Three fundamental relationships—the law of sines, the law of cosines for angles, and the law of cosines for sides—hold for all spherical triangles. These may be used to derive Napier's Rules (Section A.2) or may be derived from them. The components of a general spherical triangle as used throughout this section are shown in Fig. A-5.

Law of Sines.

Law of Cosines for Sides.

cosX=cos0cos<i>+sin0sin<i>cos A (A-26) Similar relationships hold for each side.

Law of Cosines for Angles.

cosA= — cos 0 cos <&+sin© sin $ cos A (A-27) Similar relationships hold for each angle.

Fig. A-5. Notation for Rules for Oblique Spherical Triangles

Fig. A-5. Notation for Rules for Oblique Spherical Triangles

Half-Angle Formulas. A spherical triangle is fully specified by either three sides or three angles. The remaining components are most conveniently expressed in terms of half angles. Specifically,

where and where

Similar relationships may be found for the other trigonometric functions of half angles in most spherical geometry texts.

General Solution of Oblique Spherical Triangles. Table A-l lists formulas for solving any oblique spherical triangle. In addition, in any spherical triangle, the

Table A-l. Formulas for Solving Oblique Spherical Triangles. See also Table A-2.

Was this article helpful? |

## Post a comment