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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... "
Higher Geometry and Trigonometry: Being the Third Part of a Series on ... - Page 68
by Nathan Scholfield - 1845 - 232 pages
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A Treatise of Plane Trigonometry: To which is Prefixed, a Summary View of ...

Jeremiah Day - Logarithms - 1815 - 126 pages
...equal to the sum, and FH to the difference of AC and AB. And by theorem II, [Art. 144.] the sum of the sides is to their difference ; as the tangent of half the sum of the opposite angles, to the tangent of half their difference. Therefore, R:Tan (ACH-45;::Tan tfACB+B)...
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A Practical Application of the Principles of Geometry to the Mensuration of ...

Jeremiah Day - Measurement - 1815 - 388 pages
...equal to the sum, and FH to the difference of AC and AB. And by theorem II, [Art. 144.] the sum of the sides is to their difference ; as the tangent of half the sum of the opposite angles, to the tangent of half their difference. Therefore, R:Tan (ACH-45;::Tan A(ACB +...
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The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh ...

Euclides - 1816 - 528 pages
...them, in a plane triangle, any three being given, the fourth is also given. PROP. III. FIG. 8. IN a plane triangle, the sum of any two sides is to their...difference, as the tangent of half the sum of the angles at the base, to the tangent of half their difference. . Let ABC be a plane triangle, the sum of any...
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Elements of Plane and Spherical Trigonometry: With Their Applications to ...

Olinthus Gregory - Plane trigonometry - 1816 - 276 pages
...cosines being the sines of the complements, it follows from the proposition that the sum of the cosines, is to their difference, as the tangent of half the sum of the complements, is to the tangent of halt' their difference. But half the sum of the complements of two...
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Elements of Geometry and Plane Trigonometry: With an Appendix, and Copious ...

Sir John Leslie - Geometry - 1817 - 454 pages
...cos la + 7 cos5a + 21 cos3a + 35c. ' &e. &c. &c. PROP. IV. THEOR. The sum of the sines of two arcs is to their difference, as the tangent of half the sum of those arcs to the tangent of half the difference. If A and B denote two arcs ; smA+'wB : sin A—...
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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

Euclid, John Playfair - Circle-squaring - 1819 - 348 pages
...the difference between either of them and 45o. * PROP. IV. The sum of any troo 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 ; CA+AB : CA—...
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Geometrical Problems Deducible from the First Six Books of Euclid, Arranged ...

Miles Bland - Euclid's Elements - 1819 - 444 pages
...* The preceding expressions not being easy for calculation, values i . may PROP. XIII. (88.) In any triangle, the sum of any two sides is to their difference as the tangent of the semi-sum of the angles at the base is to the tangent of their semi-difference. Let ABC be any triangle,...
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New Series of The Mathematical Repository, Volume 4

Thomas Leybourn - Mathematics - 1819 - 430 pages
...: BC* : AC*. Required a proof. 8. Prove, geometrically, that in any plane triangle, the sum of the sides is to their difference as the tangent of half the sum of the angles at the base to the tangent of half their difference. 9. Shew that tan.3 60 = 3 tan. 60 to rad. == i....
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Elements of Geometry: Containing the First Six Books of Euclid: With a ...

John Playfair - 1819 - 317 pages
...the cosine of half their difference, as the cotangent of half the angle contained between them, to the tangent of half the sum of the angles opposite to them. COR. 2. If therefore A, B, C be the three angles of a spherical triangle, a, b, c the sides opposite...
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Elements of Geometry and Trigonometry: With Notes

Adrien Marie Legendre - Geometry - 1822 - 367 pages
...principles of Art. 42 and 43 are easily deducible. XL VII. In any rectilineal 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 the difference of those same angles. From the proportion AB :...
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