Half-side formula

In spherical trigonometry, the half side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles.^[1]

For a triangle $△ A B C$ on a sphere, the half-side formula is^[2] $\begin{matrix} \tan \frac{1}{2} a & = \sqrt{\frac{- \cos (S) \cos (S - A)}{\cos (S - B) \cos (S - C)}} \end{matrix}$

where Template:Mvar are the angular lengths (measure of central angle, arc lengths normalized to a sphere of unit radius) of the sides opposite angles Template:Mvar respectively, and $S = \frac{1}{2} (A + B + C)$ is half the sum of the angles. Two more formulas can be obtained for $b$ and $c$ by permuting the labels $A, B, C .$

The polar dual relationship for a spherical triangle is the half-angle formula,

$\begin{matrix} \tan \frac{1}{2} A & = \sqrt{\frac{\sin (s - b) \sin (s - c)}{\sin (s) \sin (s - a)}} \end{matrix}$

where semiperimeter $s = \frac{1}{2} (a + b + c)$ is half the sum of the sides. Again, two more formulas can be obtained by permuting the labels $A, B, C .$

Half-tangent variant

The same relationships can be written as rational equations of half-tangents (tangents of half-angles). If $t_{a} = \tan \frac{1}{2} a,$ $t_{b} = \tan \frac{1}{2} b,$ $t_{c} = \tan \frac{1}{2} c,$ $t_{A} = \tan \frac{1}{2} A,$ $t_{B} = \tan \frac{1}{2} B,$ and $t_{C} = \tan \frac{1}{2} C,$ then the half-side formula is equivalent to:

$\begin{matrix} t_{a}^{2} & = \frac{(t_{B} t_{C} + t_{C} t_{A} + t_{A} t_{B} - 1) (- t_{B} t_{C} + t_{C} t_{A} + t_{A} t_{B} + 1)}{(t_{B} t_{C} - t_{C} t_{A} + t_{A} t_{B} + 1) (t_{B} t_{C} + t_{C} t_{A} - t_{A} t_{B} + 1)} . \end{matrix}$

and the half-angle formula is equivalent to:

$\begin{matrix} t_{A}^{2} & = \frac{(t_{a} - t_{b} + t_{c} + t_{a} t_{b} t_{c}) (t_{a} + t_{b} - t_{c} + t_{a} t_{b} t_{c})}{(t_{a} + t_{b} + t_{c} - t_{a} t_{b} t_{c}) (- t_{a} + t_{b} + t_{c} + t_{a} t_{b} t_{c})} . \end{matrix}$

References

Template:Reflist

[1] Template:Citation [1]

[2] Template:Citation.

[1]

[2]

Half-side formula

Half-tangent variant

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Half-side formula

Half-tangent variant

See also

References

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