law of sines

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The Law of Sines is a fundamental theorem in trigonometry.

Definition of the theorem

at any $\Delta ABC$ center, the angleA、B、CThe lengths of the sides opposite each other area、b、cThe radius of the outer garden of the triangle is RThe diameter is D. then there are: $\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}=2R=D$ . i.e. That is, a triangle in which the ratio of the sines of the sides and opposite angles is equal and the ratio is equal to the length of the diameter (twice the radius) of the outer garden of the triangle.

show that

Make a graph with side lengtha，b，cThe triangles whose corresponding angles areA，B，CThe From AngleCtowardcThe edge makes a perpendicular line to get a length ofhthe plumb line and two right triangles. As shown in the figure.

Obviously: $sinA=\frac{h}{b}$ ， $sinB=\frac{h}{a}$ . Therefore h=a*sinB=b*sinA as well as $\frac{sinA}{a}=\frac{sinB}{b}$ 。

Similarly, it is possible to obtain $\frac{sinB}{b}=\frac{sinC}{c}$ 。

theorem (math.)

The sine theorem states a relation between the three sides of an arbitrary triangle and the sines of the corresponding angles. From the monotonicity of the sine function on an interval, the sine theorem describes very well a quantitative relationship between sides and angles in an arbitrary triangle.

generalization of theorem

at any $\Delta ABC$ center, the angleA、B、CThe lengths of the sides opposite each other area、b、cThe radius of the outer garden of the triangle is RThe diameter is D. The sine theorem for deformation has:

1、a=2*R*sinA，b=2*R*sinB，c=2*R*sinC

2、a*sinB=b*sinA，b*sinC=c*sinB，a*sinC=c*sinA

3、a:b:c=sinA:sinB:sinC

4、 $\frac{a}{sinA}=\frac{a+b}{sinA+sinB}=\frac{a+b+c}{sinA+sinB+sinC}$ , isoperimetric, unchanged

5、S= $S=\frac{1}{2}*a*b*sinC=\frac{1}{2}*a*c*sinB=\frac{1}{2}*b*c*sinA$ The formula for the area of a triangle