Prove the sine rule.

The sine rule relates the sides and angles of a triangle.

The sine rule states that in any triangle ABC, the ratio of the length of a side to the sine of the angle opposite that side is the same for all three sides. In other words:

a/sin A = b/sin B = c/sin C

where a, b, and c are the lengths of the sides opposite angles A, B, and C, respectively.

To prove the sine rule, we start with the law of sines, which states that:

a/sin A = b/sin B = c/sin C = 2R

where R is the radius of the circumcircle of the triangle. This can be proven using the extended law of sines and the fact that the circumcenter of a triangle is equidistant from the vertices.

Next, we use the fact that the area of a triangle can be expressed as:

Area = (1/2)ab sin C

where a and b are the lengths of two sides of the triangle and C is the angle between them. This can be proven using the formula for the area of a parallelogram and the fact that the area of a triangle is half the area of a parallelogram with the same base and height.

Using the law of sines, we can express the sides of the triangle in terms of the circumradius R:

a = 2R sin A
b = 2R sin B
c = 2R sin C

Substituting these expressions into the formula for the area of a triangle, we get:

Area = (1/2)(2R sin A)(2R sin B) sin C
= R^2 sin A sin B sin C

Equating this expression for the area with the standard formula for the area of a triangle (Area = (1/2)bc sin A = (1/2)ac sin B = (1/2)ab sin C), we get:

a/sin A = b/sin B = c/sin C

which is the sine rule.