Law of sines

Summary

In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,

Law of Sines
Figure 1, With circumcircle
Figure 2, Without circumcircle
Two triangles labelled with the components of the law of sines. α, β and γ are the angles associated with the vertices at capital A, B, and C, respectively. Lower-case a, b, and c are the lengths of the sides opposite them. (a is opposite α, etc.)
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;
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.[1]

HistoryEdit

According to Ubiratàn D'Ambrosio and Helaine Selin, the spherical law of sines was discovered in the 10th century. It is variously attributed to Abu-Mahmud Khojandi, Abu al-Wafa' Buzjani, Nasir al-Din al-Tusi and Abu Nasr Mansur.[2]

Ibn Muʿādh al-Jayyānī's The book of unknown arcs of a sphere in the 11th century contains the general law of sines.[3] The plane law of sines was later stated 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.[4]

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."[5] Regiomontanus was a 15th-century German mathematician.

ProofEdit

The area T of any triangle can be written as one half of its base times its height. Selecting one side of the triangle as the base, the height of the triangle relative to that base is computed as the length of another side times the sine of the angle between the chosen side and the base. Thus depending on the selection of the base, the area of the triangle can be written as any of:

 
Multiplying these by 2/abc gives
 

The ambiguous case of triangle solutionEdit

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:

 

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′.

ExamplesEdit

The following are examples of how to solve a problem using the law of sines.

Example 1Edit

 
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

 
 

Note that the potential solution α = 147.61° is excluded because that would necessarily give α + β + γ > 180°.

Example 2Edit

 
Example 2

If the lengths of two sides of the triangle a and b are equal to x, the third side has length c, and the angles opposite the sides of lengths a, b, and c are α, β, and γ respectively then

 

Relation to the circumcircleEdit

In the identity

 
the common value of the three fractions is actually the diameter of the triangle's circumcircle. This result dates back to Ptolemy.[6][7]
 
Deriving the ratio of the sine law equal to the circumscribing diameter. Note that triangle ADB passes through the center of the circumscribing circle with diameter d.

ProofEdit

As shown in the figure, let there be a circle with inscribed   and another inscribed   that passes through the circle's center O. The   has a central angle of   and thus  . Since   is a right triangle,

 
where   is the radius of the circumscribing circle of the triangle.[7] Angles   and   have the same central angle thus they are the same:  . Therefore,
 

Rearranging yields

 

Repeating the process of creating   with other points gives

 

Relationship to the area of the triangleEdit

The area of a triangle is given by  , where   is the angle enclosed by the sides of lengths a and b. Substituting the sine law into this equation gives

 

Taking   as the circumscribing radius,[8]

 

It can also be shown that this equality implies

 
where T is the area of the triangle and s is the semiperimeter  

The second equality above readily simplifies to Heron's formula for the area.

The sine rule can also be used in deriving the following formula for the triangle's area: Denoting the semi-sum of the angles' sines as  , we have[9]

 

where   is the radius of the circumcircle:  .

The spherical law of sinesEdit

The spherical law of sines deals with triangles on a sphere, whose sides are arcs of great circles.

Suppose the radius of the sphere is 1. Let a, b, and c be the lengths of the great-arcs that are the sides of the triangle. Because it is a unit sphere, a, b, and c are the angles at the center of the sphere subtended by those arcs, in radians. Let A, B, and C be the angles opposite those respective sides. These are dihedral angles between the planes of the three great circles.

Then the spherical law of sines says:

 
 

Vector proofEdit

Consider a unit sphere with three unit vectors OA, OB and OC drawn from the origin to the vertices of the triangle. Thus the angles α, β, and γ are the angles a, b, and c, respectively. The arc BC subtends an angle of magnitude a at the centre. Introduce a Cartesian basis with OA along the z-axis and OB in the xz-plane making an angle c with the z-axis. The vector OC projects to ON in the xy-plane and the angle between ON and the x-axis is A. Therefore, the three vectors have components:

 

The scalar triple product, OA ⋅ (OB × OC) is the volume of the parallelepiped formed by the position vectors of the vertices of the spherical triangle OA, OB and OC. This volume is invariant to the specific coordinate system used to represent OA, OB and OC. The value of the scalar triple product OA ⋅ (OB × OC) is the 3 × 3 determinant with OA, OB and OC as its rows. With the z-axis along OA the square of this determinant is

 
Repeating this calculation with the z-axis along OB gives (sin c sin a sin B)2, while with the z-axis along OC it is (sin a sin b sin C)2. Equating these expressions and dividing throughout by (sin a sin b sin c)2 gives
 
where V is the volume of the parallelepiped formed by the position vector of the vertices of the spherical triangle. Consequently, the result follows.

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 sin b and sin c.
 

Geometric proofEdit

Consider a unit sphere with:

 

Construct point   and point   such that  

Construct point   such that  

It can therefore be seen that   and  

Notice that   is the projection of   on plane  . Therefore  

By basic trigonometry, we have:

 
 

But  

Combining them we have:

 
 

By applying similar reasoning, we obtain the spherical law of sine:

 

Other proofsEdit

A purely algebraic proof can be constructed from the spherical law of cosines. From the identity   and the explicit expression for   from the spherical law of cosines

 
Since the right hand side is invariant under a cyclic permutation of   the spherical sine rule follows immediately.

The figure used in the Geometric proof above is used by and also provided in Banerjee[10] (see Figure 3 in this paper) to derive the sine law using elementary linear algebra and projection matrices.

Hyperbolic caseEdit

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.

The case of surfaces of constant curvatureEdit

Define a generalized sine function, depending also on a real parameter K:

 

The law of sines in constant curvature K reads as[1]

 

By substituting K = 0, K = 1, and K = −1, one obtains respectively the Euclidean, spherical, and hyperbolic cases of the law of sines described above.

Let pK(r) indicate the circumference of a circle of radius r in a space of constant curvature K. Then pK(r) = 2π sinK r. Therefore, the law of sines can also be expressed as:

 

This formulation was discovered by János Bolyai.[11]

Higher dimensionsEdit

For an n-dimensional simplex (i.e., triangle (n = 2), tetrahedron (n = 3), pentatope (n = 4), etc.) in n-dimensional Euclidean space, the absolute value of the polar sine (psin) of the normal vectors of the facets that meet at a vertex, divided by the hyperarea of the facet opposite the vertex is independent of the choice of the vertex. Writing V for the hypervolume of the n-dimensional simplex and P for the product of the hyperareas of its (n − 1)-dimensional facets, the common ratio is

 

For example, a tetrahedron has four triangular facets. The absolute value of the polar sine of the normal vectors to the three facets that share a vertex, divided by the area of the fourth facet will not depend upon the choice of the vertex:

 

See alsoEdit

ReferencesEdit

  1. ^ a b "Generalized law of sines". mathworld.
  2. ^ Sesiano just lists al-Wafa as a contributor. Sesiano, Jacques (2000) "Islamic mathematics" pp. 137–157, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN 1-4020-0260-2
  3. ^ O'Connor, John J.; Robertson, Edmund F., "Abu Abd Allah Muhammad ibn Muadh Al-Jayyani", MacTutor History of Mathematics archive, University of St Andrews
  4. ^ Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 978-0-691-11485-9.
  5. ^ Glen Van Brummelen (2009). "The mathematics of the heavens and the earth: the early history of trigonometry". Princeton University Press. p.259. ISBN 0-691-12973-8
  6. ^ Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 1–3, 1967
  7. ^ a b "Law of Sines". www.pballew.net. Retrieved 2018-09-18.
  8. ^ Mr. T's Math Videos (2015-06-10), Area of a Triangle and Radius of its Circumscribed Circle, archived from the original on 2021-12-11, retrieved 2018-09-18
  9. ^ Mitchell, Douglas W., "A Heron-type area formula in terms of sines," Mathematical Gazette 93, March 2009, 108–109.
  10. ^ Banerjee, Sudipto (2004), "Revisiting Spherical Trigonometry with Orthogonal Projectors", The College Mathematics Journal, Mathematical Association of America, 35 (5): 375–381, doi:10.1080/07468342.2004.11922099, S2CID 122277398Text online {{citation}}: External link in |postscript= (help)CS1 maint: postscript (link)
  11. ^ Katok, Svetlana (1992). Fuchsian groups. Chicago: University of Chicago Press. p. 22. ISBN 0-226-42583-5.

External linksEdit