APPENDIX B
PLANE AND SPHERICAL TRIGONOMETRY
B-3. Properties of Plane Triangles
B-4. Formulas for Triangle Computations
B-5. Spherical Triangles: Introduction
B-6. Properties of Spherical Triangles
B-7. The Right Spherical Triangle
B-8. Formulas for Triangle Computations
B-9. Formulas Expressed in Terms of the Haversine Function
B-l. Introduction. Plane trigonometry describes relations between the sides and angles of plane triangles in terms of trigonometric functions (Secs. 21.2-1 to 21.2-4); note that all plane figures bounded by straight lines may be regarded as combinations of triangles. Since all plane triangles may be resolved into right triangles, the most important trigonometric relations are those relating the sides and angles of right triangles.
B-2. Right Triangles. In every right triangle (Fig. B-l) with sides a, b and hypotenuse c,
B-3. Properties of Plane Triangles. In every plane triangle (Fig. B-2), the sum of the angles equals 180 deg. The sum of any two sides is greater than the third, and the greater of two sides opposes the greater
of two angles. A plane triangle is uniquely determined (except for symmetric images) by
1. Three sides
2. Two sides and the included angle
3. One side and two angles
4. Two sides and the angle opposite the greater side
In every plane triangle, the three bisectors of angles intersect in the center M of the inscribed circle. The three perpendicular bisectors of the sides intersect in the center F of the circumscribed circle. The three medians intersect in the center of gravity G of the triangle. The three altitudes intersect in a point H collinear with the last two points, so that 
. The mid-points of the sides, the footpoints of the perpendiculars from the vertices to the sides, and the mid-points of the straight-line segments joining H to each vertex lie on a circle whose radius is half that of the circumscribed circle (nine-point circle, or Feuerbach circle). The center of the nine-point circle is the mid-point of the straight-line segment HF.
B-4. Formulas for Triangle Computations. In the following relations, A, B, C are the angles opposite the respective sides a, b, c of a plane triangle (Fig. B-2). The triangle area is denoted by S; r and ρ are the respective radii of the circumscribed and inscribed circles, and s= ½(a + b + c). Additional formulas are obtained by simultaneous cyclic permutation of A, B, C and a, b, c. Table B-l permits the computation of the sides and angles of any plane triangle from three suitable sides and/or angles.
Table B-1. Solution of Plane Triangles. Obtain all other cases by cyclic permutation (refer to formulas of Sec. B-4 and to Fig. B-2)
B-5. Spherical Triangles: Introduction. On the surface of a sphere, the shortest distance between two points is measured along a great circle, i.e., a circle whose plane passes through the center of the sphere (geodesic, Sec. 17.3-12). The vertices of a spherical triangle are the intersections of three directed straight lines passing through the center of the sphere and the spherical surface. The sides a, b, c of the spherical triangle are those three angles between the three directed straight lines which are less than 180 deg. Corresponding to each triangle side, there is a great-circle segment on the surface of the sphere (Fig. B-3). The angles A, B, C of the spherical triangle opposite the sides a, b, c, respectively, are the angles less than 180 deg between the great-circle segments corresponding to the triangle sides, or the corresponding angles between the three planes denned by the three given straight lines.
Spherical trigonometry is the study of relations between the sides and angles of spherical triangles (e.g., on the surface of the earth and on the celestial sphere).
In many problems, physicists and engineers will prefer the use of the rotation transformations (Sec. 14.10-1) to the use of spherical trigonometry.
B-6. Properties of Spherical Triangles. Each side or angle of a spherical triangle is, by definition, smaller than 180 deg. The geometry on the surface of a sphere is non-Euclidean (see also Sec. 17.3-13); in every spherical triangle, the sum of the sides will be between 0 and 360 deg, and the sum of the angles will be between 180 and 540 deg. In every spherical triangle, the greater of two sides opposes the greater of two angles. The sum of any two sides is greater than the third, and the sum of any two angles is less than 180 deg plus the third angle. A spherical triangle is uniquely determined (except for symmetric images) by
1. Three sides
2. Three angles
3. Two sides and the included angle
4. Two angles and the included side
5. Two sides and an opposite angle, given that the other opposite angle is less than, equal to, or greater than 90 deg
6. Two angles and an opposite side, given that the other opposite side is less than, equal to, or greater than 90 deg
NOTE: In every spherical triangle, it is possible to define great circles as perpendicular bisectors of sides, bisectors of angles, medians, and altitudes. The planes of the three great circles of each type intersect in a straight line.
In analogy to the circumscribed circle of a plane triangle, there exists a circumscribed right circular cone containing the three straight lines denning the triangle; the axis of this cone is the straight line formed by the intersection of the planes of the perpendicular bisectors. There is also an inscribed right circular cone touching the three planes corresponding to the spherical triangle; the axis of this cone is the straight line formed by the intersection of the planes of the bisectors of the angles. The “radius” r of the circumscribed circle and the “radius” ρ of the inscribed circle are angles defined as half the vertex angles of the respective cones.
Given the radius R of the sphere, the area SR of a spherical triangle is given by
where є is the spherical excess
measured in radians. The quantity d = 2π - (a + b + c) is called spherical defect.
The polar triangle corresponding to a given spherical triangle is defined by three directed straight lines perpendicular to the planes associated with the sides of the original triangle. The sides of the polar triangle are equal to the supplements of the corresponding angles of the original triangle, and conversely. Thus every theorem, or formula dealing with the sides and angles of the original triangle may be transformed into one dealing with the angles and sides of the polar triangle.
B-7. The Right Spherical Triangle. In a right spherical triangle, at least one angle, C, say, is equal to 90 deg; the opposite side, c, is called the hypotenuse. All important relations between the sides and angles of the right spherical triangle may be derived from Napier's rules, two convenient aids to memory:
Napier's Rules: In the diagram of Fig. B-4, the sine of any of the angles shown is equal
1. To the product of the tangents of the two angles adjoining it in the diagram
2. To the product of the cosines of the two angles opposing it in the diagram
EXAMPLE: To compute the sides and angles of a right spherical triangle with the hypotenuse c, given c and a.
This problem has a solution only if sin a ≤ sin c; then
NOTE: If a is less than, equal to, or greater than 90 deg, so is A, and conversely.
If b is less than, equal to, or greater than 90 deg, so is B.
If a and A are given, the problem has a solution only if the above condition is satisfied and sin a ≤ sin A; unless a = A, there are two solutions. The situation is analogous if b and B are given.
If A and B are given, the problem has a solution only if 90 < A + B < 270 deg and -90 deg < A - B < 90 deg (Sec. B-6).
A spherical triangle having a side equal to 90 deg is called a quadrantal triangle and may be treated as the polar triangle (Sec. B-6) of a right spherical triangle.
For all problems involving the spherical-triangle computations (right or oblique triangles), it is strongly recommended that a sketch be drawn which roughly indicates whether the various angles and sides will be less than, equal to, or greater than 90 deg.
B-8. Formulas for Triangle Computations (see also Fig. B-3). In the following relations, A, B, C are the angles opposite the respective sides a, b, c of a spherical triangle. The respective “radii” of the circumscribed and inscribed cones are denoted by r and ρ. Additional formulas are obtained by simultaneous cyclic permutation of A, B, C and a, b, c. Table B-2 permits the computation of the sides or angles of any spherical triangle from three suitable sides and/or angles. The inequalities noted in Sec. B-6 must be observed in order to avoid ambiguous results in triangle computations.
B-9. Formulas Expressed in Terms of the Haversine Function. Certain trigonometrical relations become particularly suitable for logarithmic computations if they are expressed in terms of the new trigonometric functions versed sine, versed cosine, and haversine, defined by
Thus, if tables of the haversine function are available, one may use the following formulas for spherical-triangle computations:
Other similar relations may be obtained by cyclic permutation.
Table B-2. Solution of Spherical Triangles (refer to formulas of Sec. B-8 and to Fig. B-3)
Kells, L. M., et al.: Plane and Spherical Trigonometry, 3d ed., McGraw-Hill, New York, 1951.
Palmer, C. I., et al.: Plane and Spherical Trigonometry, 5th ed., McGraw-Hill, New York, 1950.
