First Part of an Elementary Treatise on Spherical Trigonometry1836 - Spherical trigonometry - 71 pages |
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Page 16
... value of the third angle , it can- not but satisfy the conditions of the proposition ( 540 ) . SECTION . II ... given parts with the one to be determined are , in each case , to enter into the same equation . These three parts are ...
... value of the third angle , it can- not but satisfy the conditions of the proposition ( 540 ) . SECTION . II ... given parts with the one to be determined are , in each case , to enter into the same equation . These three parts are ...
Page 18
... given values of hand A. The problem is impossible by ( 535 ) or ( 538 ) , if the ( 564 ) given value of a differs from 90 ° while that of A is equal to 90 ° . h EXAMPLES . 1. Given in the spherical right triangle ( fig . 2. ) , 145 ...
... given values of hand A. The problem is impossible by ( 535 ) or ( 538 ) , if the ( 564 ) given value of a differs from 90 ° while that of A is equal to 90 ° . h EXAMPLES . 1. Given in the spherical right triangle ( fig . 2. ) , 145 ...
Page 20
... given value of the hypothenuse differs more from 90 ° than that of the leg . ( 570 24. Solution . When h and a are both equal to 90 ° , it may be shown , as in ( 563 ) , that the values of B and b are indeterminate . EXAMPLE . Given ...
... given value of the hypothenuse differs more from 90 ° than that of the leg . ( 570 24. Solution . When h and a are both equal to 90 ° , it may be shown , as in ( 563 ) , that the values of B and b are indeterminate . EXAMPLE . Given ...
Page 22
... values , then , of the unknown quantities , given by the tables , as in ( 562 ) , correspond to the triangle ABC , and the other set to A'BС . 27. Corollary . When the given values of a and A are equal , ( 572 ) , ( 574 ) , and ( 575 ) ...
... values , then , of the unknown quantities , given by the tables , as in ( 562 ) , correspond to the triangle ABC , and the other set to A'BС . 27. Corollary . When the given values of a and A are equal , ( 572 ) , ( 574 ) , and ( 575 ) ...
Page 33
... value of b , obtained from ( 634 ) , is the true value , regard must be had to ( 623- ( 636 ) 626 ) . But if all ... given angles which differs the most ( 637 ) from 90 ° . EXAMPLES . 1. Given in the spherical triangle ABC , a ...
... value of b , obtained from ( 634 ) , is the true value , regard must be had to ( 623- ( 636 ) 626 ) . But if all ... given angles which differs the most ( 637 ) from 90 ° . EXAMPLES . 1. Given in the spherical triangle ABC , a ...
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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.