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
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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)