Spherical trigonometry

According to the classical formula of spherical trigonometry, cos dE can be written as... 

For further information on stereographic projections and the spherical trigonometry necessary for handling goniometric data, books by Miers (1929), Tutton (1922), and Barker (1922) may be consulted. 

The quantity (cosQi,i+j) can be evaluated with the aid of the law of cosines from spherical trigonometry ... 

Analytic derivatives of the segment areas with respect to the atom coordinates can be calculated using spherical trigonometry for the partial triangles, but the expressions become very complicated due to the implicit dependence of the positions of the auxiliary spheres on several atom positions. 

It is possible to transform this expression into a form which contains the polar and azimuthal angles 0 and O as defined in the tilted coordinate frame of Fig. 1.15. Applying spherical trigonometry, one then has the following relation between these angles... 

Using spherical trigonometry, the angular functions can be expressed as... 

K. L. Nielsen and J. H. Vanlonkuyyzen, Plane and Spherical Trigonometry, Barnes and Noble, New York, 1946. 

The locus of pointsin three-space that are a fixed distance froma given point (called the center), spherical trigonometry... 

Bowser s College Algebra. A full treatment of elementary and advanced topics. 1.50. Bowser s Plane and Solid Geometry. 1.25. Plane bound separately. 75 cts. Bowser s Elements of Plane and Spherical Trigonometry. 90 cts. with tables, i.4o. Bowser s Treatise on Plane and Spherical Trigonometry. 1.50. 

Wells s New Plane and Spherical Trigonometry. For colleges and technical schno<. 1.00. With six-place tables, 1.25. With Robbins s Surveying and Navigation, <... 

This graphical technique is most useful for determining the qualitative effect of a given sequence of layers. (Quantitative calculations are possible too, but the necessary spherical trigonometry is no less cumbersome than the algebraic approach.)... 

To find more on the use of geometric algebra in spherical trigonometry, the reader should consult Ref. 50, pages 661-667. 

If h and rb are measured, 0 can be calculated from Equation (651). For large drops, h and rb can be so distorted by gravitation that Equation (651) cannot be used, and much more elaborate calculations are needed. On the other hand, the contact angle may be calculated if the drop volume can be experimentally measured by using advanced syringes. If the contact radius of the liquid drop, rb of known volume, V is measured, then 6 can be calculated from the dimensionless ratio obtained from three-dimensional spherical trigonometry... 

Based on spherical trigonometry, a relationship can be derived connecting the solar zenith angle and the azimuthal angle 0o to the sun declination A, the latitude , and the hour angle H ... 

To reach the abovementioned objective, we can easily determine from Figure 8.14(b) that the following two requirements must be satisfied (1) the arc TE should equal to the arc TA, and (2) the arc TC should equal to the arc EC. Besides, from Figure 8.14(b) we can also obtain Z.POB = zAOB = q>,TA=jt-2(p, and TC = cp. Based on spherical trigonometry, we can find the following relationships from the spherical triangles CTE and TEA ... 

Table VI-1 gives completely general expressions for elements of the types Grr, Gr, and G. " rhe formulas involve certain angles given the symbol in addition to the angles and t which have been previously defined. The s are dihedral angles and are related to the s according to the foimula of spherical trigonometry ...

---