catalogue
Related concepts of 1
1. 1? Right angle trigonometric function
1.2? triangle
2 triangle rule
3 Special value
Four important theorems
4. 1? sine law
4.2? cosine theorem
Five commonly used formulas
5. 1? Two-angle sum formula
5.2? Triangular summation formula
5.3? Sum difference product
Related concepts of 1
Right angle trigonometric function
Right trigonometric function (≈ α is acute angle)
triangle
Reciprocal relation: tanα cotα= 1+0.
sinα cscα= 1
cosα secα= 1
Relationship between businesses:
Square relation:
2 triangle rule
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.
The essence of trigonometric function:
Derive the formula according to the definition 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: x 2+y 2 =1.
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 cosθ and sinθ respectively. The triangle in the image ensures this formula; The radius is equal to the hypotenuse and the length is 1, so there are sinθ=y/ 1 and cosθ=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.
3 Special value
sin30 = 1/2
sin45 =√2/2
sin60 =√3/2
cos30 =√3/2
cos45 =√2/2
cos60 = 1/2
tan30 =√3/3
tan45 = 1
tan60 =√3[ 1]
cot30 =√3
cot45 = 1
cot60 =√3/3
Four important theorems
sine law
Sine theorem: in △ABC, a/sin A = b/sin B = c/sin C = 2R.
Where r is the radius of the circumscribed circle of △ABC.
cosine theorem
Cosine theorem: in △ABC, bb 2 = a 2+c 2-2accos θ.
Where θ is the included angle between side A and side C. ..
Five commonly used formulas
Two-angle sum formula
Triangular summation formula
sin(α+β+γ)= sinαcosβcosγ+cosαsinβcosγ+cosαcosβsinγ-sinαsinβsinγ
cos(α+β+γ)= cosαcosβcosγ-cosαsinβsinγ-sinαcosβsinγ-sinαsinαsinβcosγ-sinαsinβcosγ
Sum difference product
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
Inductive formula
Inductive formulas of trigonometric functions (six formulas)
Formula 1:
sin(α+k*2π)=sinα
cos(α+k*2π)=cosα
tan(α+k*π)=tanα
Equation 2:
Sine (π+α) =-Sine α
cos(π+α) = -cosα
tan(π+α)=tanα
Formula 3:
Sine (-α) =-Sine α
cos(-α) = cosα
tan (-α)=-tanα
Equation 4:
Sine (π-α) = Sine α
cos(π-α) = -cosα
tan(π-α) =-tanα
Formula 5:
sin(π/2-α) = cosα
cos(π/2-α) =sinα
Since π/2+α=π-(π/2-α), it can be obtained from Equation 4 and Equation 5.
Equation 6:
sin(π/2+α) = cosα
cos(π/2+α) = -sinα
Inductive formula memory skills: odd variables are unchanged, and symbols look at quadrants.
Double angle formula
Double angle
sine
sin2A=2sinA cosA
cosine
References:
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