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Law of Sines & Cosines

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Solving non-right triangles with the law of sines and cosines.

trigonometrylaw-of-sineslaw-of-cosines

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Law of sines

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  1. 1Law of sines
    a/sin α = b/sin β = c/sin γ = 2R, where a, b, c are sides, α, β, γ are opposite angles, and R is the circumradius
  2. 2Law of cosines
    c² = a² + b² - 2ab cos γ (also written for other sides)
  3. 3What does the law of cosines reduce to for right triangles?
    The Pythagorean theorem: c² = a² + b² (when γ = 90°, cos 90° = 0)
  4. 4Ambiguous case in the law of sines
    When two separate triangles can be constructed from the same data (SSA case), giving two possible solutions
  5. 5What conditions create an ambiguous case with the law of sines?
    When angle α is acute, side a < side c, and a > c sin α (SSA configuration)
  6. 6What does the term 2R represent in the law of sines?
    The diameter of the triangle's circumcircle (R is the circumradius)
  7. 7How can the law of cosines be used to find angles?
    cos γ = (a² + b² - c²)/(2ab) when all three sides are known
  8. 8Spherical law of sines
    sin a/sin A = sin b/sin B = sin c/sin C for triangles on a sphere with sides as arcs of great circles
  9. 9Law of cosines for spherical triangles
    cos c = cos a cos b + sin a sin b cos C (and cyclic permutations)
  10. 10Law of cosines sometimes called in France
    Théorème d'Al-Kashi (Al-Kashi's theorem)
  11. 11Which Euclid propositions correspond to the law of cosines?
    Propositions II.12 (obtuse triangles) and II.13 (acute triangles) from Elements Book II
  12. 12Relationship between triangle area and the law of sines
    Area T = abc/(4R) = 2R² sin α sin β sin γ, where R is the circumradius
  13. 13Heron's formula
    Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2 is the semiperimeter (can be derived from law of sines)
  14. 14Law of cosines for isosceles triangles (a = b)
    c² = 2a²(1 - cos γ), or equivalently cos γ = 1 - c²/(2a²)
  15. 15How is the law of cosines proven using Cartesian coordinates?
    By placing the triangle on a coordinate system and applying the distance formula: c² = (a - b cos γ)² + (b sin γ)²
  16. 16Version of law of cosines suited to small angles
    c² = (a - b)² + 2ab(1 - cos γ), which avoids catastrophic cancellation
  17. 17What does the law of cosines degenerate into for infinitesimal angles?
    The circular arc length formula: c = aγ
  18. 18Hyperbolic law of cosines
    cosh c = cosh a cosh b - sinh a sinh b cos C (first form)
  19. 19Second form of the hyperbolic law of cosines
    cos C = -cos A cos B + sin A sin B cosh c
  20. 20How can you derive the law of cosines using the law of sines?
    By using angle sum identities and the relation a/sin α = b/sin β = c/sin γ, then algebraic manipulation
  21. 21Ptolemy's theorem and how does it prove the law of cosines
    For a cyclic quadrilateral, the product of diagonals equals the sum of products of opposite sides; applying it proves the law of cosines
  22. 22How can the law of cosines be proven using altitudes?
    By expressing c = a cos β + b cos α and manipulating: c² = a² + b² - 2ab cos γ
  23. 23Analogue of the law of cosines for tetrahedra
    D² = A² + B² + C² - 2AB cos θAB - 2AC cos θAC - 2BC cos θBC, where A,B,C,D are face areas
  24. 24Triangulation
    The technique of computing remaining sides of a triangle when two angles and a side are known, using the law of sines
  25. 25Solve a triangle given SAS (two sides and included angle)
    Use the law of cosines to find the third side, then use law of sines to find remaining angles
  26. 26Solve a triangle given SSS (three sides)
    Use the law of cosines to find one angle: cos γ = (a² + b² - c²)/(2ab), then find other angles similarly or use law of sines
  27. 27Solve a triangle given AAS (two angles and a non-included side)
    Find the third angle (sum = 180°), then use law of sines to find the remaining sides
  28. 28Power of a point theorem
    For a point outside a circle, the power equals the product of distances along any secant, used to prove law of cosines
  29. 29Thales's theorem used in proving the law of sines
    An angle inscribed in a semicircle is a right angle, helping establish the circumradius relationship
  30. 30Inscribed angle theorem
    Angles subtending the same chord from the circle are equal, used in proving the law of sines

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