Theorem content of sine theorem formula in high school mathematics
In any △ABC, the side lengths of angles A, B, and C are A, B, and C, respectively, and the radius of the circumscribed circle of the triangle is R, as follows:
In a triangle, the ratio of sine to diagonal of each side is equal, and the ratio is equal to the diameter (twice radius) length of the circumscribed circle of the triangle.
Formula deformation
△ABC, if the sides of angles A, B and C are A, B and C, and the radius of the circumscribed circle of the triangle is R and the diameter is D, the sine theorem is deformed as follows.
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Theorem significance
The sine theorem points out the relationship between the three sides of an arbitrary triangle and the sine value of the corresponding angle. According to the monotonicity of sine function in interval, sine theorem describes a quantitative relationship between the sides and angles of any triangle.
Generally speaking, the three angles A, B and C of a triangle and their opposite sides A, B and C are called elements of a triangle. It is known that the process of finding other elements from several elements of a triangle is called triangle solution. Sine theorem is an important tool to solve triangles.
In solving triangles, there are the following application fields:
Knowing two angles and one side of a triangle, solve the triangle.
Know the angle of two sides of a triangle and one of them, and solve the triangle.
Solve the transformation relationship between angles with a:b:c=sinA:sinB:sinC.
Theorem proof of sine theorem formula in high school mathematics
Prove sine theorem with circumscribed circle
It is only necessary to prove that the ratio of the side of any angle in any triangle to its corresponding sine is the diameter of the circumscribed circle of the triangle.
Now let △ABC be its circumscribed circle with the center of O, shall we think? C and its antonym AB. Let the length of AB be C.
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