Fig. 1 - A triangle.

This article is about the law of cosines in Euclidean geometry. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. The history of Trigonometry and of Trigonometric functions may span about 4000 years Trigonometry has an enormous variety of applications The ones mentioned explicitly in textbooks and courses on trigonometry are its uses in practical endeavors such as For a more comprehensive list see the List of trigonometry topics. In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables Exact constant expressions for Trigonometric expressions are sometimes useful mainly for simplifying solutions into radical forms which allow further simplification In Mathematics, tables of Trigonometric functions are useful in a number of areas CORDIC (digit-by-digit method Volder's algorithm (for CO ordinate R otation DI gital C omputer is a simple and efficient Algorithm Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. The law of sines ( sines law sine formula sine rule) in Trigonometry, is a statement about any Triangle in a plane In Trigonometry, the law of tangents is a statement about the relationship between the lengths of the three sides of a triangle and the tangents of the angles In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Mathematics, the trigonometric integrals are a family of Integrals which involve Trigonometric functions A number of the basic trigonometric In Mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions The following is a list of Integrals ( Antiderivative functions of Trigonometric functions. The following is a list of Integrals ( Antiderivative formulas for integrands that contain inverse Trigonometric functions (also known as "arc functions" For the corresponding theorem in spherical geometry, see law of cosines (spherical). Spherical geometry is the Geometry of the two- Dimensional surface of a Sphere. In Spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles analogous For the cosine law of optics, see Lambert's cosine law. See also Lambertian reflectance Lambert's cosine law in Optics says that the Radiant intensity observed from a " Lambertian "

In trigonometry, the law of cosines (also known as Al-Kashi law or the cosine formula or cosine rule) is a statement about a general triangle which relates the lengths of its sides to the cosine of one of its angles. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. (or, Persian: غیاث‌الدین جمشید کاشانی (c A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Using notation as in Fig. 1, the law of cosines states that

or, equivalently:

Note that c is the side opposite of angle γ, and that a and b are the two sides enclosing γ. All three of the identities above say the same thing; they are listed separately only because in solving triangles with three given sides one may apply the identity three times with the roles of the three sides permuted.

The law of cosines generalizes the Pythagorean theorem, which holds only in right triangles: if the angle γ is a right angle (of measure 90° or radians), then , and thus the law of cosines reduces to

which is the Pythagorean theorem. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Two types of special right triangles appear commonly in geometry the "angle based" and the "side based" (or Pythagorean Triangles The former are characterised This article describes the unit of angle For other meanings see Degree.

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.

History

Fig. 2 - Obtuse triangle ABC with perpendicular BH

Euclid's Elements, dating back to the 3rd century BC, contains a version of the law of cosines. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek The 3rd century BC started the first day of 300 BC and ended the last day of 201 BC The case of obtuse triangle and acute triangle (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. --- Euclid's Elements, translation by Thomas L. Heath. Sir Thomas Little Heath ( October 5, 1861 &ndash March 16, 1940) was a British civil servant Mathematician, classical [1]

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 = a cos(π – γ) = −a cos(γ).

Proposition 13 contains an entirely analogous statement for acute triangles.

It was not until the development of modern trigonometry in the Middle Ages by Muslim mathematicians that the law of cosines evolved beyond Euclid's two theorems. The astronomer and mathematician al-Battani generalized Euclid's result to spherical geometry at the beginning of the 10th century, which permitted him to calculate the angular distances between stars. Historically Astronomy was more concerned with the classification and description of phenomena in the sky while Astrophysics attempted to explain these phenomena A mathematician is a person whose primary area of study and research is the field of Mathematics. Spherical geometry is the Geometry of the two- Dimensional surface of a Sphere. During the 15th century, al-Kashi in Samarcand computed trigonometric tables to great accuracy and put the theorem into a form suitable for triangulation. (or, Persian: غیاث‌الدین جمشید کاشانی (c Samarkand (Samarqand Самарқанд سمرقند UniPers: "Samarqand" is the second-largest city in Uzbekistan and the capital of In Trigonometry and Geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either In France, the law of cosines is still referred to as the theorem of Al-Kashi. This article is about the country For a topic outline on this subject see List of basic France topics.

The theorem was popularised in the Western world by François Viète, who apparently discovered it independently. The term Western world, the West or the Occident ( Latin: occidens -sunset -west as distinct from the Orient) can have multiple meanings François Viète (or Vieta) seigneur de la Bigotière ( 1540 - February 13, 1603) generally known as Franciscus Vieta, At the beginning of the 19th century modern algebraic notation allowed the law of cosines to be written in its current form. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar

Applications

Fig. 3 - Applications of the law of cosines: unknown side and unknown angle.

The theorem is used in triangulation, for solving a triangle, i. In Trigonometry and Geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either 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 (you may also use the Pythagorean Theorem to do this):

These formulas produce high round-off errors in floating point calculations if the triangle is very acute, i. For the acrobatic movement roundoff see Roundoff. A round-off error, also called rounding error, is the difference between the In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. e. , if c is small relative to a and b or γ is small compared to 1.

The third formula shown is the result of solving for a the quadratic equation a² − 2ab cos γ + b² − c² = 0. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. 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) < c < b, only one positive solution if c > b or c = b sin(C), and no solution if c < b sin(C). These different cases are also explained by the Side-Side-Angle congruence ambiguity. In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i

Proofs

Using the distance formula

Consider a triangle with sides of length a, b, c, where θ is the measurement of the angle opposite the side of length c. We can place this triangle on the coordinate system by plotting By the distance formula, we have . Now, we just work with this equation:

An advantage of this proof is that it does not require the consideration of different cases for when the triangle is acute vs. obtuse.

Using trigonometry

Fig. 4 - An acute triangle with perpendicular

Drop the perpendicular onto the side c to get (see Fig. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent 4)

(This is still true if α or β is obtuse, in which case the perpendicular falls outside the triangle. ) Multiply through by c to get

By considering the other perpendiculars obtain

Adding the latter two equations gives

Rearrange to get

Substitute this in the first equation for c² to yield the law of cosines

This proof uses trigonometry in that it treats the cosines of the various angles as quantities in their own right. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. 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 acos(γ) merely as a label for the length of a certain line segment.

Many proofs deal with the case of obtuse and acute angle γ separately.

Using the Pythagorean theorem

Fig. 5 - Obtuse triangle ABC with height BH

Case of an obtuse angle. Euclid proves this theorem by applying the Pythagorean theorem to each of the two right triangles in Fig. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry 5. Using d to denote the line segment CH and h for the height BH, triangle AHB gives us

and triangle CHB gives us

Expanding the first equation 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. In Mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek 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 γ 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 a little more trigonometry, the law of cosines by applying 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:

upon using the trigonometric identity

Remark. This proof needs a slight modification if b < a cos(γ). 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(γ) is replaced by a cos(γ) − b. As this quantity enters the calculation only through its square, the rest of the proof is unaffected. Note. This problem only occurs when β is obtuse, and may be avoided by reflecting the triangle about the bisector of γ.

Observation. Referring to Fig 6 it's worth noting that if the angle opposite side a is α 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. In Mathematics, Ptolemy's theorem is a relation in Euclidean geometry between the four sides and two diagonals or chords of a Cyclic quadrilateral. 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:

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

Plainly if angle B is 90 degrees, then ABCD is a rectangle and application of Ptolemy's theorem yields Pythagoras' theorem:

By comparing areas

One can also prove the law of cosines by calculating areas. In Geometry, a cyclic quadrilateral is a Quadrilateral whose vertices all lie on a single Circle. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The change of sign as the angle γ becomes obtuse, makes a case distinction necessary.

Recall that

• a², b², and c² are the areas of the squares with sides a, b, and c, respectively;
• if γ is acute, then ab cos(γ) is the area of the parallelogram with sides a and b forming an angle of ;
• if γ is obtuse, and so cos(γ) is negative, then −ab cos(γ) is the area of the parallelogram with sides a' and b forming an angle of . In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides

Fig. 7a - Proof of the law of cosines for acute angle γ by "cutting and pasting".

Acute case. Figure 7a shows a heptagon cut into smaller pieces (in two different ways) to yield a proof of the law of cosines. Construction A regular heptagon is not constructible with Compass and straightedge but is constructible with a marked Ruler and compass The various pieces are

• in pink, the areas a², b² on the left and the areas 2ab cos(γ) and c² on the right;
• in blue, the triangle ABC, on the left and on the right;
• in grey, auxiliary triangles, all congruent to ABC, an equal number (namely 2) both on the left and on the right. In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i

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

Fig. 7b - Proof of the law of cosines for obtuse angle γ by "cutting and pasting".

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 γ is obtuse. Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees We have

• in pink, the areas a², b², and −2ab cos(γ) on the left and c² on the right;
• in blue, the triangle ABC twice, on the left, as well as on the right.

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

The rigorous proof will have to include proofs that various shapes are congruent and therefore have equal area. In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i This will use the theory of congruent triangles. In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i

Using geometry of the circle

Using the geometry of the circle it is possible to give a more geometric proof than using the Pythagorean theorem alone. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Algebraic manipulations (in particular the binomial theorem) are avoided. Elementary algebra is a fundamental and relatively basic form of Algebra taught to students who are presumed to have little or no formal knowledge of Mathematics beyond In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says

Fig. 8a - The triangle ABC (pink), an auxiliary circle (light blue) and an auxiliary right triangle (yellow)

Case of acute angle γ, where a > 2 b cos(γ). Drop the perpendicular from A onto a = BC, creating a line segment of length b cos(γ). In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent Duplicate the right triangle to form the isosceles triangle ACP. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line Construct the circle with center A and radius b, and its tangent h = BH through B. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the 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. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the 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 Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the A secant line of a Curve is a line that (locally intersects two points on the curve In the present case: BH² = BC BP, or

Substuting into the previous equation gives the law of cosines:

Note that h² is the power of the point B with respect to the circle. In Geometry, the power of a point is a Real number h that reflects the relative distance of a given point from a given 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. In Geometry, the power of a point is a Real number h that reflects the relative distance of a given point from a given circle

Fig. 8b - The triangle ABC (pink), an auxiliary circle (light blue) and two auxiliary right triangles (yellow)

Case of acute angle γ, where a < 2 b cos γ. Drop the perpendicular from A onto a = BC, creating a line segment of length b cos(γ). In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent Duplicate the right triangle to form the isosceles triangle ACP. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line Construct the circle with center A and radius b, and a chord through B perpendicular to c = AB, half of which is h = BH. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the A chord of a Curve is a geometric Line segment whose endpoints both lie on the curve Apply the Pythagorean theorem to obtain

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 Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In the present case: BH² = BC BP, or

Substuting 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 γ. 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. A secant line of a Curve is a line that (locally intersects two points on the curve The power of the point A with respect to the circle is equal to both AB² − BC² and AC·AK. In Geometry, the power of a point is a Real number h that reflects the relative distance of a given point from a given circle 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. A negative number is a Number that is less than zero, such as −2

Vector formulation

The law of cosines is equivalent to the formula

in the theory of vectors, which expresses the dot product of two vectors in terms of their respective lengths and the angle they enclose. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called

Fig. 10 - Vector triangle

Proof of equivalence. Referring to Figure 10, note that

and so we may calculate:

The law of cosines formulated in this context states:

which is now visibly equivalent to the above formula from the theory of vectors.

Isosceles case

When a = b, i. e. , when the triangle is isosceles with the two sides incident to the angle γ equal, the law of cosines simplifies significantly. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line Namely, because a² + b² = 2a² = 2ab, the law of cosines becomes

Analog for tetrahedra

An analogous statement begins by taking to be the areas of the four faces of a tetrahedron. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. Denote the dihedral angles by etc. In Aerospace engineering, the Dihedral is the Angle between the two wings see Dihedral. Then^[1]

References

See also

• Triangulation
• Law of sines
• Law of tangents
• Law of cosines (spherical)

External links

• An Excellent Tutorial on the Law of Cosines From PlainMath. In Trigonometry and Geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either The law of sines ( sines law sine formula sine rule) in Trigonometry, is a statement about any Triangle in a plane In Trigonometry, the law of tangents is a statement about the relationship between the lengths of the three sides of a triangle and the tangents of the angles In Spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles analogous Net
• Several derivations of the Cosine Law, including Euclid's at cut-the-knot