In trigonometry, the law of sines (sometimes called the sine formula or sine rule) is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,
a sin α = b sin β = c sin γ = 2 R , {\displaystyle {\frac {a}{\sin {\alpha }}}\,=\,{\frac {b}{\sin {\beta }}}\,=\,{\frac {c}{\sin {\gamma }}}\,=\,2R,}
where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see figure 2), while R is the radius of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals;
sin α a = sin β b = sin γ c . {\displaystyle {\frac {\sin {\alpha }}{a}}\,=\,{\frac {\sin {\beta }}{b}}\,=\,{\frac {\sin {\gamma }}{c}}.}
The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data (called the ambiguous case) and the technique gives two possible values for the enclosed angle. The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines. The law of sines can be generalized to higher dimensions on surfaces with constant curvature.
Proof With the side of length a as the base, the triangle's altitude can be computed as b sin γ or as c sin β. Equating these two expressions gives
b sin β = c sin γ , {\displaystyle {\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}\,,}
and similar equations arise by choosing the side of length b or the side of length c as the base of the triangle. For a proof that these expressions are equal to 2 R {\displaystyle 2R} , see Relation to the circumcircle.
Ambiguous case of triangle solution When using the law of sines to find a side of a triangle, an ambiguous case occurs when two separate triangles can be constructed from the data provided (i.e., there are two different possible solutions to the triangle). In the case shown below they are triangles ABC and ABC′.
Given a general triangle, the following conditions would need to be fulfilled for the case to be ambiguous:
The only information known about the triangle is the angle α and the sides a and c. The angle α is acute (i.e., α < 90°). The side a is shorter than the side c (i.e., a < c). The side a is longer than the altitude h from angle β, where h = c sin α (i.e., a > h). If all the above conditions are true, then each of angles β and β′ produces a valid triangle, meaning that both of the following are true:
γ ′ = arcsin c sin α a or γ = π − arcsin c sin α a . {\displaystyle {\gamma }'=\arcsin {\frac {c\sin {\alpha }}{a}}\quad {\text{or}}\quad {\gamma }=\pi -\arcsin {\frac {c\sin {\alpha }}{a}}.}
From there we can find the corresponding β and b or β′ and b′ if required, where b is the side bounded by vertices A and C and b′ is bounded by A and C′.
Examples The following are examples of how to solve a problem using the law of sines.
Example 1
Given: side a = 20, side c = 24, and angle γ = 40°. Angle α is desired. Using the law of sines, we conclude that
sin α 20 = sin ( 40 ∘ ) 24 . {\displaystyle {\frac {\sin \alpha }{20}}={\frac {\sin(40^{\circ })}{24}}.}
α = arcsin ( 20 sin ( 40 ∘ ) 24 ) ≈ 32.39 ∘ . {\displaystyle \alpha =\arcsin \left({\frac {20\sin(40^{\circ })}{24}}\right)\approx 32.39^{\circ }.}
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