Law of sines
|Laws and theorems|
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), while d is the diameter of the triangle's circumcircle. When the last of these equations 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. Numerical calculation using this technique may result in a numerical error if an angle is close to 90 degrees. 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 Proof
- 2 The ambiguous case of triangle solution
- 3 Examples
- 4 Relation to the circumcircle
- 5 Curvature
- 6 Higher dimensions
- 7 History
- 8 See also
- 9 References
- 10 External links
Via a triangle area formula
The area T of any triangle can be written as one half of its base times its height. Depending on which side one chooses to be the base, the area can be written as any of
Multiplying these by 2/abc gives
There are three cases to consider in proving the law of sines. The first is when all angles of the triangle are acute. The second is when one angle is a right angle. The third is when one angle is obtuse.
For acute triangles
We make a triangle with the sides a, b, and c, and angles A, B, and C. Then we draw the altitude from vertex B to side b; by definition it divides the original triangle into two right angle triangles: ABR and R′BC. Mark this line h1.
Using the definition of sin α = opposite / hypotenuse we see that for angle A on the right angle triangle ABR and C on R′BC we have:
Solving for h1
Equating h1 in both expressions:
Doing the same thing from angle A to side a we call the altitude h2 and the two right angle triangles ABR and AR′C:
Solving for h2
Equating the c / sin C terms in both expressions above we have:
For right angle triangles
We make a triangle with the sides a, b, and c, and angles A, B, and C where C is a right angle.
Since we already have a right angle triangle we can use the definition of sine:
Solving for c:
For the remaining angle C we need to remember that it is a right angle and sin C = 1 in this case. Therefore we can rewrite c = c / 1 as:
Equating c in both the equations above we again have:
For obtuse triangles
We make a triangle with the sides a, b, and c, and angles A, B, and C where A is an obtuse angle. In this case if we draw an altitude from any angle other than A the point where this line will touch the base of the triangle ABC will lie outside any of the lines a, b, or c. We draw the altitude from angle B, calling it h1 and create the two extended right triangles RBA′ and RBC.
From the definition of sine we again have:
We use identity sin π − θ = sin θ to express sin A′ in terms of sin A. By definition we have:
We now draw an altitude from A calling it h2 and forming two right triangles ABR and AR′C.
From this we straightforwardly get:
Equating the c / sin C in both equations above we again get:
Proving the theorem in all cases.
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 ABC and AB′C′.
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 A and the sides a and c.
- The angle A is acute (i.e., A < 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 B, where h = c sin A (i.e., a > h).
If all the above conditions are true, then both angles C or C′ produce a valid triangle; meaning both of the following are true:
From there we can find the corresponding B and b or B′ and b′ if required, where b is the side bounded by angles A and C and b′ bounded by A and C′.
Without further information it is impossible to decide which is the triangle being asked for.
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°. Angle A is desired.
Using the law of sines, we conclude that
Another example of how to solve a problem using the law of sines is:
If two unknown sides of the triangle are equal to x and the angle C opposite the chord (opposite the third side) is given in degrees, then
Relation to the circumcircle
In the identity
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 law of sines takes on a similar form in the presence of curvature.
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 sin β and sin γ.
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.
- See also hyperbolic triangle.
Define a generalized sine function, depending also on a real parameter K:
The law of sines in constant curvature K reads as
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:
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 of the normal vectors of the faces that meet at a vertex, divided by the hyperarea of the face opposite the vertex is independent of the choice of the vertex. For example, a tetrahedron has four triangular faces. The absolute value of the polar sine of the normal vectors to three of the faces (which share a vertex) divided by the area of the fourth face will not depend upon the choice of the vertex:
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.
- Half-side formula – for solving spherical triangles
- Law of cosines
- Law of tangents
- Law of cotangents
- Mollweide's formula – for checking solutions of triangles
- Solution of triangles
- "Generalized law of sines". mathworld.
- Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 1–3, 1967
- Katok, Svetlana (1992). Fuchsian groups. Chicago: University of Chicago Press. p. 22. ISBN 0-226-42583-5.
- 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
- O'Connor, John J.; Robertson, Edmund F., "Abu Abd Allah Muhammad ibn Muadh Al-Jayyani", MacTutor History of Mathematics archive, University of St Andrews.
- 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.
- 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