First Part of an Elementary Treatise on Spherical Trigonometry1836 - Spherical trigonometry - 71 pages |
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Page 43
... cos . C But , by ( 104 ) , cos . c - cos . a cos . b + sin . a sin . b sin . a sin . b cos . ( a + b ) = cos . a ... half the sum of the sides or s = ( a + b + c ) ; M = ( a + b + c ) = s , N = ( 679 ) if we make in ( 678 ) { N = 1 ...
... cos . C But , by ( 104 ) , cos . c - cos . a cos . b + sin . a sin . b sin . a sin . b cos . ( a + b ) = cos . a ... half the sum of the sides or s = ( a + b + c ) ; M = ( a + b + c ) = s , N = ( 679 ) if we make in ( 678 ) { N = 1 ...
Page 50
... cos . ( M - N ) y + x Demonstration . The equations ( 738 ) substituted in ( 743 ) give sin . M sin . N x ( 745 ) ... half the sum of two angles of a spherical triangle is to the sine of half their difference , as the tangent of half ...
... cos . ( M - N ) y + x Demonstration . The equations ( 738 ) substituted in ( 743 ) give sin . M sin . N x ( 745 ) ... half the sum of two angles of a spherical triangle is to the sine of half their difference , as the tangent of half ...
Page 51
... cosine of half the sum of two angles of a spherical triangle is to the cosine of half their difference , as the tangent of half the side to ( 758 ) which they are both adjacent is to the tangent of half the sum of the other two sides ...
... cosine of half the sum of two angles of a spherical triangle is to the cosine of half their difference , as the tangent of half the side to ( 758 ) which they are both adjacent is to the tangent of half the sum of the other two sides ...
Page 57
... cos . ( M + N ) - cos . ( M — N ) 2 cos . M cos . N ; and letting S denote half the sum of the angles or S = + ( A + B + C ) ; ( 795 ) ( 796 ) if we make in ( 795 ) , { NA M = ( A + B + C ) = S , ( 797 ) N = 1 ( A + B + C ) = S – A ; we ...
... cos . ( M + N ) - cos . ( M — N ) 2 cos . M cos . N ; and letting S denote half the sum of the angles or S = + ( A + B + C ) ; ( 795 ) ( 796 ) if we make in ( 795 ) , { NA M = ( A + B + C ) = S , ( 797 ) N = 1 ( A + B + C ) = S – A ; we ...
Page 61
... sin . a and sin . 16 ( 803 ) and ( 804 ) are identical , and therefore a b . ( 837 ) : ( 838 ) 73. Corollary . An ... half the sum of. cotan . ( A ' - B ' ) 6 CH . II . § II . ] SPHERICAL OBLIQUE TRIANGLES . 61 or, by transposition, ...
... sin . a and sin . 16 ( 803 ) and ( 804 ) are identical , and therefore a b . ( 837 ) : ( 838 ) 73. Corollary . An ... half the sum of. cotan . ( A ' - B ' ) 6 CH . II . § II . ] SPHERICAL OBLIQUE TRIANGLES . 61 or, by transposition, ...
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Common terms and phrases
A'BC AC the perpendicular adjacent angles angle are known angles are given angles respectively equal AP and PC ar.co B'OC Corollary cosec cosine of half cotan Demonstration differs from 90 equal to 90 fall on AC given angle given sides given value greater than 90 h tang half the sum Hence hypothenuse included angle legs are known Lemma less than 90 Let ABC fig let fall logarithm lunary surface means of 496 middle Napier's Rules negative obtuse opposite angle opposite side perpendicular BP perpendicular to OA planes BOC Problem quotient right angle right triangle fig right triangle PBC Scholium second member Secondly side BC sides and angles sides equal Solution of Spherical solve a spherical solve the triangle Spherical Oblique Triangles spherical right triangle spherical triangle ABC SPHERICAL TRIGONOMETRY substituted surface ABC tang.b tangent of half Thirdly trian triangle ABC figs
Popular passages
Page 1 - A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.
Page 69 - The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. D c A' D' Hyp. In triangles ABC and A'B'C', ZA = ZA'. To prove AABC = ABxAC. A A'B'C' A'B'xA'C' Proof. Draw the altitudes BD and B'D'.
Page 69 - THEOREM. The surface of a spherical triangle is measured by the excess of the sum of its three angles above two right angles, multiplied by the tri-rectangular triangle.
Page 8 - I. The sine of the middle part is equal to the product of the tangents of the adjacent parts.
Page 8 - II. The sine of the middle part is equal to the product of the cosines of the opposite parts.
Page 30 - Any angle is greater than the difference between 180° and the sum of the other two angles.
Page 63 - The cosine of half the sum of two sides of a spherical triangle is to the cosine of half their difference as the cotangent of half the included angle is to the tangent of half the sum of the other two angles. The sine of half the sum of two sides of a spherical...
Page 62 - The sine of half the sum of two sides of a spherical triangle is to the sine of half their difference as the cotangent of half the included angle is to the tangent of half the difference of the other two angles.
Page 71 - ... and the sum of the angles in all the triangles is evidently the same as that of all the angles of the polygon ; hence, the surface of the polygon is equal to the sum of all its angles, diminished by twice as many right angles as it has sides less two, into the tri-rectangular triangle.