Tangent of Half Angle for Spherical Triangles

Theorem

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.

Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.

Then:

$\tan \dfrac A 2 = \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin s \, \map \sin {s - a} } }$

where $s = \dfrac {a + b + c} 2$.

Proof

\(\ds \tan \dfrac A 2\)

\(=\)

\(\ds \dfrac {\sqrt {\dfrac {\sin \paren {s - b} \sin \paren {s - c} } {\sin b \sin c} } } {\sqrt {\dfrac {\sin s \, \map \sin {s - a} } {\sin b \sin c} } }\)

Sine of Half Angle for Spherical Triangles, Cosine of Half Angle for Spherical Triangles

\(\ds \)

\(=\)

\(\ds \dfrac {\sqrt {\sin \paren {s - b} \sin \paren {s - c} } } {\sqrt {\sin s \, \map \sin {s - a} } }\)

simplification

The result follows.

$\blacksquare$

Also see

• The other Half Angle Formulas for Spherical Triangles:
  • Sine of Half Angle for Spherical Triangles
  • Cosine of Half Angle for Spherical Triangles

• Half Side Formulas for Spherical Triangles:
  • Sine of Half Side for Spherical Triangles
  • Cosine of Half Side for Spherical Triangles
  • Tangent of Half Side for Spherical Triangles

Sources

• 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $5$. The cosine-formula.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): half-angle formulae: 3.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): half-angle formulae: 3.