In trigonometry, the law of sines (also known as the sines law, sine formula, or sine rule) is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angle. According to the law,
where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the opposite angles (see the figure to the right). Sometimes the law is stated using the reciprocal of this equation:
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 nonenclosed angles are known. In some such cases, the formula gives two possible values for the enclosed angle, leading to an ambiguous case.
The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in a general triangle, the other being the law of cosines.
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The following are examples of how to solve a problem using the law of sines:
Given: side a = 20, side c = 24, and angle C = 40°
Using the law of sines, we conclude that
Or another example of how to solve a problem using the law of sines:
If two sides of the triangle are equal to R and the length of the third side, the chord, is given as 100 feet and the angle C opposite the chord is given in degrees, then
and
When using the law of sines to solve triangles, under special conditions there exists an ambiguous case where two separate triangles can be constructed (i.e., there are two different possible solutions to the triangle).
Given a general triangle ABC, the following conditions would need to be fulfilled for the case to be ambiguous:
Given all of the above premises are true, the angle B may be acute or obtuse; meaning, one of the following is true:
OR
In the identity
the common value of the three fractions is actually the diameter of the triangle's circumcircle. It can be shown that this quantity is equal to
where S is the area of the triangle and s is the semiperimeter
The second equality above is essentially Heron's formula.
In the spherical case, the formula is:
Here, α, β, and γ are the angles at the center of the sphere subtended by the three arcs of the spherical surface triangle a, b, and c, respectively. A, B, and C are the surface angles opposite their respective arcs.
The spherical law of sines was discovered in the 10th century by the Persian mathematician, Abu Nasr Mansur.^{[1]} It was later applied practically in the 11th century by Abu Rayhan Biruni in order to accurately determine the Earth radius in his Masudic Canon.^{[2]}
AlJayyani's The book of unknown arcs of a sphere in the 11th century introduced the general law of sines.^{[3]} The plane law of sines was later described in the 13th century by Nasīr alDīn alTūsī. In his On the Sector Figure, he stated the law of sines for plane and spherical triangles, and provided proofs for this law.^{[4]}
Make a triangle with the sides a, b, and c, and angles A, B, and C. Draw the altitude from vertex C to the side across c; by definition it divides the original triangle into two right angle triangles. Mark the length of this line h.
It can be observed that:
Therefore
and
Doing the same thing with the line drawn between vertex A and side a will yield:
For 2nd angle:
for 3rd angle:
A corollary of the law of sines as stated above is that in a tetrahedron with vertices O, A, B, C, we have
One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface.
Putting any of the four vertices in the role of O yields four such identities, but in a sense at most three of them are independent: If the "clockwise" sides of three of them are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity. One reason to be interested in this "independence" relation is this: It is widely known that three angles are the angles of some triangle if and only if their sum is a halfcircle. What condition on 12 angles is necessary and sufficient for them to be the 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be a halfcircle. Since there are four such triangles, there are four such constraints on sums of angles, and the number of degrees of freedom is thereby reduced from 12 to 8. The four relations given by this sines law further reduce the number of degrees of freedom, not from 8 down to 4, but only from 8 down to 5, since the fourth constraint is not independent of the first three. Thus the space of all shapes of tetrahedra is 5dimensional.
