A Treatise on Elementary Geometry: With Appendices Containing a Collection of Exercises for Students and an Introduction to Modern Geometry

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J. B. Lippincott, 1896 - Geometry - 368 pages
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Page 150 - In a spherical triangle, the sines of the sides are proportional to the sines of the opposite angles. Let ABC, Fig. 1, be a spherical triangle, 0 the center of the sphere. The angles...
Page 152 - ... cos a = cos b cos с + sin b sin с cos A ; (2) cos b = cos a cos с + sin a sin с cos в ; ^ A. (3) cos с = cos a cos b + sin a sin b cos C.
Page 169 - I. The sine of the middle part is equal to the product of the tangents of the adjacent parts.
Page 58 - THE SUM OF THE OPPOSITE ANGLES', To THE TANGENT OF HALF THEIR DIFFERENCE. Thus, the sum of AB and AC, (Fig. 25.) is to their difference ; as the tangent of half the sum of the angles ACB and ABC, to the tangent of half their difference.
Page 229 - This problem is solved in geometry, where it is proved that the surface of a spherical triangle is measured by the excess of the sum of its three angles over two right angles...
Page 153 - A cos 6 = cos a cos c + sin a sin c cos B cos c = cos a cos 6 + sin a sin 6 cos C Law of Cosines for Angles cos A = — cos B...
Page 64 - Art. 117, and state the proportions thus : the sine of the angle opposite the given side is to the sine of the angle opposite the required side, as the given side is to the required side.
Page 191 - It is easily seen, also, that all the formulae above given for this case might have been obtained by these considerations. 84. CASE III. G-iven two sides and an angle opposite one of them ; or a, b, and A. Fig, 9. First Solution, in which each required part is deduced directly from fundamental formula independently of the other two parts. To find c. We have, by (4...
Page 151 - ... might therefore be considered as general, without requiring a special examination of the various positions of the lines of the diagram. 5. In a spherical triangle^ the cosine of any side is equal to the product of the cosines of the other two sides, plus the continued product of the sines of those sides and the cosine of the included angle.
Page 65 - Art. 117, and state the proportion thus : the side opposite the given angle is to the side opposite the required angle as the sine of the given angle is to the sine of the required angle.

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