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 $\triangle ABC$ on a sphere, the half-side formula is^[2]

{\begin{aligned}\tan {\tfrac {1}{2}}a&={\sqrt {\frac {-\cos(S)\,\cos(S-A)}{\cos(S-B)\,\cos(S-C)}}}\end{aligned}}

where $a, b, c$ are the angular lengths (measure of central angle, arc lengths normalized to a sphere of unit radius) of the sides opposite angles $A, B, C$ respectively, and $S={\tfrac {1}{2}}(A+B+C)$ 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{aligned}\tan {\tfrac {1}{2}}A&={\sqrt {\frac {\sin(s-b)\,\sin(s-c)}{\sin(s)\,\sin(s-a)}}}\end{aligned}}

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

Half-tangent variant[edit]

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

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

and the half-angle formula is equivalent to:

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

References[edit]

^ Bronshtein, I. N.; Semendyayev, K. A.; Musiol, Gerhard; Mühlig, Heiner (2007), Handbook of Mathematics, Springer, p. 165, ISBN 9783540721222[1]
^ Nelson, David (2008), The Penguin Dictionary of Mathematics (4th ed.), Penguin UK, p. 529, ISBN 9780141920870.

[1] Bronshtein, I. N.; Semendyayev, K. A.; Musiol, Gerhard; Mühlig, Heiner (2007), Handbook of Mathematics, Springer, p. 165, ISBN 9783540721222[1]

[2] Nelson, David (2008), The Penguin Dictionary of Mathematics (4th ed.), Penguin UK, p. 529, ISBN 9780141920870.

[1]

[2]

Half-tangent variant[edit]

See also[edit]

References[edit]