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" In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. In any plane triangle, the sum of any two sides is to their difference as the tangent of... "
Euclid's Elements: Or, Second Lessons in Geometry,in the Order of Simson's ... - Page 117
by Dennis M'Curdy - 1846 - 138 pages
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Annual Report of the Trustees, Volumes 4-8

1853 - 476 pages
...each other as the sides opposite. 5. Find the value of tang. (a+b). 6. In a plane triangle, show that the sum of two sides is to their difference as the tangent of half the sum is to tangent of half the difference of the angles opposite. 7. Find the sin. \ A, and prove that 1...
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A Course of Mathematics: Containing the Principles of Plane Trigonometry ...

Jeremiah Day - Mathematics - 1853 - 5 pages
...therefore, from the preceding proposition, (Alg. 38'.>.) that the sum of any two sides of a. triangle, is to their difference ; as the tangent of half the sum of tin; opposite angles, to the tangent of half their difference. This is the second theorem npplied to...
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Report of twenty-one years' experience of the Dick bequest for elevating the ...

Allan Menzies - 1854
...Suppose AC, CB, and angle C to be given, then rule is, — Sum of the two sides (containing given angle) is to their difference as the tangent of half the...the base is to the tangent of half their difference ; half the sum = ^ (180 — angle C), then having found the half sum, J sum + £ diff. = greater angle,...
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Elements of Geometry and Trigonometry from the Works of A.M. Legendre ...

Charles Davies - Geometry - 1854 - 436 pages
...sides. 90. We also have (Art. 22), a + b : ab :: tan $(A + B) : ta.n$(A — B): tha| is, the sum of any two sides is to their difference, as the tangent of half the sum of the opposite angles to the tangent of half their difference. 91. In case of a right•angled triangle,...
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Elements of Surveying, and Navigation: With Descriptions of the Instruments ...

Charles Davies - Navigation - 1854 - 444 pages
...AC :: sin G : sin B. THEOREM II. In any triangle, the sum of the two sides containing either *ngle, is to their difference, as the tangent of half the sum of the two oilier angles, to the tangent of half their difference. 22. Let ACS be a triangle: then will AB+AC...
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A Treatise on Land Surveying: Comprising the Theory Developed from Five ...

William Mitchell Gillespie - Surveying - 1855 - 436 pages
...triangle, the sines of the angles are to each other as the opposite sides. THEOREM II. — In every plane triangle, the sum of two sides is to their difference as the tangent of half the sum of the angles opposite those sides is to the tangent of half their difference. THEOREM III. — In every plane triangle,...
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Elements of Plane Trigonometry, Surveying and Navigation

William Smyth - Navigation - 1855 - 236 pages
...tan — ~ ; lU —4 a proportion, which we may thus enunciate ; the sum of two sides of a triangle is to their difference, as the tangent of half the sum of the opposite angles is to the tangent of half their difference. Ex. 1. Let AC (fig. 30) be 52. 96 -yds,...
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Navigation

James Pryde - Navigation - 1867 - 458 pages
...add the sides a and b and also subtract them, this will give a + b and a — b/ then the sum of the sides is to their difference as the tangent of half the sum of the remaining angles to the tangent of half their difference. The half sum and half difference being added,...
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Elements of Trigonometry, Plane and Spherical

Lefébure de Fourcy (M., Louis Etienne) - Trigonometry - 1868 - 288 pages
...tang } (A — B) Therefore, a + I _ tang } (A + B) a — b tang} (A — B) *• ; which shows that, in any triangle, the sum of two sides is to their...difference as the tangent of half the sum of the angles opposite to those sides is to the tangent of half their difference. We have A + B=180° — C; hence...
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A Treatise on Land-surveying: Comprising the Theory Developed from Five ...

William Mitchell Gillespie - Electronic book - 1868 - 530 pages
...triangle, the sines of the angles are to each other as the opposite sides. THEOREM II. — In every plane triangle, the sum of two sides is to their difference as the tangent of half the sum of the angles opposite those sides is to the tangent of half their difference. THEOREM III.— In every plane triangle,...
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