Law of cosines explained

In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides

a, b, and c, opposite respective angles \alpha, \beta, and \gamma (see Fig. 1), the law of cosines states:\begin{align} c ^{2} & = a ^{2} + b ^{2} - 2 ab\cos\gamma, \\[ 3 mu] a ^{2} & = b ^{2+c} ^{2-2bc\cos\alpha,} \\[ 3 mu] b ^{2} & = a ^{2+c} ^{2-2ac\cos\beta. \end{align}}

The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if

\gamma is a right angle then \cos\gamma = 0 , and the law of cosines reduces to c ^{2} = a ^{2} + b ^{2.}

The law of cosines is useful for solving a triangle when all three sides or two sides and their included angle are given.

Use in solving triangles

The theorem is used in solution of triangles, i.e., to find (see Figure 3):

• the third side of a triangle if two sides and the angle between them is known: $$c\; =\; \backslash sqrt\backslash ,;$$
• the angles of a triangle if the three sides are known: $$\backslash gamma\; =\; \backslash arccos\backslash left(\backslash frac\backslash right)\backslash ,;$$
• the third side of a triangle if two sides and an angle opposite to one of them is known (this side can also be found by two applications of the law of sines): $$a=b\backslash cos\backslash gamma\; \backslash pm\; \backslash sqrt\backslash ,.$$

These formulas produce high round-off errors in floating point calculations if the triangle is very acute, i.e., if is small relative to and or is small compared to 1. It is even possible to obtain a result slightly greater than one for the cosine of an angle.

The third formula shown is the result of solving for a in the quadratic equation . This equation can have 2, 1, or 0 positive solutions corresponding to the number of possible triangles given the data. It will have two positive solutions if, only one positive solution if, and no solution if . These different cases are also explained by the side-side-angle congruence ambiguity.

History

Book II of Euclid's Elements, compiled c. 300 BC from material up to a century or two older, contains a geometric theorem corresponding to the law of cosines but expressed in the contemporary language of rectangle areas; Hellenistic trigonometry developed later, and sine and cosine per se first appeared centuries afterward in India.

The cases of obtuse triangles and acute triangles (corresponding to the two cases of negative or positive cosine) are treated separately, in Propositions II.12 and II.13:

Proposition 13 contains an analogous statement for acute triangles. In his (now-lost and only preserved through fragmentary quotations) commentary, Heron of Alexandria provided proofs of the converses of both II.12 and II.13.^[1]

Using notation as in Fig. 2, Euclid's statement of proposition II.12 can be represented more concisely (though anachronistically) by the formula

AB ^{2} = CA ^{2} + CB ^{2} + 2 (CA)(CH).

To transform this into the familiar expression for the law of cosines, substitute

AB = c, CA = b, CB = a, and CH = a \cos(\pi - \gamma) = - a \cos \gamma.

Proposition II.13 was not used in Euclid's time for the solution of triangles, but later it was used that way in the course of solving astronomical problems by al-Bīrūnī (11th century) and Johannes de Muris (14th century).^[2] Something equivalent to the spherical law of cosines was used (but not stated in general) by al-Khwārizmī (9th century), al-Battānī (9th century), and Nīlakaṇṭha (15th century).^[3]

The 13th century Persian mathematician Naṣīr al-Dīn al-Ṭūsī, in his Arabic: Kitāb al-Shakl al-qattāʴ (Book on the Complete Quadrilateral, c. 1250), gave a method for finding the third side of a general scalene triangle given two sides and the included angle by dropping a perpendicular from the vertex of one of the unknown angles to the opposite base, reducing the problem to solving a right-angled triangle by the Pythagorean theorem. If written out using modern mathematical notation, the resulting relation can be algebraically manipulated into the modern law of cosines.^[4]

About two centuries later, another Persian mathematician, Jamshīd al-Kāshī, who computed the most accurate trigonometric tables of his era, wrote about various methods of solving triangles in his Arabic: Miftāḥ al-ḥisāb (Key of Arithmetic, 1427), and repeated essentially al-Ṭūsī's method, including more explicit details, as follows:^[5]

Using modern algebraic notation and conventions this might be written

c = \sqrt{(b - a \cos \gamma) ^{2} + (a \sin \gamma) ^{2}}

when

\gamma is acute orc = \sqrt{\left(b + a \left|\cos \gamma\right| \right) ^{2} + \left(a \sin \gamma\right) ^{2}}

when

\gamma is obtuse. (When \gamma is obtuse, the modern convention is that \cos\gamma is negative and \cos(\pi - \gamma) = - \cos\gamma is positive; historically sines and cosines were considered to be line segments with non-negative lengths.) By squaring both sides, expanding the squared binomial, and then applying the Pythagorean trigonometric identity \cos ^{2\gamma} + \sin ^{2\gamma} = 1 , we obtain the familiar law of cosines:\begin{align} c ^{2 &=} b ^{2} - 2 ba\cos \gamma + a ^{2\cos} ^{2} \gamma + a ^{2\sin} ^{2\gamma} \\[ 5 mu] & = a ^{2} + b ^{2} - 2 ab\cos \gamma. \end{align}

In France, the law of cosines is sometimes referred to as the théorème d'Al-Kashi.^{[6] [7]}

The theorem was first written using algebraic notation by François Viète in the 16th century. At the beginning of the 19th century, modern algebraic notation allowed the law of cosines to be written in its current symbolic form.^[8]

Proofs

Using the Pythagorean theorem

Case of an obtuse angle

Euclid proved this theorem by applying the Pythagorean theorem to each of the two right triangles in Fig. 2 (and). Using to denote the line segment and for the height, triangle gives us

c ^{2} = (b + d) ^{2} + h ^{2,}

and triangle gives

d ^{2} + h ^{2} = a ^{2.}

Expanding the first equation gives

c ^{2} = b ^{2} + 2 bd + d ^{2} + h ^{2.}

Substituting the second equation into this, the following can be obtained:

c ^{2} = a ^{2} + b ^{2} + 2 bd.

This is Euclid's Proposition 12 from Book 2 of the Elements.^[9] To transform it into the modern form of the law of cosines, note that

d = a\cos(\pi - \gamma) = - a\cos\gamma.

Case of an acute angle

Euclid's proof of his Proposition 13 proceeds along the same lines as his proof of Proposition 12: he applies the Pythagorean theorem to both right triangles formed by dropping the perpendicular onto one of the sides enclosing the angle and uses the square of a difference to simplify.

Another proof in the acute case

Using more trigonometry, the law of cosines can be deduced by using the Pythagorean theorem only once. In fact, by using the right triangle on the left hand side of Fig. 6 it can be shown that:

\begin{align} c ^{2} & = (b - a\cos\gamma) ^{2} + (a\sin\gamma) ^{2} \\ & = b ^{2} - 2 ab\cos\gamma + a ^{2\cos} ^{2\gamma} + a ^{2\sin} ^{2\gamma} \\ & = b ^{2} + a ^{2} - 2 ab\cos\gamma, \end{align} \cos ^{2\gamma} + \sin ^{2\gamma} = 1 .

This proof needs a slight modification if . In this case, the right triangle to which the Pythagorean theorem is applied moves outside the triangle . The only effect this has on the calculation is that the quantity is replaced by As this quantity enters the calculation only through its square, the rest of the proof is unaffected. However, this problem only occurs when is obtuse, and may be avoided by reflecting the triangle about the bisector of .

Referring to Fig. 6 it is worth noting that if the angle opposite side is then:

\tan\alpha = (a\sin\gamma)/(b - a\cos\gamma) .

This is useful for direct calculation of a second angle when two sides and an included angle are given.

From three altitudes

The altitude through vertex is a segment perpendicular to side . The distance from the foot of the altitude to vertex plus the distance from the foot of the altitude to vertex is equal to the length of side (see Fig. 5). Each of these distances can be written as one of the other sides multiplied by the cosine of the adjacent angle,^[10]

c = a\cos\beta + b\cos\alpha.

(This is still true if or is obtuse, in which case the perpendicular falls outside the triangle.) Multiplying both sides by yields

c ^{2} = ac\cos\beta + bc\cos\alpha.

The same steps work just as well when treating either of the other sides as the base of the triangle:

\begin{align} a ^{2} & = ac\cos\beta + ab\cos\gamma, \\[ 3 mu] b ^{2} & = bc\cos\alpha + ab\cos\gamma. \end{align}

Taking the equation for

c ^{2} and subtracting the equations for b ^{2} and a ^{2,} \begin{align} c ^{2} - a ^{2} - b ^{2} & = {\color{BlueGreen}\cancel{\color{Black}ac\cos\beta}} + {\color{Peach}\cancel{\color{Black}bc\cos\alpha}} - {\color{BlueGreen}\cancel{\color{Black}ac\cos\beta}} - {\color{Peach}\cancel{\color{Black}bc\cos\alpha}} - 2 ab\cos\gamma \\ c ^{2} & = a ^{2} + b ^{2-} 2 ab\cos\gamma. \end{align}

This proof is independent of the Pythagorean theorem, insofar as it is based only on the right-triangle definition of cosine and obtains squared side lengths algebraically. Other proofs typically invoke the Pythagorean theorem explicitly, and are more geometric, treating as a label for the length of a certain line segment.^[10]

Unlike many proofs, this one handles the cases of obtuse and acute angles in a unified fashion.

Cartesian coordinates

Consider a triangle with sides of length,,, where is the measurement of the angle opposite the side of length . This triangle can be placed on the Cartesian coordinate system with side aligned along the x axis and angle placed at the origin, by plotting the components of the 3 points of the triangle as shown in Fig. 4:

A = (b \cos\theta, b \sin\theta), B = (a, 0 ), and C = ( 0 , 0 ).

By the distance formula,^[11]

c = \sqrt{(a - b \cos\theta) ^{2} + ( 0 - b \sin\theta) ^{2}.}

Squaring both sides and simplifying

\begin{align} c ^{2} & = (a - b \cos\theta) ^{2} + ( - b \sin\theta) ^{2} \\ & = a ^{2} - 2 a b \cos\theta + b ^{2} \cos ^{2} \theta + b ^{2} \sin ^{2} \theta\\ & = a ^{2} + b ^{2} (\sin ^{2} \theta + \cos ^{2} \theta) - 2 a b \cos\theta\\ & = a ^{2} + b ^{2} - 2 a b \cos\theta. \end{align}

An advantage of this proof is that it does not require the consideration of separate cases depending on whether the angle is acute, right, or obtuse. However, the cases treated separately in Elements II.12–13 and later by al-Ṭūsī, al-Kāshī, and others could themselves be combined by using concepts of signed lengths and areas and a concept of signed cosine, without needing a full Cartesian coordinate system.

Using Ptolemy's theorem

Referring to the diagram, triangle ABC with sides =, = and = is drawn inside its circumcircle as shown. Triangle is constructed congruent to triangle with = and = . Perpendiculars from and meet base at and respectively. Then:

\begin{align} & BF = AE = BC\cos\hat{B} = a\cos\hat{B} \\ ⇒ & DC = EF = AB -2 BF = c -2 a\cos\hat{B}. \end{align}

Now the law of cosines is rendered by a straightforward application of Ptolemy's theorem to cyclic quadrilateral :

\begin{align} & AD x BC + AB x DC = AC x BD \\ ⇒ & a ^{2} + c(c -2 a\cos\hat{B}) = b ^{2} \\ ⇒ & a ^{2+c} ^{2-2ac} \cos\hat{B} = b ^{2. \end{align}}

Plainly if angle is right, then is a rectangle and application of Ptolemy's theorem yields the Pythagorean theorem:

a ^{2+c} ^{2=b} ^{2.}

By comparing areas

One can also prove the law of cosines by calculating areas. The change of sign as the angle becomes obtuse makes a case distinction necessary.

Recall that

• ,, and are the areas of the squares with sides,, and, respectively;
• if is acute, then is the area of the parallelogram with sides and forming an angle of ;
• if is obtuse, and so is negative, then is the area of the parallelogram with sides a and b forming an angle of .

Acute case. Figure 7a shows a heptagon cut into smaller pieces (in two different ways) to yield a proof of the law of cosines. The various pieces are

• in pink, the areas, on the left and the areas and on the right;
• in blue, the triangle, on the left and on the right;
• in grey, auxiliary triangles, all congruent to, an equal number (namely 2) both on the left and on the right.

The equality of areas on the left and on the right gives

a ^{2} + b ^{2} = c ^{2} + 2 ab\cos\gamma.

Obtuse case. Figure 7b cuts a hexagon in two different ways into smaller pieces, yielding a proof of the law of cosines in the case that the angle is obtuse. We have

• in pink, the areas,, and on the left and on the right;
• in blue, the triangle twice, on the left, as well as on the right.

The equality of areas on the left and on the right gives

a ^{2} + b ^{2} - 2 ab\cos(\gamma) = c ^{2.}

The rigorous proof will have to include proofs that various shapes are congruent and therefore have equal area. This will use the theory of congruent triangles.

Using circle geometry

Using the geometry of the circle, it is possible to give a more geometric proof than using the Pythagorean theorem alone. Algebraic manipulations (in particular the binomial theorem) are avoided.

Case of acute angle, where . Drop the perpendicular from onto =, creating a line segment of length . Duplicate the right triangle to form the isosceles triangle . Construct the circle with center and radius, and its tangent through . The tangent forms a right angle with the radius (Euclid's Elements: Book 3, Proposition 18; or see here), so the yellow triangle in Figure 8 is right. Apply the Pythagorean theorem to obtain

c ^{2} = b ^{2} + h ^{2.}

Then use the tangent secant theorem (Euclid's Elements: Book 3, Proposition 36), which says that the square on the tangent through a point outside the circle is equal to the product of the two lines segments (from) created by any secant of the circle through . In the present case:, or

h ^{2} = a(a - 2 b\cos\gamma).

Substituting into the previous equation gives the law of cosines:

c ^{2} = b ^{2} + a(a - 2 b\cos\gamma).

Note that is the power of the point with respect to the circle. The use of the Pythagorean theorem and the tangent secant theorem can be replaced by a single application of the power of a point theorem.

Case of acute angle, where . Drop the perpendicular from onto =, creating a line segment of length . Duplicate the right triangle to form the isosceles triangle . Construct the circle with center and radius, and a chord through perpendicular to half of which is Apply the Pythagorean theorem to obtain

b ^{2} = c ^{2} + h ^{2.}

Now use the chord theorem (Euclid's Elements: Book 3, Proposition 35), which says that if two chords intersect, the product of the two line segments obtained on one chord is equal to the product of the two line segments obtained on the other chord. In the present case: or

h ^{2} = a( 2 b\cos\gamma - a).

Substituting into the previous equation gives the law of cosines:

b ^{2} = c ^{2} + a( 2 b\cos\gamma - a).

Note that the power of the point with respect to the circle has the negative value .

Case of obtuse angle . This proof uses the power of a point theorem directly, without the auxiliary triangles obtained by constructing a tangent or a chord. Construct a circle with center and radius (see Figure 9), which intersects the secant through and in and . The power of the point with respect to the circle is equal to both and . Therefore,

\begin{align} c ^{2} - a ^{2} & {} = b(b + 2 a\cos(\pi - \gamma)) \\ & {} = b(b - 2 a\cos\gamma), \end{align}

which is the law of cosines.

Using algebraic measures for line segments (allowing negative numbers as lengths of segments) the case of obtuse angle and acute angle can be treated simultaneously.

Using the law of sines

The law of cosines can be proven algebraically from the law of sines and a few standard trigonometric identities.^[12] To start, three angles of a triangle sum to a straight angle (

\alpha + \beta + \gamma = \pi radians). Thus by the angle sum identities for sine and cosine,\begin{alignat}{ 3 } \sin \gamma & = \phantom{ - }\sin(\pi - \gamma) && = \phantom{ - }\sin(\alpha + \beta) && = \sin\alpha\cos\beta + \cos\alpha\sin\beta, \\[ 5 mu] \cos \gamma & = - \cos(\pi - \gamma) && = - \cos(\alpha + \beta) && = \sin\alpha\sin\beta - \cos\alpha\cos\beta. \end{alignat}

Squaring the first of these identities, then substituting

\cos\alpha\cos\beta = {} \sin\alpha\sin\beta - \cos \gamma from the second, and finally replacing \cos ^{2} \alpha + \sin ^{2} \alpha = {} \cos ^{2} \beta + \sin ^{2} \beta = 1 , the Pythagorean trigonometric identity, we have:\begin{align} \sin ^{2} \gamma & = (\sin\alpha\cos\beta + \cos\alpha\sin\beta) ^{2} \\[ 3 mu] & = \sin ^{2\alpha\cos} ^{2\beta} + 2 \sin\alpha\sin\beta\cos\alpha\cos\beta + \cos ^{2\alpha\sin} ^{2\beta} \\[ 3 mu] & = \sin ^{2\alpha\cos} ^{2\beta} + 2 \sin\alpha\sin\beta(\sin\alpha\sin\beta - \cos \gamma) + \cos ^{2\alpha\sin} ^{2\beta} \\[ 3 mu] & = \sin ^{2\alpha(\cos} ^{2\beta} + \sin ^{2\beta)} + \sin ^{2\beta(\cos} ^{2\alpha} + \sin ^{2\alpha)} - 2 \sin\alpha\sin\beta\cos \gamma \\[ 3 mu] & = \sin ^{2\alpha} + \sin ^{2\beta} - 2 \sin\alpha\sin\beta\cos \gamma. \end{align}

The law of sines holds that

(a)/(\sin \alpha\vphantom{\beta) } = \frac = \frac = k,

so to prove the law of cosines, we multiply both sides of our previous identity by

k ^{2\colon} \begin{align} \sin ^{2} \gamma (c ^{2})/(\sin ^{2} \gamma) & = \sin ^{2} \alpha (a ^{2})/(\sin ^{2} \alpha) + \sin ^{2} \beta (b ^{2})/(\sin ^{2} \beta) -2 \sin\alpha\sin\beta\cos\gamma (ab)/(\sin\alpha\sin\beta\vphantom{\sin ^{2}) } \\[10mu]c^2 &= a^2 + b^2 - 2ab\cos\gamma.\end

This concludes the proof.

Using vectors

Denote

\overrightarrow{CB} = \vec{a}, \overrightarrow{CA} = \vec{b}, \overrightarrow{AB} = \vec{c}

Therefore,

\vec{c} = \vec{a} - \vec{b}

Taking the dot product of each side with itself:

\vec{c} ⋅ \vec{c} = (\vec{a} - \vec{b}) ⋅ (\vec{a} - \vec{b}) \Vert\vec{c}\Vert ^{2} = \Vert\vec{a}\Vert ^{2} + \Vert\vec{b}\Vert ^{2} - 2 \vec{a} ⋅ \vec{b}

Using the identity

\vec{u} ⋅ \vec{v} = \Vert\vec{u}\Vert\Vert\vec{v}\Vert \cos\angle(\vec{u}, \vec{v})

leads to

\Vert\vec{c}\Vert ^{2} = \Vert\vec{a}\Vert ^{2} + {\Vert\vec{b}\Vert} ^{2} - 2 \Vert\vec{a}\Vert \Vert\vec{b}\Vert \cos\angle(\vec{a}, \vec{b})

The result follows.

Isosceles case

When, i.e., when the triangle is isosceles with the two sides incident to the angle equal, the law of cosines simplifies significantly. Namely, because, the law of cosines becomes

\cos\gamma = 1 - (c ^{2})/(2 a ^{2})

or

c ^{2} = 2 a ^{2} ( 1 - \cos\gamma).

Analogue for tetrahedra

\varphi _{ab} between faces and, etc., a higher-dimensional analogue of the law of cosines is:^[13] A ^{2} = B ^{2} + C ^{2} + D ^{2} - 2 \left(BC \cos \varphi _{bc} + CD \cos\varphi _{cd} + DB\cos \varphi _{db} \right).

Version suited to small angles

When the angle,, is small and the adjacent sides, and, are of similar length, the right hand side of the standard form of the law of cosines is subject to catastrophic cancellation in numerical approximations. In situations where this is an important concern, a mathematically equivalent version of the law of cosines, similar to the haversine formula, can prove useful:

\begin{align} c ^{2} & = (a - b) ^{2} + 4 ab\sin ^{2 \left( (\gamma)/(2)} \right) \\ & = (a - b) ^{2} + 4 ab\operatorname{haversin}(\gamma). \end{align} In the limit of an infinitesimal angle, the law of cosines degenerates into the circular arc length formula, .

In spherical and hyperbolic geometry

See main article: article, Spherical law of cosines and Hyperbolic law of cosines. Versions similar to the law of cosines for the Euclidean plane also hold on a unit sphere and in a hyperbolic plane. In spherical geometry, a triangle is defined by three points,, and on the unit sphere, and the arcs of great circles connecting those points. If these great circles make angles,, and with opposite sides,, then the spherical law of cosines asserts that both of the following relationships hold:

\begin{align} \cos a & = \cos b\cos c + \sin b\sin c\cos A\\ \cos A & = - \cos B\cos C + \sin B\sin C\cos a. \end{align}

In hyperbolic geometry, a pair of equations are collectively known as the hyperbolic law of cosines. The first is

\cosh a = \cosh b\cosh c - \sinh b \sinh c \cos A

where and are the hyperbolic sine and cosine, and the second is

\cos A = - \cos B \cos C + \sin B\sin C\cosh a.

As in Euclidean geometry, one can use the law of cosines to determine the angles,, from the knowledge of the sides,, . In contrast to Euclidean geometry, the reverse is also possible in both non-Euclidean models: the angles,, determine the sides,, .

Polyhedra

The Law of Cosines can be generalized to all polyhedra by considering any polyhedron with vector sides and invoking the Divergence Theorem.^[14]

See also

• Half-side formula
• Law of sines
• Law of tangents
• Law of cotangents
• List of trigonometric identities
• Mollweide's formula

External links

• • Several derivations of the Cosine Law, including Euclid's at cut-the-knot
• Interactive applet of Law of Cosines

Notes and References

1. Book: Heath, Thomas . Introduction . The Thirteen Books of Euclid's Elements . Thomas Heath (classicist) . 2nd . https://archive.org/details/thirteenbooksofe0001eucl_e2h2/page/23 . limited . 1908 . 1956.
2. Kennedy . E.S. . Muruwwa . Ahmad . 1958 . Bīrūnī on the Solar Equation . Journal of Near Eastern Studies . 17 . 2 . 112-121 . 542617. Johannes de Muris credits an anonymous author for the relevant section of his work De Arte Mesurandi. See Book: Van Brummelen, Glen . 2009 . The Mathematics of the Heavens and the Earth . Princeton University Press . 240–241.
3. Book: Van Brummelen, Glen . 2012 . Heavenly mathematics: The forgotten art of spherical trigonometry . Princeton University Press . 98.
4. Book: Traité du quadrilatère attribué a Nassiruddinel-Toussy . Ch. 3.2: Sur la manière de calculer les côtés et les angles d'un triangle les uns par les autres . 69 . fr . Naṣīr al-Dīn al-Ṭūsī . Caratheodory . Alexandre Pacha . Alexander Karatheodori Pasha . https://books.google.com/books?id=gVE7AQAAIAAJ&pg=PA69 . Typographie et Lithographie Osmanié . 1891 . On donne deux côtés et un angle. [...] Que si l'angle donné est compris entre les deux côtés donnés, comme l'angle A est compris entre les deux côtés AB AC, abaissez de B sur AC la perpendiculaire BE. Vous aurez ainsi le triangle rectangle [BEA] dont nous connaissons le côté AB et l'angle A; on en tirera BE, EA, et l'on retombera ainsi dans un des cas précédents; c. à. d. dans le cas où BE, CE sont connus; on connaîtra dès lors BC et l'angle C, comme nous l'avons expliqué . Given [...] the angle A is included between the two sides AB AC, drop from B to AC the perpendicular BE. You will thus have the right triangle [BEA] of which we know the side AB and the angle A; in that triangle compute BE, EA, and the problem is reduced to one of the preceding cases; that is, to the case where BE, CE are known; we will thus know BC and the angle C, as we have explained..
5. Azarian . Mohammad K. . 2000 . Meftab Al-Hesab: A Summary . Missouri Journal of Mathematical Sciences . 12 . 2 . 75-95 . 10.35834/2000/1202075 . free.
6. Book: Pickover, Clifford A. . The Math Book: From Pythagoras to the 57th Dimension . 2009 . Sterling Publishing Company, Inc. . 9781402757969 . 106.
7. Book: Programme de mathématiques de première générale . Ministère de l'Éducation nationale et de la Jeunesse . 2022 . 11,12 . fr.
8. For example in Book: Carnot, Lazare . Géométrie de position . 1803 . J.B.M Duprat . 202 .
9. http://aleph0.clarku.edu/~djoyce/java/elements/bookII/propII12.html Java applet version
10. [Alexander Bogomolny]
11. Book: Wylie, Clarence Raymond . 1955 . Plane Trigonometry . McGraw-Hill . §9.1 The Law of Cosines, . 54-11278 . limited .
12. Burton . L. J. . 1949 . The Laws of Sines and Cosines . The American Mathematical Monthly . 56 . 8 . 550–551 . 2305533.
13. Book: Casey , John . A Treatise on Spherical Trigonometry: And Its Application to Geodesy and Astronomy with Numerous Examples. London. Longmans, Green, & Company. 1889. 133.
14. Collins . L . Osler . T . 2011 . Law of Cosines Generalised for any Polygon and any Polyhedron . The Mathematical Gazette . 95 . 533 . 240-243 . 23248682.