Sine theorem formula and its inference
Sine theorem: In a triangle, the sine of each side is equal to the angle it faces.
Sine formula
A, sine theorem formula
a/sinA=b/sinB=c/sinC=2R .
Note 1 where "r" is the radius of triangle △ABC circumscribed circle. The same below.
Note 2 Sine theorem applies to all triangles. In junior high school mathematics, the sine value of the inner angle of a triangle is equal to the "contrast angle", which is only applicable to right-angled triangles.
Second, the sine theorem inference formula
1 、( 1)a = 2r Sina；
(2)b = 2r sinb；
(3)c=2RsinC .
2. (1)a:b = Sina: sinB
(2)a:c = Sina: sinC
(3)b:c = sinB:sinC；
(4)a:b:c=sinA:sinB:sinC .
Annotations are mainly used for the relationship between "edge" and "angle"
The angular relation of triangle also satisfies sine and cosine theorems.
3. From "a/sinA=b/sinB=c/sinC=2R":
( 1)(a+b)/(sinA+sinB)= 2R；
(2)(a+c)/(sinA+sinC)= 2R；
(3)(b+c)/(sin b+sinC)= 2R；
(4)(a+b+c)/(sinA+sinB+sinC)=2R .
Inference formula of sine theorem
4. Several commonly used equivalent inequalities in triangle ABC.
( 1)" a & gt； B ","A>b "and" Sina & gtSinB "are equivalent.
(2)“a+b & gt； C' is equivalent to' ‘Sina+sinb >；; sinC .
(3)“a+c & gt； B "is equivalent to" Sina+SINC >; Simbo.
(4)“b+ c >； A' is equivalent to' ‘sin b+ sinc >；; Sina.
5. The area of triangle △ABC is S=(abc)/4R. Where "r" is the radius of the circumscribed circle of the triangle △ABC.
Partial formulas of trigonometric functions
Cosine Theorem Formula and Its Inference
Cosine theorem: the square of any side in a triangle is equal to the sum of the squares of the other two sides MINUS the product of the cosine of the angle between these two sides.
I. Formula of Cosine Theorem
( 1)a^2=b^2+c^2-2bccosa；
(2)b^2=a^2+c^2-2accosb；
(3)c^2=a^2+b^2-2abcosC。
Cosine theorem and its inference apply to all triangles. In junior high school mathematics, the cosine of the inner angle of a triangle is equal to "adjacent oblique", which is only applicable to right-angled triangles.