Page 69 - Intro to Tensor Calculus

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I 52. (Spherical Trigonometry) Construct a spherical triangle ABC on the surface of a unit sphere with
sides and angles less than 180 degrees. Denote by a,bc the unit vectors from the origin of the sphere to the
vertices A,B and C. Make the construction such that a·(b×c) is positive with a, b, c forming a right-handed
system. Let α, β, γ denote the angles between these unit vectors such that

a · b =cos γ c · a =cos β b · c =cos α. (1)

The great circles through the vertices A,B,C then make up the sides of the spherical triangle where side α
is opposite vertex A, side β is opposite vertex B and side γ is opposite the vertex C. The angles A,B and C
between the various planes formed by the vectors a, b and c are called the interior dihedral angles of the
spherical triangle. Note that the cross products

a × b =sin γ c b × c =sin α a c × a =sin β b (2)

define unit vectors a, b and c perpendicular to the planes determined by the unit vectors a, b and c. The
dot products
a · b =cos γ b · c =cos α c · a =cos β (3)
define the angles α,β and γ which are called the exterior dihedral angles at the vertices A,B and C and are
such that
α = π − A β = π − B γ = π − C. (4)

(a) Using appropriate scaling, show that the vectors a, b, c and a, b, c form a reciprocal set.
(b) Show that a · (b × c)=sin α a · a =sin β b · b =sin γ c · c
(c) Show that a · (b × c)= sin α a · a =sin β b · b =sin γ c · c
(d) Using parts (b) and (c) show that
sin α sin β sin γ
= =
sin α sin β sin γ
(e) Use the results from equation (4) to derive the law of sines for spherical triangles
sin α sin β sin γ
= =
sin A sin B sin C
(f) Using the equations (2) show that

sin β sin γb · c =(c × a) · (a × b)= (c · a)(a · b) − b · c

and hence show that
cos α =cos β cos γ − sin β sin γ cos α.
In a similar manner show also that

cos α =cos β cos γ − sin β sin γ cos α.

(g) Using part (f) derive the law of cosines for spherical triangles
cos α =cos β cos γ +sin β sin γ cos A

cos A = − cos B cos C +sin B sin C cos α
A cyclic permutation of the symbols produces similar results involving the other angles and sides of the
spherical triangle.