Spherical Triangles

Excerpts

• Spherical Triangles

  Spherical Trigonometry investigates the relations which subsist between the angles of the plane faces which form a solid angle and the angles at which the plane faces are inclined to each other.

• Spherical Triangles

  Thus a figure will be formed on the surface of the sphere which is called a *spherical triangle* if it is bounded by *three* arcs of great circles; this will be the case when the solid angle is formed by the meeting of *three* plane angles.

• Spherical Triangles

  It will be seen that what are called *sides* of a spherical triangle are really *arcs* of great circles, and these arcs are proportional to the three plane angles which form the solid angle corresponding to the spherical triangle.

• Spherical Triangles

  As in the case of plane triangles, $A$, $B$, and $C$ may be used to denote the numerical values of the angles expressed in *terms of any unit*, provided we understand distinctly what the unit is.

• Spherical Triangles

  In spherical triangles each side is restricted to be less than a semicircle; this is of course a *convention*, and it is adopted because it is found convenient.

• Spherical Triangles

  From the restriction of the preceding Article it will follow that *any angle of a spherical triangle is less than two right angles*.

Equations

• Spherical Triangles

  \dfrac{\operatorname{arc} AB} {\operatorname{radius} OA}

  The plane angle AOB at the centre O is measured by the arc AB divided by the radius OA, so an arc is proportional to its angle on the same sphere.

• Spherical Triangles

  C = 90^\circ

  If the angle C of a spherical triangle is a right angle, its numerical value is 90 when the unit is the degree.

• Spherical Triangles

  C = \dfrac{\pi}{2}

  If the angle C of a spherical triangle is a right angle, its numerical value is pi/2 when the unit is the angle subtended at the centre by an arc equal to the radius.

Problems

No exercises in this chapter.