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Solution of triangles (Latin: solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere .
In the case that the side c has length π /2 on the unit sphere the equations governing the remaining sides and angles may be obtained by applying the rules for the right spherical triangle of the previous section to the polar triangle A'B'C' with sides a', b', c' such that A' = π − a, a' = π − A etc.
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles.According to the law, = = =, where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see figure 2), while R is the radius of the triangle's circumcircle.
In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle. [1] [2]A variant in more geometrical style was first published by Isaac Newton in 1707 and then by Friedrich Wilhelm von Oppel [] in 1746.
Fig. 1 – A triangle. The angles α (or A), β (or B), and γ (or C) are respectively opposite the sides a, b, and c.. In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles.
Other equations, known as triangle identities, [84] relate both the sides and angles of a given triangle. Triangle identities In the following identities, A , B and C are the angles of a triangle and a , b and c are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram).
The quaternions q, r, and s are used to represent rotations with axes of rotation w′, u′, and v′, respectively, and angles of rotation 2a, 2b, and 2c, respectively. Because these are double angles, each of q, r, and s represents two applications of the rotation implied by an edge of the spherical triangle. From the definitions, it follows ...
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