From Wikipedia, the free encyclopedia - View original article
|Laws and theorems|
|Law of sines|
Law of cosines
Law of tangents
Law of cotangents
Integrals of functions
Derivatives of functions
Integrals of inverse functions
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), and D is the diameter of the triangle's circumcircle. When the last part of the equation is not used, sometimes the law is stated using the reciprocal:
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 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, with the other being the law of cosines.
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 x 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
Like the law of cosines, although the law of sines is mathematically true, it has problems for numeric use. Much precision may be lost if an arcsine is computed when the sine of an angle is close to one.
When using the law of sines to solve triangles, 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:
In the identity
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.
It is easy to see how for small spherical triangles, when the radius of the sphere is much greater than the sides of the triangle, this formula becomes the planar formula at the limit, since
and the same for and .
In hyperbolic geometry when the curvature is −1, the law of sines becomes
In the special case when B is a right angle, one gets
which is the analog of the formula in Euclidean geometry expressing the sine of an angle as the opposite side divided by the hypotenuse.
Define a generalized sine function, depending also on a real parameter :
The law of sines in constant curvature reads as
By substituting , , and , one obtains respectively the euclidian, spherical, and hyperbolic cases of the law of sines described above.
Let indicate the circumference of a circle of radius in a space of constant curvature . Then . Therefore the law of sines can also be expressed as:
According to Ubiratàn D'Ambrosio and Helaine Selin, the spherical law of sines was discovered in the 10th century. It is variously attributed to al-Khujandi, Abul Wafa Bozjani, Nasir al-Din al-Tusi and Abu Nasr Mansur.
Al-Jayyani's The book of unknown arcs of a sphere in the 11th century introduced the general law of sines. The plane law of sines was later described in the 13th century by Nasīr al-Dīn al-Tūsī. In his On the Sector Figure, he stated the law of sines for plane and spherical triangles, and provided proofs for this law.
According to Glen Van Brummelen, "The Law of Sines is really Regiomontanus's foundation for his solutions of right-angled triangles in Book IV, and these solutions are in turn the bases for his solutions of general triangles." Regiomontanus was a 15th-century German mathematician.
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:
Doing the same thing with the line drawn between vertex A and side a will yield:
Observe that the area of the triangle can be written as any of
Multiplying these by gives
An equation involving sine functions and tetrahedra is as follows. For a tetrahedron with vertices O, A, B, C, it is true that
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 half-circle. 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 half-circle. 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 5-dimensional.