Editing Law of sines

{{Short description|Property of all triangles on a Euclidean plane}}
{{About|the law of sines in trigonometry|the law of sines in physics|Snell's law}}
{{multiple image
| direction         = horizontal
| total_width       = 350px
| title             = Law of Sines
| image1            = Law_of_sines_in_plane_trigonometry-20201220.svg
| caption1          = Figure 1, With [[circumcircle]]
| image2            = Law of sines (simple).svg
| caption2          = Figure 2, Without circumcircle
| footer            = Two triangles labelled with components of the law of sines. The angles {{math|''α''}}, {{math|''β''}} and {{math|''γ''}} are associated with the respective vertices {{math|''A''}}, {{math|''B''}}, and {{math|''C''}}; the respective sides of lengths {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} are opposite these (e.g., side {{math|''a''}} is opposite vertex {{math|''A''}} with angle {{math|''α''}}).
}}
{{Trigonometry}}
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 [[sine]]s of its [[angle]]s. According to the law,
<math display="block"> \frac{a}{\sin{\alpha}} \,=\, \frac{b}{\sin{\beta}} \,=\, \frac{c}{\sin{\gamma}} \,=\, 2R, </math>
where {{math|''a'', ''b''}}, and {{math|''c''}} are the lengths of the sides of a triangle, and {{math|''α'', ''β''}}, and {{math|''γ''}} are the opposite angles (see figure 2), while {{math|''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 [[Multiplicative inverse|reciprocals]];
<math display="block"> \frac{\sin{\alpha}}{a} \,=\, \frac{\sin{\beta}}{b} \,=\, \frac{\sin{\gamma}}{c}. </math>
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 triangle]]s, with the other being the [[law of cosines]].

The law of sines can be generalized to higher dimensions on surfaces with constant curvature.<ref name=mathworld>{{cite web|url=http://mathworld.wolfram.com/GeneralizedLawofSines.html|website=mathworld|title=Generalized law of sines}}</ref>

==Proof==
With the side of length {{mvar|a}} as the base, the triangle's [[Altitude (triangle)|altitude]] can be computed as {{math|''b'' sin ''γ''}} or as {{math|''c'' sin ''β''}}.  Equating these two expressions gives
<math display=block>\frac{b}{\sin \beta} = \frac{c}{\sin \gamma}\,,</math>
and similar equations arise by choosing the side of length {{mvar|b}} or the side of length {{mvar|c}} as the base of the triangle.

==The 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 {{math|''ABC''}} and {{math|''ABC′''}}.

{{bi|left=1.3|[[File:PictureAmbitext (Greek angles).svg|frameless|left|upright=2]]}}
{{clear}}

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 {{math|''α''}} and the sides {{math|''a''}} and {{math|''c''}}.
* The angle {{math|''α''}} is [[Angle#Types of angles|acute]] (i.e., {{math|''α''}} < 90°).
* The side {{math|''a''}} is shorter than the side {{math|''c''}} (i.e., {{math|''a'' < ''c''}}).
* The side {{math|''a''}} is longer than the altitude {{math|''h''}} from angle {{math|''β''}}, where {{math|1=''h'' = ''c'' sin ''α''}} (i.e., {{math|''a'' > ''h''}}).

If all the above conditions are true, then each of angles {{math|''β''}} and {{math|''β′''}} produces a valid triangle, meaning that both of the following are true:
<math display="block"> {\gamma}' = \arcsin\frac{c \sin{\alpha}}{a} \quad \text{or} \quad {\gamma} = \pi - \arcsin\frac{c \sin{\alpha}}{a}.</math>

From there we can find the corresponding {{math|''β''}} and {{math|''b''}} or {{math|''β′''}} and {{math|''b′''}} if required, where {{math|''b''}} is the side bounded by vertices {{math|''A''}} and {{math|''C''}} and {{math|''b′''}} is bounded by {{math|''A''}} and {{math|''C′''}}.

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

===Example 1===
[[File:Law of sines (example 01).svg|thumb|right|upright|Example 1]]
Given: side {{math|1=''a'' = 20}}, side {{math|1=''c'' = 24}}, and angle {{math|1=''γ'' = 40°}}. Angle {{math|''α''}} is desired.

Using the law of sines, we conclude that
<math display="block">\frac{\sin \alpha}{20} = \frac{\sin (40^\circ)}{24}.</math>
<math display="block"> \alpha = \arcsin\left( \frac{20\sin (40^\circ)}{24} \right) \approx 32.39^\circ. </math>

Note that the potential solution {{math|1=''α'' = 147.61°}} is excluded because that would necessarily give {{math|1=''α'' + ''β'' + ''γ'' > 180°}}.

===Example 2===
[[File:Law of sines (example 02).svg|thumb|right|Example 2]]
If the lengths of two sides of the triangle {{math|''a''}} and {{math|''b''}} are equal to {{math|''x''}}, the third side has length {{math|''c''}}, and the angles opposite the sides of lengths {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} are {{math|''α''}}, {{math|''β''}}, and {{math|''γ''}} respectively then
<math display="block">\begin{align}
& \alpha = \beta = \frac{180^\circ-\gamma}{2}= 90^\circ-\frac{\gamma}{2} \\[6pt]
& \sin \alpha = \sin \beta = \sin \left(90^\circ-\frac{\gamma}{2}\right) = \cos \left(\frac{\gamma}{2}\right) \\[6pt]
& \frac{c}{\sin \gamma}=\frac{a}{\sin \alpha}=\frac{x}{\cos \left(\frac{\gamma}{2}\right)} \\[6pt]
& \frac{c \cos \left(\frac{\gamma}{2}\right)}{\sin \gamma} = x
\end{align}</math>

==Relation to the circumcircle==
In the identity
<math display="block"> \frac{a}{\sin{\alpha}} = \frac{b}{\sin{\beta}} = \frac{c}{\sin{\gamma}},</math>
the common value of the three fractions is actually the [[diameter]] of the triangle's [[circumcircle]]. This result dates back to [[Ptolemy]].<ref>Coxeter, H. S. M. and Greitzer, S. L. ''Geometry Revisited''. Washington, DC: Math. Assoc. Amer., pp. 1–3, 1967</ref><ref name=":0">{{Cite web|url=http://www.pballew.net/lawofsin.html|archive-url=https://web.archive.org/web/20021229104823/http://www.pballew.net/lawofsin.html|url-status=usurped|archive-date=December 29, 2002|title=Law of Sines|website=www.pballew.net|access-date=2018-09-18}}</ref>

=== Proof ===
[[File:Sinelaw radius (Greek angles).svg|thumb|upright=1.25|Deriving the ratio of the sine law equal to the circumscribing diameter. Note that triangle {{math|''ADB''}} passes through the center of the circumscribing circle with diameter {{math|''d''}}.]]

As shown in the figure, let there be a circle with inscribed <math> \triangle ABC</math> and another inscribed <math> \triangle ADB</math> that passes through the circle's center {{math|'''O'''}}. The <math> \angle AOD</math> has a [[central angle]] of <math> 180^\circ</math> and thus {{nowrap|<math> \angle ABD = 90^\circ</math>,}} by [[Thales's theorem]]. Since <math> \triangle ABD</math> is a right triangle,
<math display="block"> \sin{\delta}= \frac{\text{opposite}}{\text{hypotenuse}}= \frac{c}{2R},</math>
where <math display="inline"> R= \frac{d}{2}</math> is the radius of the circumscribing circle of the triangle.<ref name=":0" /> Angles <math>{\gamma}</math> and <math>{\delta}</math> lie on the same circle and [[subtend]] the same [[Chord (geometry)|chord]] {{math|''c''}}; thus, by the [[Inscribed angle#Theorem|inscribed angle theorem]], {{nowrap|<math>{\gamma} = {\delta}</math>.}} Therefore,
<math display="block"> \sin{\delta} = \sin{\gamma} = \frac{c}{2R}.</math>

Rearranging yields
<math display="block"> 2R = \frac{c}{\sin{\gamma}}.</math>

Repeating the process of creating <math> \triangle ADB</math> with other points gives

{{equation box 1|equation=<math> \frac{a}{\sin{\alpha}} = \frac{b}{\sin{\beta}} = \frac{c}{\sin{\gamma}}=2R.</math>}}

=== Relationship to the area of the triangle ===
The area of a triangle is given by {{nowrap|<math display="inline">T = \frac{1}{2}ab \sin \theta</math>,}} where <math>\theta</math> is the angle enclosed by the sides of lengths {{math|''a''}} and {{math|''b''}}. Substituting the sine law into this equation gives
<math display="block">T=\frac{1}{2}ab \cdot \frac {c}{2R}.</math>

Taking <math>R</math> as the circumscribing radius,<ref>{{Citation|last=Mr. T's Math Videos|title=Area of a Triangle and Radius of its Circumscribed Circle|date=2015-06-10|url=https://www.youtube.com/watch?v=t6QNGDPG4Og| archive-url=https://ghostarchive.org/varchive/youtube/20211211/t6QNGDPG4Og| archive-date=2021-12-11 | url-status=live|access-date=2018-09-18}}{{cbignore}}</ref>

{{equation box 1|equation=<math>T=\frac{abc}{4R}.</math>}}

It can also be shown that this equality implies
<math display="block">\begin{align}
\frac{abc} {2T}
& = \frac{abc} {2\sqrt{s(s-a)(s-b)(s-c)}} \\[6pt]
& = \frac {2abc} {\sqrt{{(a^2+b^2+c^2)}^2-2(a^4+b^4+c^4) }},
\end{align}</math>
where {{math|''T''}} is the area of the triangle and {{math|''s''}} is the [[semiperimeter]] {{nowrap|<math display="inline">s = \frac{1}{2}\left(a+b+c\right).</math>}}

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 {{nowrap|<math display="inline">S =\frac{1}{2}\left(\sin A + \sin B + \sin C\right)</math>,}} we have<ref>Mitchell, Douglas W., "A Heron-type area formula in terms of sines," ''Mathematical Gazette'' 93, March 2009, 108–109.</ref>

{{equation box 1|equation=<math>T = 4R^{2} \sqrt{S \left(S - \sin A\right) \left(S - \sin B\right) \left(S - \sin C\right)}</math>}}

where <math>R</math> is the radius of the circumcircle: {{nowrap|<math>2R = \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}</math>.}}

==Spherical law of sines==

The spherical law of sines deals with triangles on a sphere, whose sides are arcs of [[great circle]]s.

Suppose the radius of the sphere is&nbsp;1. Let {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} be the lengths of the great-arcs that are the sides of the triangle. Because it is a unit sphere, {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} are the angles at the center of the sphere subtended by those arcs, in radians. Let {{math|''A''}}, {{math|''B''}}, and {{math|''C''}} be the angles opposite those respective sides. These are [[dihedral angle]]s between the planes of the three great circles.

Then the spherical law of sines says:
<math display="block">\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}.</math>

[[File:Spherical trigonometry vectors.svg|thumb|right|200px]]

=== Vector proof ===
Consider a unit sphere with three unit vectors {{math|'''OA'''}}, {{math|'''OB'''}} and {{math|'''OC'''}} drawn from the origin to the vertices of the triangle. Thus the angles {{math|''α''}}, {{math|''β''}}, and {{math|''γ''}} are the angles {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}, respectively. The arc {{math|BC}} subtends an angle of magnitude {{math|''a''}} at the centre. Introduce a Cartesian basis with {{math|'''OA'''}} along the {{math|''z''}}-axis and {{math|'''OB'''}} in the {{math|''xz''}}-plane making an angle {{math|''c''}} with the {{math|''z''}}-axis. The vector {{math|'''OC'''}} projects to {{math|ON}} in the {{math|''xy''}}-plane and the angle between {{math|ON}} and the {{math|''x''}}-axis is {{math|''A''}}. Therefore, the three vectors have components:
<math display="block">\mathbf{OA} = \begin{pmatrix}0 \\ 0 \\ 1\end{pmatrix}, \quad
\mathbf{OB} = \begin{pmatrix}\sin c \\ 0 \\ \cos c\end{pmatrix}, \quad
\mathbf{OC} = \begin{pmatrix}\sin b\cos A \\ \sin b\sin A \\ \cos b\end{pmatrix}.</math>

The [[scalar triple product]], {{math|'''OA''' ⋅ ('''OB''' × '''OC''')}} is the volume of the [[parallelepiped]] formed by the position vectors of the vertices of the spherical triangle {{math|'''OA'''}}, {{math|'''OB'''}} and {{math|'''OC'''}}. This volume is invariant to the specific coordinate system used to represent {{math|'''OA'''}}, {{math|'''OB'''}} and {{math|'''OC'''}}. The value of the [[scalar triple product]] {{math|'''OA''' ⋅ ('''OB''' × '''OC''')}} is the {{math|3 × 3}} determinant with {{math|'''OA'''}}, {{math|'''OB'''}} and {{math|'''OC'''}} as its rows. With the {{math|''z''}}-axis along {{math|'''OA'''}} the square of this determinant is
<math display="block"> \begin{align}
\bigl(\mathbf{OA} \cdot (\mathbf{OB} \times \mathbf{OC})\bigr)^2
& = \left(\det \begin{pmatrix}\mathbf{OA} & \mathbf{OB} & \mathbf{OC}\end{pmatrix}\right)^2 \\[4pt]
& =  \begin{vmatrix}
0 & 0 & 1 \\   
\sin c & 0 & \cos c \\ 
\sin b \cos A & \sin b \sin A & \cos b 
\end{vmatrix} ^2
= \left(\sin b \sin c \sin A\right)^2.
\end{align}</math>
Repeating this calculation with the {{math|''z''}}-axis along {{math|'''OB'''}} gives {{math|(sin ''c'' sin ''a'' sin ''B'')<sup>2</sup>}}, while with the {{math|''z''}}-axis along {{math|'''OC'''}} it is {{math|(sin ''a'' sin ''b'' sin ''C'')<sup>2</sup>}}. Equating these expressions and dividing throughout by {{math|(sin ''a'' sin ''b'' sin ''c'')<sup>2</sup>}} gives
<math display="block">
\frac{\sin^2 A}{\sin^2 a}
= \frac{\sin^2 B}{\sin^2 b}
= \frac{\sin^2 C}{\sin^2 c}
= \frac{V^2}{\sin^2 (a) \sin^2 (b) \sin^2 (c)},
</math>
where {{mvar|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
<math display="block">\lim_{a \to 0} \frac{\sin a}{a} = 1</math>
and the same for {{math|sin ''b''}} and {{math|sin ''c''}}.

[[File:Sine law spherical small.svg|thumb|378x378px]]

=== Geometric proof ===
Consider a unit sphere with:
<math display="block">OA = OB = OC = 1</math>

Construct point <math>D</math> and point <math>E</math> such that <math>\angle ADO = \angle AEO = 90^\circ</math>

Construct point <math>A'</math> such that <math>\angle A'DO = \angle A'EO = 90^\circ</math>

It can therefore be seen that <math>\angle ADA' = B</math> and <math>\angle AEA' = C</math>

Notice that <math>A'</math> is the projection of <math>A</math> on plane <math>OBC</math>. Therefore <math>\angle AA'D = \angle AA'E = 90^\circ</math>

By basic trigonometry, we have:
<math display="block">\begin{align}
  AD &= \sin c \\
  AE &= \sin b
\end{align}</math>

But <math>AA' = AD \sin B = AE \sin C </math>

Combining them we have:
<math display="block">\begin{align}
\sin c \sin B &= \sin b \sin C \\
\Rightarrow \frac{\sin B}{\sin b} &=\frac{\sin C}{\sin c}
\end{align}</math>

By applying similar reasoning, we obtain the spherical law of sines:
<math display="block">\frac{\sin A}{\sin a} =\frac{\sin B}{\sin b} =\frac{\sin C}{\sin c} </math>

{{see also|Spherical trigonometry|Spherical law of cosines|Half-side formula}}

=== Other proofs ===
A purely algebraic proof can be constructed from the [[spherical law of cosines]]. From the identity <math>\sin^2 A = 1 - \cos^2 A</math> and the explicit expression for <math>\cos A</math> from the spherical law of cosines
<math display="block">\begin{align}
   \sin^2\!A &= 1-\left(\frac{\cos a  - \cos b\, \cos c}{\sin b \,\sin c}\right)^2\\
   &=\frac{\left(1-\cos^2\!b\right) \left(1-\cos^2\!c\right)-\left(\cos a  - \cos b\, \cos c\right)^2}
          {\sin^2\!b \,\sin^2\!c}\\[8pt]
 \frac{\sin A}{\sin a}
 &= \frac{\left[1-\cos^2\!a-\cos^2\!b-\cos^2\!c + 2\cos a\cos b\cos c\right]^{1/2}}{\sin a\sin b\sin c}.
\end{align}</math>
Since the right hand side is invariant under a cyclic permutation of <math>a,\;b,\;c</math> the spherical sine rule follows immediately.

The figure used in the Geometric proof above is used by and also provided in Banerjee<ref name="banerjee">{{Citation | last = Banerjee | first = Sudipto | date = 2004 | title = Revisiting Spherical Trigonometry with Orthogonal Projectors | journal = The College Mathematics Journal | volume = 35 | issue = 5 | pages = 375–381 | publisher = Mathematical Association of America| doi = 10.1080/07468342.2004.11922099 | url = http://www.biostat.umn.edu/~sudiptob/ResearchPapers/banerjee.pdf | archive-url = https://web.archive.org/web/20041029141245id_/http://www.biostat.umn.edu/~sudiptob/ResearchPapers/banerjee.pdf | url-status = dead | archive-date = 2004-10-29 }}</ref> (see Figure 3 in this paper) to derive the sine law using elementary linear algebra and projection matrices.

==Hyperbolic case==
In [[hyperbolic geometry]] when the curvature is −1, the law of sines becomes
<math display="block">\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c} \,.</math>

In the special case when {{math|''B''}} is a right angle, one gets
<math display="block">\sin C = \frac{\sinh c}{\sinh b} </math>

which is the analog of the formula in Euclidean geometry expressing the sine of an angle as the opposite side divided by the hypotenuse.

{{See also|Hyperbolic triangle}}

== The case of surfaces of constant curvature ==

Define a generalized sine function, depending also on a real parameter <math>\kappa</math>:
<math display="block">\sin_\kappa(x) = x - \frac{\kappa}{3!}x^3 + \frac{\kappa^2}{5!}x^5 - \frac{\kappa^3}{7!}x^7 + \cdots = \sum_{n=0}^\infty \frac{(-1)^n \kappa^n}{(2n+1)!}x^{2n+1}.</math>

The law of sines in constant curvature <math>\kappa</math> reads as<ref name="mathworld"/>
<math display="block">\frac{\sin A}{\sin_\kappa a} = \frac{\sin B}{\sin_\kappa b} = \frac{\sin C}{\sin_\kappa c} \,.</math>

By substituting <math>\kappa=0</math>, <math>\kappa=1</math>, and <math>\kappa=-1</math>, one obtains respectively <math>\sin_{0}(x) = x</math>, <math>\sin_{1}(x) = \sin x</math>, and <math>\sin_{-1}(x) = \sinh x</math>, that is, the Euclidean, spherical, and hyperbolic cases of the law of sines described above.<ref name="mathworld"/>

Let <math>p_\kappa(r)</math> indicate the circumference of a circle of radius <math>r</math> in a space of constant curvature <math>\kappa</math>. Then <math>p_\kappa(r)=2\pi\sin_\kappa(r)</math>. Therefore, the law of sines can also be expressed as:
<math display="block">\frac{\sin A}{p_\kappa(a)} = \frac{\sin B}{p_\kappa(b)} = \frac{\sin C}{p_\kappa(c)} \,.</math>

This formulation was discovered by [[János Bolyai]].<ref>{{cite book|last=Katok|first=Svetlana|author-link=Svetlana Katok| title=Fuchsian groups|url=https://archive.org/details/fuchsiangroups00kato|url-access=limited|year=1992|publisher=University of Chicago Press|location=Chicago|isbn=0-226-42583-5|page=[https://archive.org/details/fuchsiangroups00kato/page/n31 22]}}</ref>

==Higher dimensions==
A [[tetrahedron]] has four triangular [[facet (geometry)|facet]]s. The [[absolute value]] of the [[polar sine]] ({{math|psin}}) of the [[normal vector]]s to the three facets that share a [[vertex (geometry)|vertex]] of the tetrahedron, divided by the area of the fourth facet will not depend upon the choice of the vertex:<ref>{{cite journal |last=Eriksson |first=Folke |year=1978 |title=The law of sines for tetrahedra and n-simplices |journal=Geometriae Dedicata |volume=7 |issue=1 |pages=71–80 |doi=10.1007/bf00181352}}</ref>

<math display="block">\begin{align}
& \frac{\left|\operatorname{psin}(\mathbf{b}, \mathbf{c}, \mathbf{d})\right|}{\mathrm{Area}_a} =
  \frac{\left|\operatorname{psin}(\mathbf{a}, \mathbf{c}, \mathbf{d})\right|}{\mathrm{Area}_b} =
  \frac{\left|\operatorname{psin}(\mathbf{a}, \mathbf{b}, \mathbf{d})\right|}{\mathrm{Area}_c} =
  \frac{\left|\operatorname{psin}(\mathbf{a}, \mathbf{b}, \mathbf{c})\right|}{\mathrm{Area}_d} \\[4pt]
= {} & \frac{(3~\mathrm{Volume}_\mathrm{tetrahedron})^2}{2~\mathrm{Area}_a \mathrm{Area}_b \mathrm{Area}_c \mathrm{Area}_d}\,.
\end{align}</math>

More generally, for an {{math|''n''}}-dimensional [[simplex]] (i.e., [[triangle]] ({{math|1=''n'' = 2}}), [[tetrahedron]] ({{math|1=''n'' = 3}}), [[pentatope]] ({{math|1=''n'' = 4}}), etc.) in {{math|''n''}}-dimensional [[Euclidean space]], the absolute value of the polar sine 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 {{math|''V''}} for the hypervolume of the {{math|''n''}}-dimensional simplex and {{math|''P''}} for the product of the hyperareas of its {{math|(''n'' − 1)}}-dimensional facets, the common ratio is
<math display="block">\frac{\left|\operatorname{psin}(\mathbf{b}, \ldots, \mathbf{z})\right|}{\mathrm{Area}_a} = \cdots = \frac{\left|\operatorname{psin}(\mathbf{a}, \ldots, \mathbf{y})\right|}{\mathrm{Area}_z} = \frac{(nV)^{n-1}}{(n-1)! P}.</math>

Note that when the vectors {{math|'''v'''{{sub|1}}, ..., '''v'''{{sub|''n''}}}}, from a selected vertex to each of the other vertices, are the columns of a matrix {{mvar|V}} then the columns of the matrix <math>N = -V (V^TV)^{-1} \sqrt{\det{V^TV}} / (n-1)!</math> are outward-facing normal vectors of those facets that meet at the selected vertex.  This formula also works when the vectors are in a {{mvar|m}}-dimensional space having {{math|''m'' > ''n''}}.  In the {{math|1=''m'' = ''n''}} case that {{mvar|V}} is square, the formula simplifies to <math>N = -(V^T)^{-1} |\det{V}| / (n-1)!\,.</math>

==History==
An equivalent of the law of sines, that the sides of a triangle are proportional to the [[chord (trigonometry)|chords]] of double the opposite angles, was known to the 2nd century Hellenistic astronomer [[Ptolemy]] and used occasionally in his ''[[Almagest]]''.<ref>{{cite book |editor-last=Toomer |editor-first=Gerald J. |editor-link=Gerald J. Toomer |title=Ptolemy's Almagest |publisher=Princeton University Press |year=1998 |pages=[https://archive.org/details/ptolemys-almagest-toomer/page/7/mode/1up 7, fn. 10]; [https://archive.org/details/ptolemys-almagest-toomer/page/462/mode/1up 462, fn. 96] }}</ref>

Statements related to the law of sines appear in the astronomical and trigonometric work of 7th century Indian mathematician [[Brahmagupta]]. In his ''[[Brāhmasphuṭasiddhānta]]'', Brahmagupta expresses the circumradius of a triangle as the product of two sides divided by twice the [[Altitude (triangle)|altitude]]; the law of sines can be derived by alternately expressing the altitude as the sine of one or the other base angle times its opposite side, then equating the two resulting variants.<ref>{{cite book|last=Winter |first=Henry James Jacques |title=Eastern Science |publisher=[[John Murray (publishing house)|John Murray]] |year=1952 |page=46 |url=https://archive.org/details/easternscienceou0000wint/page/46/}} {{pb}} {{cite book|last=Colebrooke |first=Henry Thomas |title=Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bhascara |location=London |publisher=[[John Murray (publishing house)|John Murray]] |year=1817 |pages=299–300 |url=https://archive.org/details/algebrawitharith00brahuoft/page/298/mode/2up}}</ref> An equation even closer to the modern law of sines appears in Brahmagupta's ''[[Khaṇḍakhādyaka]]'', in a method for finding the distance between the Earth and a planet following an [[epicycle]]; however, Brahmagupta never treated the law of sines as an independent subject or used it systematically for solving triangles.<ref>{{cite book |last=Van Brummelen |first=Glen |authorlink=Glen Van Brummelen |year=2009 |title=The Mathematics of the Heavens and the Earth |publisher=Princeton University Press |pages=109–111 |isbn=978-0-691-12973-0 }} {{pb}} {{cite book |last=Brahmagupta |title=The Khandakhadyaka: An Astronomical Treatise of Brahmagupta |translator=Sengupta |translator-first=Prabodh Chandra |year=1934 |publisher=University of Calcutta }}</ref>

The spherical law of sines is sometimes credited to 10th century scholars [[Abu-Mahmud Khujandi]] or [[Abū al-Wafāʾ]] (it appears in his ''Almagest''), but it is given prominence in [[Abū Naṣr Manṣūr]]'s ''Treatise on the Determination of Spherical Arcs'', and was credited to Abū Naṣr Manṣūr by his student [[al-Bīrūnī]] in his ''Keys to Astronomy''.<ref>{{cite book|last=Sesiano |first=Jacques |chapter=Islamic mathematics |pages=137–157 |title=Mathematics Across Cultures: The History of Non-western Mathematics|editor-first1=Helaine|editor-last1=Selin| editor-first2=Ubiratan|editor-last2=D'Ambrosio |year=2000 |publisher=Springer |isbn=1-4020-0260-2}} {{pb}} {{cite book |last=Van Brummelen |first=Glen |authorlink=Glen Van Brummelen |year=2009 |title=The Mathematics of the Heavens and the Earth |publisher=Princeton University Press |pages=183–185 |isbn=978-0-691-12973-0 }}</ref> [[Ibn Muʿādh al-Jayyānī]]'s 11th-century ''Book of Unknown Arcs of a Sphere'' also contains the spherical law of sines.<ref name="MacTutor Al-Jayyani">{{MacTutor|id=Al-Jayyani|title=Abu Abd Allah Muhammad ibn Muadh Al-Jayyani}}</ref> 

The 13th-century Persian mathematician [[Naṣīr al-Dīn al-Ṭūsī]] stated and proved the planar law of sines:<ref>{{Cite web |title=Nasir al-Din al-Tusi - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Al-Tusi_Nasir/ |access-date=2025-03-10 |website=Maths History |language=en}}</ref> <blockquote>In any plane triangle, the ratio of the sides is equal to the ratio of the sines of the angles opposite to those sides. That is, in triangle ABC, we have AB : AC = Sin(∠ACB) : Sin(∠ABC)</blockquote> By employing the law of sines, al-Tusi could solve triangles where either two angles and a side were known or two sides and an angle opposite one of them were given. For triangles with two sides and the included angle, he divided them into right triangles that he could then solve. When three sides were given, he dropped a perpendicular line and then used Proposition II-13 of Euclid's ''Elements'' (a geometric version of the [[law of cosines]]). Al-Tusi established the important result that if the sum or difference of two arcs is provided along with the ratio of their sines, then the arcs can be calculated.<ref>{{Cite book |last=Katz |first=Victor J. |url=https://books.google.com/books?id=7rP2MAAACAAJ |title=A History of Mathematics: An Introduction |date=2017-03-21 |publisher=Pearson |isbn=978-0-13-468952-4 |pages=315 |language=en}}</ref>

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."<ref>{{cite book|first=Glen |last=Van Brummelen |year=2009 |url=https://books.google.com/books?id=bHD8IBaYN-oC |title=The Mathematics of the Heavens and the Earth: The Early History of Trigonometry |publisher=[[Princeton University Press]] |page=259 |isbn=978-0-691-12973-0}}</ref> Regiomontanus was a 15th-century German mathematician.

==See also==
* {{annotated link|Gersonides}}
* [[Half-side formula]]{{snd}} for solving [[spherical triangles]]
* [[Law of cosines]]
* [[Law of tangents]]
* [[Law of cotangents]]
* [[Mollweide's formula]]{{snd}} for checking solutions of triangles
* [[Solution of triangles]]
* [[Surveying]]

==References==
{{Reflist}}

==External links==
{{Commons category}}
* {{springer|title=Sine theorem|id=p/s085520}}
* [http://www.cut-the-knot.org/proofs/sine_cosine.shtml#law The Law of Sines] at [[cut-the-knot]]
* [http://mysite.du.edu/~jcalvert/railway/degcurv.htm Degree of Curvature]
* [https://web.archive.org/web/20160815080014/http://www.efnet-math.org/Meta/sine1.htm Finding the Sine of 1 Degree]
* [http://mathworld.wolfram.com/GeneralizedLawofSines.html Generalized law of sines to higher dimensions]

{{Ancient Greek mathematics}}

{{DEFAULTSORT:Law Of Sines}}
[[Category:Trigonometry]]
[[Category:Angle]]
[[Category:Articles containing proofs]]
[[Category:Theorems about triangles]]