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Law of sines

In trigonometry, the law of sines (or sine law) is a statement about arbitrary triangles in the plane. If the sides of the triangle are (lower-case) a, b and c and the angles opposite those sides are (capital) A, B and C, then the law of sines states

${\sin A \over a}={\sin B \over b}={\sin C \over c}.\,$

This formula is useful to compute the remaining sides of a triangle if two angles and a side is known, a common problem in the technique of triangulation. It can also be used when two sides and one of the non-enclosed angles are known; in this case, the formula may give two possible values for the enclosed angle. When this happens, often only one result will cause all angles to be less than 180°; in other cases, there are two valid solutions to the triangle.

The reciprocal of the number described by the sine law (i.e. a/sin(A)) is equal to the diameter D of the triangle's circumcircle (the unique circle through the three points A, B and C). The law can therefore be written

${a \over \sin A }={b \over \sin B }={c \over \sin C }=D.$

Derivation

Make a triangle with sides a, b, and c, and opposite angles A, B, and C. Make a line from the angle C to its opposite side c that cuts the figure into two right triangles, and call the length of this line h.

It can be observed that:

$\sin A = \frac{h}{b}$ and $\; \sin B = \frac{h}{a}$

Therefore:

$h = b\,\sin A = a\,\sin B$

and

$\frac{\sin A}{a} = \frac{\sin B}{b}.$

Doing the same thing with the line drawn between angle A and side a will yield:

$\frac{\sin B}{b} = \frac{\sin C}{c}$