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Law of sines

The law of sines relates the lengths of the sides of any triangle to the sines of its angles.

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$\frac{a}{\sin α} = \frac{b}{\sin β} = \frac{c}{\sin γ} = 2 r,$

where $r$ is the radius of the circle circumscribing the triangle.

The law is a corollary of the statement $h = b \sin γ = c \sin β$ , where $h$ is the altitude of the triangle.

Ambiguity

A triangle may have more than one solution when applying the law of sines. Nondeterminism occurs when all of the following is true:

The only information known about the triangle is the angle $α$ and the sides $a$ and $c$ .
The angle $α$ is acute.
The side $a$ is shorter than the side $c$ .
The side $a$ is longer than the altitude $h$ from angle $β$ .

The complementary statement is that the law of sines only has a unique solution when all of the following is true:

The measure of an angle is known.
The length of the side opposite to a known angle is known.
At least one more side or angle is known.

When sides $a$ and $b$ are given and if the angle $α$ is known, then ambiguity may be understood via the value of $\sin β$ :

In the case of two oblique solutions, the possible values for $β$ are $\arcsin (\sin β)$ (in quadrant I) and $π - \arcsin (\sin β)$ (in quadrant II).

References

[cite:@enwiki:1292486449]
[cite:@enwiki:1289587190]