FACTOID # 6: Michigan is ranked 22nd in land area, but since 41.27% of the state is composed of water, it jumps to 11th place in total area.

 Home Encyclopedia Statistics States A-Z Flags Maps FAQ About

 WHAT'S NEW

SEARCH ALL

FACTS & STATISTICS    Advanced view

Search encyclopedia, statistics and forums:

(* = Graphable)

Encyclopedia > Law of cosines
Fig. 1 - A triangle.
This article is about the law of cosines in Euclidean geometry. For the corresponding theorem in spherical geometry, see law of cosines (spherical). For the cosine law of optics, see Lambert's cosine law.

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. Using notation as in Fig. 1, the law of cosines states that Image File history File links Triangle_with_notations_2. ... Image File history File links Triangle_with_notations_2. ... 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) refers to a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. ... Lamberts cosine law says that the total radiant power observed from a Lambertian surface is directly proportional to the cosine of the angle Î¸ between the observers line of sight and the surface normal. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigÅnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... A triangle. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... âˆ , the angle symbol. ...

$c^2 = a^2 + b^2 - 2abcos(gamma) , ,$

or, equivalently:

$b^2 = c^2 + a^2 - 2cacos(beta) , ,$
$a^2 = b^2 + c^2 - 2bccos(alpha) . ,$

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 $frac{pi}{2}$ radians), then cos(γ) = 0, and thus the law of cosines reduces to In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... This article describes the unit of angle. ...

$c^2 = a^2 + b^2 ,$

which is the Pythagorean theorem.

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

Euclid's Elements, dating back to the 3rd century BC, contains a version of the law of cosines. 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: Image File history File links Obtuse-triangle-with-altitude. ... Image File history File links Obtuse-triangle-with-altitude. ... For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... The 3rd century BC started the first day of 300 BC and ended the last day of 201 BC. It is considered part of the Classical era, epoch, or historical period. ...

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.[1]

Using notation as in Fig. 2, Euclid's statement can be represented by the formula Thomas Little Heath (October 5, 1861 - March 16, 1940) was a mathematician, classical scholar, historian of ancient Greek mathematics, and translator. ...

$AB^2 = CA^2 + CB^2 + 2 (CA)(CH),.$

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. During the 15th century, al-Kashi in Samarcand computed trigonometric tables to great accuracy and put the theorem into a form suitable for triangulation. In France, the law of cosines is still referred to as the theorem of Al-Kashi. The Middle Ages formed the middle period in a traditional schematic division of European history into three ages: the classical civilization of Antiquity, the Middle Ages, and modern times, beginning with the Renaissance. ... Islamic mathematics is the profession of Muslim Mathematicians. ... An astronomer or astrophysicist is a person whose area of interest is astronomy or astrophysics. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Al Battani (c. ... Spherical geometry is the geometry of the two-dimensional surface of a sphere. ... As a means of recording the passage of time, the 10th century was that century which lasted from 901 to 1000. ... (14th century - 15th century - 16th century - other centuries) As a means of recording the passage of time, the 15th century was that century which lasted from 1401 to 1500. ... Kashani, dubbed, the Second Ptolemy, was an outstanding Persian mathematician of the middle ages. ... Colour photograph of Ulugh Beg Madrasa taken in Samarkand ca. ... Triangulation can be used to find the distance from the shore to the ship. ...

The theorem was popularised in the Western world by François Viète, who apparently discovered it independently. At the beginning of the 19th century modern algebraic notation allowed the law of cosines to be written in its current form. The term Western world, the West or the Occident (Latin occidens -sunset, -west, as distinct from the Orient) [1] can have multiple meanings dependent on its context (e. ... FranÃ§ois ViÃ¨te. ... Alternative meaning: Nineteenth Century (periodical) (18th century &#8212; 19th century &#8212; 20th century &#8212; more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of 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.e., to find (see Figure 3) Image File history File links Triangle-with-an-unknown-angle-or-side. ... Image File history File links Triangle-with-an-unknown-angle-or-side. ... Triangulation can be used to find the distance from the shore to the ship. ...

• the third side of a triangle if one knows an angle and its adjacent sides:
$,c = sqrt{a^2+b^2-2abcos(gamma)},;$
• the angles of a triangle if one knows the three sides:
$,gamma = cos^{-1} frac{-c^2+a^2+b^2}{2ab}.,$

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 γ is small compared to 1.
A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ...

Proofs

Using trigonometry

Fig. 4 - An acute triangle with perpendicular

Drop the perpendicular onto the side c to get (see Fig. 4) Image File history File links Download high resolution version (1173x1029, 26 KB) Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... Image File history File links Download high resolution version (1173x1029, 26 KB) Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... Fig. ...

$c=acos(beta)+bcos(alpha),.$

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

$c^2 = accos(beta) + bccos(alpha),.$

By considering the other perpendiculars obtain

$a^2 = accos(beta) + abcos(gamma),,$
$b^2 = bccos(alpha) + abcos(gamma),.$

Adding the latter two equations gives the law of cosines

$a^2 + b^2 = accos(beta) + abcos(gamma), + bccos(alpha) + abcos(gamma),$
$a^2 + b^2 = [accos(beta) + bccos(alpha)] + [abcos(gamma), + abcos(gamma),]$
$a^2 + b^2 = c^2 + 2abcos(gamma),.$

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 acos(γ) mereley as a label for the length of a certain line segment.
Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigÅnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with...

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. 5. Using d to denote the line segment CH and h for the height BH, triangle AHB gives us Image File history File links Obtuse-triangle-with-altitude. ... Image File history File links Obtuse-triangle-with-altitude. ... For other uses, see Euclid (disambiguation). ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

$c^2 = (b+d)^2 + h^2,,$

and triangle CHB gives us

$d^2 + h^2 = a^2.,$

Expanding the first equation gives us 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. ...

$c^2 = b^2 + 2bd + d^2 +h^2.,$

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

$c^2 = a^2 + b^2 + 2bd.,$

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 The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems...

$d = acos(pi-gamma)= -acos(gamma).,$

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 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: Image File history File links Triangle_with_trigonometric_proof_of_the_law_of_cosines. ... Image File history File links Triangle_with_trigonometric_proof_of_the_law_of_cosines. ...

begin{align} c^2 & {} = (b-acos(gamma))^2 + (asin(gamma))^2 & {} = b^2 - 2abcos(gamma) + a^2(cos^2(gamma))+a^2(sin^2(gamma)) & {} = b^2 + a^2 - 2abcos(gamma), end{align}

upon using the trigonometric identity

$cos^2(gamma) + sin^2(gamma) = 1. ,$

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.

By comparing areas

One can also prove the law of cosines by calculating areas. The change of sign as the angle γ becomes obtuse, makes a case distinction necessary. Area is a physical quantity expressing the size of a part of a surface. ...

Recall that

• a2, b2, and c2 are the areas of the squares with sides a, b, and c, respectively;
• if γ is acute, than ab cos(γ) is the area of the parallelogram with sides a and b forming an angle of $scriptstylegamma', =, pi/2 - gamma$;
• 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 $scriptstylegamma' ,=, gamma - pi/2$.
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. The various pieces are A parallelogram. ... A parallelogram. ... Image File history File links Law_of_cosines_with_acute_angles. ... In geometry, a heptagon is a polygon with seven sides and seven angles. ...

• 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.

The equality of areas on the left and on the right gives An example of congruence. ...

$,a^2 + b^2 = c^2 + 2abcos(gamma),.$

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. We have Image File history File links Law_of_cosines_with_an_obtuse_angle. ... For other uses, see Hexagon (disambiguation). ...

• 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

$,a^2 + b^2 - 2abcos(gamma) = c^2.$

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.
An example of congruence. ... An example of congruence. ...

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. Algebraic manipulations (in particular the binomial theorem) are avoided. Circle illustration This article is about the shape and mathematical concept of circle. ... Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... 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 arithmetic. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

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

$c^2 = b^2 + h^2,.$

Now 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 Circle illustration This article is about the shape and mathematical concept of circle. ... A secant line of a curve is a line that intersects two or more points on the curve. ...

$h^2 = a(a - 2bcos(gamma)),.$

Substuting into the previous equation gives the law of cosines:

$c^2 = b^2 + a(a - 2bcos(gamma)) ,.$

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.
The power of a point A (circle power,power of a circle) with respect to a circle with center 0 and radius r is defined as Therefore points inside the circle have negative power, points outside have positive power, and points on the circle have power zero. ... The power of a point A (circle power,power of a circle) with respect to a circle with center 0 and radius r is defined as Therefore points inside the circle have negative power, points outside have positive power, and points on the circle have power zero. ...

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

$b^2 = c^2 + h^2,.$

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: BH2 = BC BP, or Circle illustration This article is about the shape and mathematical concept of circle. ...

$h^2 = a(2bcos(gamma) - a),.$

Substuting into the previous equation gives the law of cosines:

$b^2 = c^2 + a(2bcos(gamma) - a) ,.$

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. The power of the point A with respect to the circle is equal to both AB2 − BC2 and AC·AK. Therefore, Image File history File links Triangle_with_circle_of_center_B_and_radius_BC.png This image (or all images in this article or category) should be recreated using vector graphics as an SVG file. ... A secant line of a curve is a line that intersects two or more points on the curve. ... The power of a point A (circle power,power of a circle) with respect to a circle with center 0 and radius r is defined as Therefore points inside the circle have negative power, points outside have positive power, and points on the circle have power zero. ...

begin{align} c^2 - a^2 & {} = b(b + 2acos(pi - gamma)) & {} = b(b - 2acos(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 (CK > 0) and acute angle (CK < 0) can be treated simultaneously.
A negative number is a number that is less than zero, such as &#8722;3. ...

Vector formulation

The law of cosines is equivalent to the formula

$vec bcdot vec c = Vert vec bVertVertvec cVertcos theta$

in the theory of vectors, which expresses the dot product of two vectors in terms of their respective lengths and the angle they enclose. A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... âˆ , the angle symbol. ...

Fig. 10 - Vector triangle

Proof of equivalence. Referring to Figure 10, note that Vector triangle drawn using xpaint. ... Vector triangle drawn using xpaint. ...

$vec a=vec b-vec c,,$

and so we may calculate:

 $Vertvec aVert^2,$ $= Vertvec b - vec cVert^2 ,$ $= (vec b - vec c)cdot(vec b - vec c),$ $= Vertvec b Vert^2 + Vertvec c Vert^2 - 2 vec bcdotvec c ,.$

The law of cosines formulated in this context states:

$Vertvec aVert^2 = Vertvec b Vert^2 + Vertvec c Vert^2 - 2 Vert vec bVertVertvec cVertcos(theta) ,,$

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. Namely, because a2 + b2 = 2a2 = 2ab, the law of cosines becomes A triangle. ...

$cos(gamma) = 1 - frac{c^2}{2a^2}. ;$

Analog for tetrahedra

An analogous statement begins by taking $scriptstyle{alpha, beta, gamma, delta }$ to be the areas of the four faces of a tetrahedron. Denote the dihedral angles by $scriptstyle{ widehat{betagamma}, }$ etc. Then A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ...

$alpha^2 = beta^2 + gamma^2 + delta^2 + 2betagammacosleft(widehat{betagamma}right) + 2gammadeltacosleft(widehat{gammadelta}right) + 2deltabetacosleft(widehat{deltabeta}right).,$

See A Treatise on Spherical Trigonometry: And Its Application to Geodesy and Astronomy with Numerous Examples by John Casey, Longmans, Green, & Company, London, 1889, page 133. Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... Geodetic pillar (1855); Ostend, Belgium Archive with lithography plates for maps of Bavaria in the Landesamt fÃ¼r Vermessung und Geoinformation in Munich Geodesy (IPA North American English ; British, Australian English etc. ... For other uses, see Astronomy (disambiguation). ...

Triangulation can be used to find the distance from the shore to the ship. ... In trigonometry, the law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. ... In trigonometry, the law of tangents is a statement about arbitrary triangles in the plane. ... Lamberts cosine law says that the total radiant power observed from a Lambertian surface is directly proportional to the cosine of the angle Î¸ between the observers line of sight and the surface normal. ... In spherical trigonometry, the law of cosines (also called the cosine rule for sides) refers to a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. ...

Results from FactBites:

 Law of cosines - Wikipedia, the free encyclopedia (1773 words) In trigonometry, the law of cosines (also known as 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. 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. It was not until the development of modern trigonomotry in the middle ages by muslim mathematicians that the law of cosines evolved beyond Euclid's two theorems.
 Law of cosines - definition of Law of cosines in Encyclopedia (329 words) In trigonometry, the law of cosines is a statement about arbitrary triangles which generalizes the Pythagorean theorem by correcting it with a term proportional to the cosine of the opposing angle. Let a, b, and c be the sides of the triangle and A, B, and C the angles opposite those sides. Although the law of cosines is a broader statement of the Pythagorean theorem, it isn't a proof of the Pythagorean theorem, because the law of cosines derivation given below depends on the Pythagorean theorem.
More results at FactBites »

Share your thoughts, questions and commentary here