spherical trigonometry

Cosine law.

In the following we deduce the cosine law for a spherical trihedron.

Let $\\mathbf{e_{1},e_{2},e_{3}}$ be the vertex unitary vectors as shownin the figure.

The cosine of the angle $\\alpha$ formed by the planedefined by $\\mathbf{e_{1},e_{2}}$ and the plane defined by $\\mathbf{e_{1},e_{3}}$ is:

\\cos\\alpha=\\frac{(\\mathbf{e_{1}}\\times\\mathbf{e_{3}})\\cdot(\\mathbf{e_{1}}%\\times\\mathbf{e_{2}})}{\\left\\|\\mathbf{e_{1}}\\times\\mathbf{e_{3}}\\right\\|\\left%\\|\\mathbf{e_{1}}\\times\\mathbf{e_{2}}\\right\\|}=\\frac{(\\mathbf{e_{1}}\\times%\\mathbf{e_{3}})\\cdot(\\mathbf{e_{1}}\\times\\mathbf{e_{2}})}{\\sin b\\sin c}

Now, using the cyclic property of the triple vector product and Lagrange’s formula (http://planetmath.org/TripleCrossProduct), we can write:

\\cos\\alpha=\\frac{\\mathbf{e_{1}}\\cdot(\\mathbf{e_{3}}\\times(\\mathbf{e_{1}}\\times%\\mathbf{e_{2}}))}{\\sin b\\sin c}=\\frac{\\mathbf{e_{1}}\\cdot((\\mathbf{e_{3}}\\cdot%\\mathbf{e_{2}})\\mathbf{e_{1}}-(\\mathbf{e_{3}}\\cdot\\mathbf{e_{1}})\\mathbf{e_{2}%})}{\\sin b\\sin c}=\\frac{\\cos a-\\cos b\\cos c}{\\sin b\\sin c}

Hence:

\\cos a=\\cos b\\cos c+\\sin b\\sin c\\cos\\alpha