. The vectors and have lengths and respectively and the angle between them is , so , using cross products, dot products, and the Binet–Cauchy identity . Third proof Let , and denote the unit vectors from the center of the sphere to those corners of the triangle. Consider the following rotational sequence where we first rotate the vector to by an angle , followed another rotation of vector to by an angle , after which we rotate the vector back to by an angle . The composition of these three rotations will form an identity transform. That is, the composite rotation maps the point to itself. These three rotational operations can be represented by quaternions: where , , and are the unit vectors representing the axes of rotations, as defined by the right-hand rule, respectively. The composition of these three rotations is unity, Right multiplying both sides by conjugates , we have , where and . This gives us the identity The quaternion product on the right-hand side of this identity is given by Equating the scalar parts on both sides of the identity, we have Here . Since this identity is valid for any angles, suppressing the halves, we have We can also recover the sine law by first noting that and then equating the vector parts on both sides of the identity as The vector is orthogonal to both the vectors and , and as such . Taking dot product with respect to on both sides, and suppressing the halves, we have Now and so we have Dividing each side by , we have Since the right-hand side of the above expression is unchanged by cyclic permutation, we have Rearrangements The first and second spherical laws of cosines can be rearranged to put the sides () and angles () on opposite sides of the equations: Planar limit: small angles For small spherical triangles, i.e. for small , and , the spherical law of cosines is approximately the same as the ordinary planar law of cosines, To prove this, we will use the small-angle approximation obtained from the Maclaurin series for the cosine and sine functions: Substituting these expressions into the spherical law of cosines nets: or after simplifying: The big O terms for and are dominated by as and get small, so we can write this last expression as: See also Half-side formula Hyperbolic law of cosines