Spherical Right Triangles
Spherical Right Triangles
Draw a spherical triangle on the surface of a unit sphere centered at . Denote the arcs opposite the corresponding vertices as , , . Let , , be the angles at the vertices , , . Suppose , so that these become spherical right triangles. The following relations are then valid:
ABC
O=(0,0,0)
a
b
c
α
β
γ
A
B
C
γ=π/2
sina=sincsinα
sinb=sincsinβ
tana=sinbtanα
tanb=sinatanβ
cosc=cosacosb
cosα=cosasinβ
cosβ=cosbsinα
sina=sincsinαsina=sincsinαsina=sincsinα
Details
Details
Let be the orthogonal projection of onto the edge . Let be the orthogonal projection of onto . Then , , but also . Thus .
P
A
OC
Q
P
OB
AP=sinb
AQ=sinc
AP=sincsinβ
sinb=sincsinβ
OP=cosb
OQ=cosc
OQ=OPcosa
cosc=cosbcosa
References
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
External Links
Permanent Citation
Permanent Citation
Izidor Hafner
"Spherical Right Triangles"
http://demonstrations.wolfram.com/SphericalRightTriangles/
Wolfram Demonstrations Project
Published: March 22, 2017