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Law of Cosines
Fig. 1 - A triangle. The angles ? (or A), ? (or B), and ? (or C) are respectively opposite the sides a, b, and c.
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states
where ? denotes the angle contained between sides of lengths a and b and opposite the side of length c. For the same figure, the other two relations are analogous:
The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known.
Fig. 2 - Obtuse triangle ABC with perpendicular BH
Though the notion of the cosine was not yet developed in his time, Euclid's Elements, dating back to the 3rd century BC, contains an early geometric theorem almost equivalent to the law of cosines. The cases of obtuse triangles and acute triangles (corresponding to the two cases of negative or positive cosine) are treated separately, in Propositions 12 and 13 of Book 2. Trigonometric functions and algebra (in particular negative numbers) being absent in Euclid's time, the statement has a more geometric flavor:
Proposition 12 In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle.
Using notation as in Fig. 2, Euclid's statement can be represented by the formula
This formula may be transformed into the law of cosines by noting that CH = (CB) cos(? - ?) = -(CB) cos ?. Proposition 13 contains an entirely analogous statement for acute triangles.
Euclid's Elements paved the way for the discovery of law of cosines. In the 15th century, Jamsh?d al-K?sh?, a Persian mathematician and astronomer, provided the first explicit statement of the law of cosines in a form suitable for triangulation. He provided accurate trigonometric tables and expressed the theorem in a form suitable for modern usage. As of the 1990s, in France, the law of cosines is still referred to as the Théorème d'Al-Kashi.
The theorem was popularized in the Western world 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.
Fig. 3 - Applications of the law of cosines: unknown side and unknown angle.
The theorem is used in triangulation, for solving a triangle or circle, i.e., to find (see Figure 3):
the third side of a triangle if one knows two sides and the angle between them:
the angles of a triangle if one knows the three sides:
the third side of a triangle if one knows two sides and an angle opposite to one of them (one may also use the Pythagorean theorem to do this if it is a right triangle):
These formulas produce high round-off errors in floating point calculations if the triangle is very acute, i.e., if c is small relative to a and b 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 equationa2 - 2ab cos ? + b2 - c2 = 0. 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 b sin ? < c < b, only one positive solution if c = b sin ?, and no solution if c < b sin ?. These different cases are also explained by the side-side-angle congruence ambiguity.
Using the distance formula
Fig. 4 - Coordinate geometry proof
Consider a triangle with sides of length a, b, c, where ? is the measurement of the angle opposite the side of length c. This triangle can be placed on the Cartesian coordinate system aligned with edge a with origin at C, by plotting the components of the 3 points of the triangle as shown in Fig. 4:
An advantage of this proof is that it does not require the consideration of different cases for when the triangle is acute, right, or obtuse.
Fig. 5 - An acute triangle with perpendicular
Dropping the perpendicular onto the side c through point C, an altitude of the triangle, shows (see Fig. 5)
(This is still true if ? or ? is obtuse, in which case the perpendicular falls outside the triangle.) Multiplying through by c yields
Considering the two other altitudes of the triangle yields
Adding the latter two equations gives
Subtracting the first equation from the last one results in
which simplifies to
This proof uses trigonometry in that it treats the cosines of the various angles as quantities in their own right. It uses the fact that the cosine of an angle expresses the relation between the two sides enclosing that angle in any right triangle. Other proofs (below) are more geometric in that they treat an expression such as a cos ? merely as a label for the length of a certain line segment.
Many proofs deal with the cases of obtuse and acute angles ? separately.
Using the Pythagorean theorem
Obtuse triangle ABC with height BH
Cosine theorem in plane trigonometry, proof based on Pythagorean theorem.
Case of an obtuse angle
Euclid proved this theorem by applying the Pythagorean theorem to each of the two right triangles in the figure shown (AHB and CHB). Using d to denote the line segment CH and h for the height BH, triangle AHB gives us
Substituting the second equation into this, the following can be obtained:
This is Euclid's Proposition 12 from Book 2 of the Elements. To transform it into the modern form of the law of cosines, note that
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 binomial theorem to simplify.
Fig. 6 - A short proof using trigonometry for the case of an acute angle
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:
This proof needs a slight modification if b < a cos(?). In this case, the right triangle to which the Pythagorean theorem is applied moves outside the triangle ABC. The only effect this has on the calculation is that the quantity b - a cos(?) is replaced by a cos(?) - b. 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 a is ? then:
This is useful for direct calculation of a second angle when two sides and an included angle are given.
Using Ptolemy's theorem
Proof of law of cosines using Ptolemy's theorem
Referring to the diagram, triangle ABC with sides AB = c, BC = a and AC = b is drawn inside its circumcircle as shown. Triangle ABD is constructed congruent to triangle ABC with AD = BC and BD = AC. Perpendiculars from D and C meet base AB at E and F respectively. Then:
Fig. 8a - The triangle ABC (pink), an auxiliary circle (light blue) and an auxiliary right triangle (yellow)
Case of acute angle ?, where a > 2b cos ?. Drop the perpendicular from A onto a = BC, creating a line segment of length b cos ?. Duplicate the right triangle to form the isosceles triangleACP. Construct the circle with center A and radius b, and its tangenth = BH through B. The tangent h forms a right angle with the radius b (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
Then use the tangent secant theorem (Euclid's Elements: Book 3, Proposition 36), which says that the square on the tangent through a point B outside the circle is equal to the product of the two lines segments (from B) created by any secant of the circle through B. In the present case: BH2 = BC·BP, or
Substituting into the previous equation gives the law of cosines:
Note that h2 is the power of the point B 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.
Fig. 8b - The triangle ABC (pink), an auxiliary circle (light blue) and two auxiliary right triangles (yellow)
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: BH2 = BC·BP, or
Substituting into the previous equation gives the law of cosines:
Note that the power of the point B with respect to the circle has the negative value -h2.
Fig. 9 - Proof of the law of cosines using the power of a point theorem.
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 B and radius a (see Figure 9), which intersects the secant through A and C in C and K. The power of the point A with respect to the circle is equal to both AB2 - BC2 and AC·AK. Therefore,
which is the law of cosines.
Using algebraic measures for line segments (allowing negative numbers as lengths of segments) the case of obtuse angle (CK > 0) and acute angle (CK < 0) can be treated simultaneously.
Using the law of sines
By using the law of sines and knowing that the angles of a triangle must sum to 180 degrees, we have the following system of equations (the three unknowns are the angles):
Then, by using the third equation of the system, we obtain a system of two equations in two variables:
where we have used the trigonometric property that the sine of a supplementary angle is equal to the sine of the angle.
When a = b, i.e., when the triangle is isosceles with the two sides incident to the angle ? equal, the law of cosines simplifies significantly. Namely, because a2 + b2 = 2a2 = 2ab, the law of cosines becomes
Analogue for tetrahedra
An analogous statement begins by taking ?, ?, ?, ? to be the areas of the four faces of a tetrahedron. Denote the dihedral angles by etc. Then
Version suited to small angles
When the angle, ?, is small and the adjacent sides, a and b, are of similar length, the right hand side of the standard form of the law of cosines can lose a lot of accuracy to numerical loss of significance. 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:
In the limit of an infinitesimal angle, the law of cosines degenerates into the circular arc length formula, c = a?.
In spherical and hyperbolic geometry
Spherical triangle solved by the 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 u, v, and w on the unit sphere, and the arcs of great circles connecting those points. If these great circles make angles A, B, and C with opposite sides a, b, c then the spherical law of cosines asserts that both of the following relationships hold:
As in Euclidean geometry, one can use the law of cosines to determine the angles A, B, C from the knowledge of the sides a, b, c. In contrast to Euclidean geometry, the reverse is also possible in both non-Euclidean models: the angles A, B, C determine the sides a, b, c.
Unified formula for surfaces of constant curvature