Common trigonometric functions are sine function, cosine function and tangent function. Other trigonometric functions, such as cotangent function, secant function, cotangent function, dyadic function, cofactor function, semidyadic function and semifactorial function, are also used in other disciplines, such as navigation, surveying and engineering. The relationship between different trigonometric functions can be obtained by geometric intuition or calculation, which is called trigonometric identity.
Trigonometric functions are generally used to calculate the sides and angles of triangles with unknown lengths, and are widely used in navigation, engineering and physics. In addition, taking trigonometric functions as templates, we can define a class of similar functions, which are called hyperbolic functions. Common hyperbolic functions are also called hyperbolic sine functions, hyperbolic cosine function and so on. Trigonometric function (also called circular function) is a function of angle; They are very important in studying triangles, simulating periodic phenomena and many other applications. Trigonometric function is usually defined as the ratio of two sides of a right triangle containing this angle, and it can also be equivalently defined as the lengths of various line segments on the unit circle. More modern definitions express them as infinite series or solutions of specific differential equations, allowing them to be extended to any positive and negative values, even complex values.
Definition of trigonometric function with arbitrary angle;
As shown in the figure: Let O-x be the starting edge of any angle α in the plane rectangular coordinate system, and take any point P(x, y) on the final edge of the angle α, so that OP = R. 。
sinα=y/r cosα=x/r
cscα=r/y secα=r/x
tanα=y/x cotα=x/y
Definition of unit circle:
You can also define six trigonometric functions according to the unit circle with the radius of 1 and the center of the circle as the origin. 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: For any point on the circle (x, y), x? +y？ = 1。
In trigonometric functions, there are some special angles, such as 30, 45 and 60. The trigonometric function values of these angles are simple monomials, and the specific values can be obtained directly in the calculation.
Trigonometric identity:
Sum and difference of two angles
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cos(α+β)=cosα cosβ-sinα sinβ
cos(α-β)=cosα cosβ+sinα sinβ
sin(α+β)=sinα cosβ+cosα sinβ
sin(α-β)=sinα cosβ-cosα sinβ
tan(α+β)=(tanα+tanβ)/( 1-tanαtanβ)
tan(α-β)=(tanα-tanβ)/( 1+tanαtanβ)
certificate
Take a rectangular coordinate system and make a unit circle.
Take point A, connect OA, and the included angle with X axis is α. Take a point B, connect OB, the included angle with X axis is β, and the included angle between OA and OB is α-β.
A(cosα，sinα)，B(cosβ，sinβ) OA=(cosα，sinα) OB=(cosβ，sinβ)
OA OB
= | OA | | OB | cos(α-β)= cosαcosβ+sinαsinβ
|OA|=|OB|= 1
cos(α-β)=cosαcosβ+sinαsinβ
Sum difference product
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]
Sum and difference of products
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)]
Double angle formula
sin(2α)=2sinα cosα=2/(tanα+cotα)
cos(2α)=cos？ α-sin？ α=2cos？ α- 1= 1-2sin？ α
tan(2α)=2tanα/[ 1-(tanα)？ ]
cot(2α)=(cot？ α- 1)/(2cotα)
sec(2α)=sec？ α/( 1-tan？ α)
csc(2α)= 1/2secα cscα
Triple angle formula
Symplectic (3α) = 3s in α-4s in 3α = 4 symplectic α symplectic (60 +α) symplectic (60 -α)
cos(3α)=4cos^3α-3cosα= 4 cosαcos(60+α)cos(60-α)
Tan (3 α) = (3 tan α-tan 3 α)/(1-3 tan? α) = tanαtan(π/3+α)tan(π/3-α)
cot(3α)=(cot^3α-3cotα)/(3cot？ α- 1)
N times angle formula
According to Euler formula (cosθ+is innθ)n = cosθ+is innθ.
Expand the left side with binomial theorem, and sort out the real part and imaginary part respectively, and get the following two groups of formulas.
sin(nα)=ncos^(n- 1)αsinα-c(n,3)cos^(n-3)αsin^3α+c(n,5)cos^(n-5)αsin^5α-…
cos(nα)=cos^nα-c(n,2)cos^(n-2)αsin^2α+c(n,4)cos^(n-4)αsin^4α
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= CSCα-cotα
cot(α/2)=√[( 1+cosα)/( 1-cosα)]=( 1+cosα)/sinα= sinα/( 1-cosα)= csα+cotα
sec(α/2)=√[(2 secα/(secα+ 1)]
CSC(α/2)=√[(2 secα/(secα- 1)]
Auxiliary angle formula
asinα+bcosα=√a^2+b^2(sinαcosβ+cosαsinβ)=√a^2+b^2sin(α+β)=√a^2+b^2sin(α+arctanb/a)
General formula of trigonometric function
Sina =[2tan(a/2)]/[ 1+ tan? (a/2)]
cosa=[ 1-tan？ (a/2)]/[ 1+ Tan? (a/2)]
tana=[2tan(a/2)]/[ 1-tan？ (a/2)]
Reduced power formula
Sin? α=[ 1-cos(2α)]/2
Because? α=[ 1+cos(2α)]/2
Tan? α=[ 1-cos(2α)]/[ 1+cos(2α)]
Triangular 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)
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
Taylor expansion is also called power series expansion.
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:
e^x = 1+x+x？ /2! +x^3/3！ +……+x^n/n！ +……(-∞& lt； x & lt∞)
ln( 1+x)=x-x^2/2+x^3/3-……+(- 1)^(k- 1)*(x^k)/k(|x|<； 1)
Sin x = x-x 3/3! +x^5/5！ -……+(- 1)^(k- 1)*(x^(2k- 1))/(2k- 1)！ +………(-∞& lt； x & lt∞)
Because x =1-x 2/2! +x^4/4！ -……+(- 1)k*(x^(2k))/(2k)！ +……(-∞& lt； x & lt∞)
arcsin x = x+x^3/(2*3)+( 1*3)x^5/(2*4*5)+ 1*3*5(x^7)/(2*4*6*7)……+(2k+ 1)！ ! *x^(2k+ 1)/(2k！ ! *(2k+ 1))+……(| x | & lt； 1) ! ! Representing double factorial
arccos x =π-(x+x^3/(2*3)+( 1*3)x^5/(2*4*5)+ 1*3*5(x^7)/(2*4*6*7)……)(|x|<； 1)
arctan x = x-x^3/3+x^5/5-……(x≤ 1)
sinh x = x+x^3/3！ +x^5/5！ +……+(x^(2k- 1))/(2k- 1)！ +……(-∞& lt； x & lt∞)
cosh x = 1+x^2/2！ +x^4/4！ +……+(x^(2k))/(2k)！ +……(-∞& lt； x & lt∞)
arcsinh x = x-x^3/(2*3)+( 1*3)x^5/(2*4*5)- 1*3*5(x^7)/(2*4*6*7)……(|x|<； 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.