sina = b/sinb = c/sinc = 2r.

Cosine theorem means that the square of any side in a triangle is equal to the sum of the squares of the other two sides MINUS twice the product of the cosine of the angle between these two sides, that is, A 2 = B 2+C 2-2BC COSA.

Edit some advanced content in this paragraph.

Exponential representation of trigonometric functions in higher algebra (easily obtained from Taylor series);

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

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

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

Taylor expansion has infinite series, e z = exp (z) = 1+z/ 1! +z^2/2！ +z^3/3！ +z^4/4！ +…+z^n/n！ +…

At this time, the domain of trigonometric function has been extended to the whole complex set.

Trigonometric function as the solution of differential equation;

For the differential equation y =-y ""; Y=y'', there is a general solution q, which can be proved.

Q=Asinx+Bcosx, so trigonometric functions can also be defined from this angle.

Supplement: Represented by the corresponding exponent, a similar function-hyperbolic function can be defined, which has many similar properties with trigonometric function, and both are very interesting.

Edit the special trigonometric function value of this paragraph.

0 ' 30 ' 45 ' 60 ' 90 '

Sina 0 1/2 √2/2 √3/2 1

cosa 1 √3/2 √2/2 1/2 0

Tana 0 √3/3 1 √3 None.

cota None √3 1 √3/3 0

Edit the calculation of trigonometric functions in this paragraph.

power series

c0+c 1x+c2x2+...+cnxn+...=∑cnxn (n=0..∞)

c0+c 1(x-a)+c2(x-a)2+...+cn(x-a)n+...=∑cn(x-a)n (n=0..∞)

Their terms are power functions of positive integer powers, where c0, c 1, c2, ... communication network (abbreviation of Communicating Net) ... and A are constants, and this series is called power series.

Taylor expansion (power series expansion method);

f(x)=f(a)+f'(a)/ 1！ *(x-a)+f''(a)/2！ *(x-a)2+...f(n)(a)/n！ *(x-a)n+ ...

Practical power series:

ex = 1+x+x2/2！ +x3/3！ +...+xn/n！ + ...

ln( 1+x)= x-x2/3+x3/3-...(- 1)k- 1*xk/k+...(| x | & lt 1)

sin x = x-x3/3！ +x5/5！ -...(- 1)k- 1 * x2k- 1/(2k- 1)！ +...(-∞& lt； x & lt∞)

cos x = 1-x2/2！ +x4/4！ -...(- 1)k*x2k/(2k)！ +...(-∞& lt； x & lt∞)

arcsin x = x+ 1/2 * x3/3+ 1 * 3/(2 * 4)* X5/5+...(| x | & lt 1)

arccos x =π-(x+ 1/2 * x3/3+ 1 * 3/(2 * 4)* X5/5+...)(| x | & lt 1)

arctan x = x - x^3/3 + x^5/5 -...(x≤ 1)

sinh x = x+x3/3！ +x5/5！ +...(- 1)k- 1 * x2k- 1/(2k- 1)！ +...(-∞& lt； x & lt∞)

cosh x = 1+x2/2！ +x4/4！ +...(- 1)k*x2k/(2k)！ +...(-∞& lt； x & lt∞)

arcsinh x = x- 1/2 * x3/3+ 1 * 3/(2 * 4)* X5/5-...(| x | & lt 1)

arctanh x = x + x^3/3 + x^5/5 +...(| x | & lt 1)

When solving elementary trigonometric functions, you can easily solve them as long as you remember the formula. In competitions, you often use the method of combining images to find trigonometric function values, trigonometric function inequalities, areas and so on.

-

Fourier series (trigonometric series)

f(x)=a0/2+∑(n=0..∞)(ancos NX+bns inx)

a0= 1/π∫(π..-π) (f(x))dx

an= 1/π∫(π..-π) (f(x)cosnx)dx

bn= 1/π∫(π..-π) (f(x)sinnx)dx

Digital symbol of trigonometric function

The first and second quadrants of sine are positive, and the third and fourth quadrants are negative.

Cosine is positive in the first quadrant and negative in the second and third quadrants.

Tangent first, three quadrants are positive second, and four quadrants are negative.

Edit the definition and value fields of trigonometric functions in this paragraph.

The domain of sin (x) and cos (x) is r, and the range of values is [- 1, 1].

The definition domain of tan(x) is that x is not equal to π/2+kπ, and the value domain is r.

Cot(x) is defined as x but not equal to kπ, and its value range is r.