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  2. Law of cosines - Wikipedia

    en.wikipedia.org/wiki/Law_of_cosines

    The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if is a right angle then ⁡ =, and the law of cosines reduces to = +. The law of cosines is useful for solving a triangle when all three sides or two sides and their included angle are given.

  3. Spherical law of cosines - Wikipedia

    en.wikipedia.org/wiki/Spherical_law_of_cosines

    If the law of cosines is used to solve for c, the necessity of inverting the cosine magnifies rounding errors when c is small. In this case, the alternative formulation of the law of haversines is preferable. [3] A variation on the law of cosines, the second spherical law of cosines, [4] (also called the cosine rule for angles [1]) states:

  4. Spherical trigonometry - Wikipedia

    en.wikipedia.org/wiki/Spherical_trigonometry

    There are many ways of deriving the fundamental cosine and sine rules and the other rules developed in the following sections. For example, Todhunter [1] gives two proofs of the cosine rule (Articles 37 and 60) and two proofs of the sine rule (Articles 40 and 42). The page on Spherical law of cosines gives

  5. Solution of triangles - Wikipedia

    en.wikipedia.org/wiki/Solution_of_triangles

    To find an unknown angle, the law of cosines is safer than the law of sines. The reason is that the value of sine for the angle of the triangle does not uniquely determine this angle. For example, if sin β = 0.5, the angle β can equal either 30° or 150°. Using the law of cosines avoids this problem: within the interval from 0° to 180° the ...

  6. Trigonometry - Wikipedia

    en.wikipedia.org/wiki/Trigonometry

    The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-". [32] With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. [33]

  7. Trigonometry of a tetrahedron - Wikipedia

    en.wikipedia.org/wiki/Trigonometry_of_a_tetrahedron

    By the spherical law of cosines: ⁡, = ⁡, ⁡, + ⁡, ⁡, ⁡ Take the spherical triangle of the tetrahedron X {\displaystyle X} at the point P i {\displaystyle P_{i}} . The sides are given by α i , l , α k , j , λ {\displaystyle \alpha _{i,l},\alpha _{k,j},\lambda } and the only known opposite angle is that of λ {\displaystyle \lambda ...

  8. Eisenstein triple - Wikipedia

    en.wikipedia.org/wiki/Eisenstein_triple

    Triangles with an angle of 60° are a special case of the Law of Cosines: [1] [2] [3] = +. When the lengths of the sides are integers, the values form a set known as an Eisenstein triple. [4] Examples of Eisenstein triples include: [5]

  9. Heron's formula - Wikipedia

    en.wikipedia.org/wiki/Heron's_formula

    See the law of cotangents for the reasoning behind this. If ⁠ r {\displaystyle r} ⁠ is the radius of the incircle of the triangle, then the triangle can be broken into three triangles of equal altitude ⁠ r {\displaystyle r} ⁠ and bases ⁠ a , {\displaystyle a,} ⁠ ⁠ b , {\displaystyle b,} ⁠ and ⁠ c . {\displaystyle c.}

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