Spherical Right Triangles

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point A
1.0472
point B
1.0472
show sphere
Draw a spherical triangle
ABC
on the surface of a unit sphere centered at
O=(0,0,0)
. Denote the arcs opposite the corresponding vertices as
a
,
b
,
c
. Let
α
,
β
,
γ
be the angles at the vertices
A
,
B
,
C
. Suppose
γ=π/2
, so that these become spherical right triangles. The following relations are then valid:
sina=sincsinα
sinb=sincsinβ
tana=sinbtanα
tanb=sinatanβ
cosc=cosacosb
cosα=cosasinβ
cosβ=cosbsinα
sina=sincsinαsina=sincsinαsina=sincsinα

Details

Let
P
be the orthogonal projection of
A
onto the edge
OC
. Let
Q
be the orthogonal projection of
P
onto
OB
. Then
AP=sinb
,
AQ=sinc
, but also
AP=sincsinβ
. Thus
sinb=sincsinβ
.
OP=cosb
,
OQ=cosc
, but also
OQ=OPcosa
. Thus
cosc=cosbcosa
.

References

[1] Wikipedia. "Spherical Law of Cosines." (Mar 21, 2017) en.wikipedia.org/wiki/Spherical_law_of _cosines.
[2] Wikipedia. "Spherical Trigonometry." (Mar 21, 2017) en.wikipedia.org/wiki/Spherical_trigonometry.

External Links

Spherical Trigonometry (Wolfram MathWorld)
Spherical Triangle Solutions

Permanent Citation

Izidor Hafner
​
​"Spherical Right Triangles"​
​http://demonstrations.wolfram.com/SphericalRightTriangles/​
​Wolfram Demonstrations Project​
​Published: March 22, 2017