Law of Cosines
Let a, b, and c be the lengths of the legs of a triangle opposite angles A, B, and C. Then the law of cosines states
Solving for the cosines yields the equivalent formulas
This law can be derived in a number of ways. The definition of the dot product incorporates the law of cosines, so that the length of the vector from X to Y is given by
where theta is the angle between X and Y.
The formula can also be derived using a little geometry and simple algebra. From the above diagram,
The law of cosines for the sides of a spherical triangle states that
(Beyer 1987). The law of cosines for the angles of a spherical triangle states that
(Beyer 1987).
For similar triangles, a generalized law of cosines is given by
| aa^'=bb^'+cc^'-(bc^'+b^'c)cosA |
(19)
|
(Lee 1997). Furthermore, consider an arbitrary tetrahedron A_1A_2A_3A_4 with triangles T_1=DeltaA_2A_3A_4, T_2=DeltaA_1A_3A_4, T_3=DeltaA_1A_2A_4, and T_4=A_1A_2A_3. Let the areas of these triangles be s_1, s_2, s_3, and s_4, respectively, and denote the dihedral angle with respect to T_i and T_j for i!=j=1,2,3,4 by theta_(ij). Then
which gives the law of cosines in a tetrahedron,
(Lee 1997). A corollary gives the nice identity
See also
Law of Sines, Law of Tangents Explore this topic in the MathWorld classroomExplore with Wolfram|Alpha
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References
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 79, 1972.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 148-149, 1987.Lee, J. R. "The Law of Cosines in a Tetrahedron." J. Korea Soc. Math. Ed. Ser. B: Pure Appl. Math. 4, 1-6, 1997.Referenced on Wolfram|Alpha
Law of CosinesCite this as:
Weisstein, Eric W. "Law of Cosines." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LawofCosines.html