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" 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. "
A Treatise of Plane and Spherical Trigonometry: In Theory and Practice ... - Page 12
by Francis Nichols - 1811 - 128 pages
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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

John Playfair - Euclid's Elements - 1842 - 332 pages
...parallel to FG, CE : CF : : BE ; BG, (2. 6.) that is, the sum of the two sides of the triangle ABC is to their difference as the tangent of half the sum of the angles opposite to those sides to the tangent of half their difference. PROP. V. THEOR. If a perpendicular...
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A Treatise on Plane and Spherical Trigonometry: Including the Construction ...

Enoch Lewis - Conic sections - 1844 - 240 pages
...sine of A ; these sines being suited to any radius whatever (Art. 27). QED ART. 30. In any right lined triangle, the sum of any two sides is, to their difference, as the tangent of half the sum of the angles, opposite to those sides, to the tangent of half their difference. Let ABC be the triangle;...
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A Series on Elementary and Higher Geometry, Trigonometry, and Mensuration ...

Nathan Scholfield - Conic sections - 1845 - 542 pages
...have, by the proposition, a sin. A ' b a sin. B sin. A c sin. C sin. B b PROPOSITION III. In any plane triangle, the sum of any two sides, is to their difference, as the tangent of half the sum of the angles opposite to them, is to the tangent of half their difference. Let ABC be any plane triangle,...
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Higher Geometry and Trigonometry: Being the Third Part of a Series on ...

Nathan Scholfield - Conic sections - 1845 - 244 pages
...According to this, we shall have, by the proposition, a sin. A.~ c b sin. 68 FROPOSITION III. In any plane triangle, the sum of any two sides, is to their difference, as the tangent of half the sum of the angles opposite to them, is to the tangent of half their difference. Let ABC be any plane triangle,...
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Elements of plane (solid) geometry (Higher geometry) and trigonometry (and ...

Nathan Scholfield - 1845 - 896 pages
...proposition, sin. A' a ~b a c b sin. B sin. A sin. C sin. B sin. C. 68 PROFOSITION in. In any plane triangle, the sum of any two sides, is to their difference, as the tangent of half the sum of the angles opposite to them, is to the tangent of half their difference. Let ABC be any plane triangle,...
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The First Six, and the Eleventh and Twelfth Books of Euclid's Elements: With ...

Euclid, James Thomson - Geometry - 1845 - 380 pages
...proposition is a particular case of this PROP. III. THEOR. — The sum of any two sides of a triangle 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. Let ABC be a triangle,...
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Plane Trigonometry and Mensuration for the Use of the Royal Military College

William Scott - Measurement - 1845 - 290 pages
...b : a — b :: tan. | (A + в) : tan. (A — в).* Hence the sum of any two sides of a triangle, is to their difference, as the tangent of half the sum of the angles oppo-* site to those sides, to the tangent of half their difference. SECT. T. EESOLUTION OF...
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Euclid's Elements: Or, Second Lessons in Geometry,in the Order of Simson's ...

Dennis M'Curdy - Geometry - 1846 - 168 pages
...AC+sin. AB : sin. AC—sin. AB : : tan. J(AC-(AB): tan. J(AC—AB). QED 4 Th. In any triangle, the sum of two sides is to their difference, as the tangent of half the sum of the angles at the base is to the tangent of half their difference. Given the triangle ABC, the side AB...
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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

Euclid, John Playfair - Euclid's Elements - 1846 - 332 pages
...difference between either of them and 45. PROP. IV. THEOR. The sum of any two sides of a triangle is to their difference, as the tangent of half the sum of the angles opposite to those sides, to the tangent of half their difference. Let ABC be any plane triangle...
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Pantology; or, A systematic survey of human knowledge

Roswell Park - 1847 - 622 pages
...an oblique angled triangle, the sides are proportional to the sines of the opposite angles : also, the sum of any two sides is to their difference, as the tangent of the half sum of the two opposite angles, is to the tangent of their half difference : and finally,...
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