Cosine (cos): The adjacent side of angle α is the upper hypotenuse.
Tangent (tan): The opposite side of angle α is greater than the adjacent side.
Cotangent: The adjacent side of angle α is higher than the opposite side.
Secant: the hypotenuse of angle α is larger than the adjacent side.
Cotangent: The hypotenuse of angle α is higher than the edge.
The basic relationship between trigonometric functions with the same angle
? Square relation:
(sinx)^2+(cosx)^2= 1
1+(tanx)^2=(secx)^2
1+(cotx)^2=(cscx)^2
? Product relationship:
sinα=tanα×cosα
cosα=cotα×sinα
tanα=sinα×secα
cotα= cosα×csα
secα=tanα×cscα
cscα=secα×cotα
? Reciprocal relationship:
tanα？ cotα= 1
sinα？ cscα= 1
cosα？ secα= 1
Relationship between businesses:
sinα/cosα=tanα=secα/cscα
cosα/sinα=cotα=cscα/secα
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,
symmetrical
The terminal edge of 180 degrees-α and the terminal edge of α are symmetrical about y axis.
The terminal edge of-α and the terminal edge of α are symmetric about X. ..
The terminal edge of 180+α and the terminal edge of α are symmetrical about the origin.
The terminal edge of 180 degrees /2-α is symmetrical about y = X.
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-tanα？ tanβ)
tan(α-β)=(tanα-tanβ)/( 1+tanα？ tanβ)
? Trigonometric function of trigonometric sum:
sin(α+β+γ)=sinα？ cosβ？ cosγ+cosα？ sinβ？ cosγ+cosα？ cosβ？ sinγ-sinα？ sinβ？ sinγ
cos(α+β+γ)=cosα？ cosβ？ cosγ-cosα？ sinβ？ sinγ-sinα？ cosβ？ sinγ-sinα？ sinβ？ cosγ
tan(α+β+γ)=(tanα+tanβ+tanγ-tanα？ tanβ？ tanγ)/( 1-tanα？ tanβ-tanβ？ tanγ-tanγ？ tanα)
? Auxiliary angle formula:
Asinα+Bcosα=√(A？ +B？ ) sin(α+arctan(B/A)), where
sint=B/√(A？ +B？ )
Cost =A/√(A? +B？ )
tant=B/A
Asinα-Bcosα=√(A？ +B？ )cos(α-t)，tant=A/B
? 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-tan？ α)
? Triple angle formula:
sin(3α) = 3sinα-4sin？ α = 4sinα？ sin(60 +α)
cos(3α) = 4cos？ α-3cosα = 4cosα？ cos(60 +α)cos(60 -α)
tan(3α) = (3tanα-tan？ α)/( 1-3tan？ α) = tanαtan(π/3+α)tan(π/3-α)
? 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
? Reduced power formula
Sin? α=( 1-cos(2α))/2 = versin(2α)/2
Because? α=( 1+cos(2α))/2 = covers(2α)/2
Tan? α=( 1-cos(2α))/( 1+cos(2α))
? General formula:
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)]
? Product sum and 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= 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？ α
1-cos2α=2sin？ α
1+sinα=[sin(α/2)+cos(α/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? (α)+sin？ (α-2π/3)+sin？ (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
Derivation of triple angle formula
sin3a
=sin(2a+a)
=sin2acosa+cos2asina
=2sina( 1-sin？ a)+( 1-2sin？ A) Sina
=3sina-4sin？ a
cos3a
=cos(2a+a)
=cos2acosa-sin2asina
=(2cos？ a- 1)cosa-2( 1-cos？ a)cosa
=4cos？ a-3cosa
sin3a=3sina-4sin？ a
=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？ a-3cosa
=4cosa(cos？ a-3/4)
= 4c OSA【cos？ a-(√3/2)？ ]
=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)
Inductive formula of trigonometric function
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)
Properties of triangles
1. The sum of any two sides of the triangle must be greater than the third side, which also proves that the difference between any two sides of the triangle must be less than the third side.
2. The sum of the internal angles of the triangle is equal to 180 degrees.
3. The bisector of the vertex, the midline of the bottom and the height of the bottom of the isosceles triangle coincide, that is, the three lines are one.
4. The square sum of two right angles of a right triangle is equal to the square-pythagorean theorem of the hypotenuse. The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.
5. A triangle has six centers: inner center, outer center, center of gravity, vertical center and Euler line.
Heart: The intersection of bisectors of three angles is also the center of the inscribed circle of a triangle.
Attribute: The distances to three sides are equal.
Eccentricity: the intersection of three perpendicular lines is also the center of the circumscribed circle of the triangle.
Attribute: The distances to the three vertices are equal.
Center of gravity: the intersection of three midlines.
Property: The distance from the bisector of the three median lines to the vertex is twice the distance from the midpoint of the opposite side.
Vertical center: the intersection of straight lines of three heights.
Attribute: This point is divided into two parts of each high line.
Paracenter: the intersection of the bisector of the outer corner of any two angles of a triangle and the bisector of the inner corner of the third angle.
Attribute: The distances to three sides are equal.
Centroid: Through a vertex of a triangle, divide the perimeter of the triangle into the intersection of 1: 1 and a straight line of one side of the triangle.
Properties: A triangle * * * has three boundary centers, and three straight lines connecting these three boundary centers and their corresponding triangle vertices intersect at one point.
Euler Line: The outer center, center of gravity, center of nine points and vertical center of a triangle are located on the same straight line in turn. This straight line is called the Euler line of triangle.
6. The outer angle of a triangle (the angle formed by one side of the inner angle of the triangle and the extension line of the other side) is equal to the sum of the inner angles that are not adjacent to it.
7. A triangle has at least two acute angles.
8. Angle bisector of triangle: the bisector of an angle of triangle intersects with the opposite side of this angle, and the line segment between the vertex and the intersection of this angle.
9. In an isosceles triangle, the bisector of the vertex of the isosceles triangle bisects the base and is perpendicular to the base.
10. Pythagorean inverse theorem: If three sides of a triangle have the following relationship, then a? +b？ =c？
Then this triangle must be a right triangle.
1 1. The sum of the outer angles of the triangle is 360 degrees.
12. The area of the triangle with equal base and height is equal.
13. The area ratio of equilateral triangles is equal to their height ratio, and the area ratio of equilateral triangles is equal to their base ratio.
14. The sum of squares of the lengths of the three center lines of a triangle is equal to 3/4 of the sum of squares of the lengths of its three sides.
15. tanatantbank = tana+tanb+tanc is always satisfied in △ABC.
16. The outer angle of a triangle is larger than any inner angle that is not adjacent to it.
17. The corresponding edges of congruent triangles are equal, and the corresponding angles are equal.
The area formula of triangle: (1)S△= 1/2*ah(a is the base of triangle, and H is the height corresponding to the base).
(2) s △ =1/2 * AC * sinb =1/2 * BC * Sina =1/2 * ab * sinc (the three angles are ∠A∠B∠C, respectively, and the opposite side
(3) s △ = √ [s * (s-a) * (s-b) * (s-c)] s =1/2 (a+b+c) (Helen-Qin Jiushao formula).
(4)S△=abc/(4R)R is the radius of the circumscribed circle.
(5)S△= 1/2*(a+b+c)*r r is the radius of the inscribed circle.
(6)...........| a b 1 |
S△= 1/2 * | c d 1 |
................| e f 1 |
. | a b 1 |
| c d 1 | is a third-order determinant, and this triangle ABC is in a plane rectangular coordinate system, A(a, b), B(c, d), C(e, f).
....| e f 1 |
It's best to select the constituencies in counterclockwise order from the upper right corner, because the results obtained in this way are generally positive. If you don't follow this rule, you may get a negative value, but it doesn't matter. As long as you take the absolute value, it won't affect the triangular surface.