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sin (0 r6)  cot @ tan A sin Tg
tan Tg » cos A tan 0 2 VALID SOLUTIONS
section as presents an alternative formula.
following rules are sufficient to determine the quadrant of any component:
1. If one side (angle) differs From 90 deg by more than another side (angle), it is in the same quadrant as its opposite angle (side).
2. Half the sum of any two sides is in the same quadrant as half the sum of the opposite angles.
A.4 Differential Spherical Trigonometry
The development here follows that of Newcomb [1960], which contains a more extended discussion of the subject.
Differential Relations Between the Parts of a Spherical Triangle. In general, any part of a spherical triangle may be determined from three other parts. Thus, it is of interest to determine the error in any part produced by infinitesimal errors in the three given parts. This may be done by determining the partial derivatives relating any four parts of a spherical triangle from the following differentials, where the notation of Fig. AS is retained. Given three angles and one side:
 sin X sin4» dfl + d© + cos«J> d A + cos X d$ = 0 (A30)
Given three sides and one angle:
Given two sides and the opposite angles:
cos0sin$d0—cos«fsin©d<f+sin0cos$d$—sin<#»[email protected]©=0 (A32)
Given two sides, the included angle, and one opposite angle:
sin$d0+cosXsin0d<j>[email protected]+cos$!sin0dA = O (A33)
As an example of the determination of partial derivatives, consider a triangle in which the three independent variables are the three sides. Then, from Eq. (A31),
Infinitesimal Triangles. The simplest infinitesimal spherical triangle is one in which the entire triangle is small relative to the radius of the sphere. In this case, the spherical triangle may be treated as a plane triangle if the three rotation angles remain finite quantities. If one of the rotation angles is infinitesimal, the analysis presented below should be used.
Figure A6 shows a spherical triangle in which two sides are of arbitrary, but nearly equal, length and the included rotation angle is infinitesimal. Then the
Fig. A6. Spherical Triangle With One Infinitesimal Angle
Fig. A6. Spherical Triangle With One Infinitesimal Angle change in the angle by which the two sides intercept a great circle is given by
The perpendicular separation, o, between the two long arcs is given by o = 80sinA (A35)
If two angles are infinitesimal (such that the third angle is nearly 180 deg), the triangle may be divided into two triangles and treated as above.
A.5 Haversines
A convenient computational tool for spherical trigonometry is the haversine, defined as haversine 9=hav 0=^(1 cos0) (A36)
The principal advantage of the haversine is that a given value of the function corresponds to only one angle over the range from 0 deg to 180 deg, in contrast to the sine function for which there is an ambiguity as to whether the angle corresponding to a given value of the sine falls in the range 0 deg to 90 deg or 90 deg to 180 deg. Given the notation of Fig. A5, two fundamental haversine relations in any spherical triangle are as follows:
hav A=ha v(0  <j>)+sin 0 sin <J> hav A (A37)
sin0sm<f>
The first three formulas from Table Al can be expressed in a simpler form to evaluate in terms of haversines, as shown in Table A2. Most spherical geometry

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