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" In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. "
An Easy Introduction to the Mathematics: In which the Theory and Practice ... - Page 405
by Charles Butler - 1814
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New American Practical Navigator

Nautical astronomy - 1821 - 708 pages
...(supposing arty tide to be the bast, and calling the other two the sides) the sum of the sides is to th-ir difference, as the tangent of half the sum of the angles at the base is to the tangent of half the difference of the same angle*. Thus, in the triangle ABC, if we call...
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Elements of Geometry and Trigonometry: With Notes

Adrien Marie Legendre - Geometry - 1822 - 394 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...
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Euclid's Elements of Geometry: The Six First Books. To which are Added ...

Rev. John Allen - Astronomy - 1822 - 516 pages
...legs AC and CB, and AD their difference ; therefore the sum of the legs AC, CB of the triangle ABC is to their difference, as the tangent of half the sum of the angles CAB and CBA at the Ijase is tQ the tangent of half their difference. PROP. VII. THEOR. If to half the...
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A Course of Mathematics for the Use of Academies, as Well as Private Tuition

Charles Hutton - Mathematics - 1822 - 680 pages
...that the base is to the sum of the other two sides as ^ SPHERICAL TRIGONOMETRY. 25 | as the cosine of half the sum of the angles at the base, to the if cosine of half their difference. . I Ex. 14. How must three trees, A, B, c, be planted, so that...
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The New Practical Builder and Workman's Companion, Containing a Full Display ...

Peter Nicholson - Architecture - 1823 - 210 pages
...BC : : AC - BC : AD - BD. TRIGONOMETRY. — THEOREM 2. 234. The sum of the two sides of a triangle 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. Let ABC be a triangle ; then, of the two sides, CA and...
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Glasgow Mechanics' Magazine, and Annals of Philosophy, Volume 1

Industrial arts - 1824 - 492 pages
...because DA C = AC B, (Euc. 1. 29.) Therefore, DAC+ DCA = 130o, and consequently ADC = of any triangle is to their difference, as the tangent of half the sum of the angles opposite them, is to the tangent of half their difference. Therefore, by logarithms, As, CD + DA =...
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The Second [-fifth and Sixth] Part of A Course of Mathematics: Adapted to ...

Jeremiah Day - Geometry - 1824 - 440 pages
...equal to the sum, and FH to the di/erencc 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^(ACB-fB)...
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A treatise on navigation, and nautical astronomy

Edward Riddle - Nautical astronomy - 1824 - 572 pages
...angle. Shew that radius is to the tangent of the difference between this angle and half a right angle, as the tangent of half the sum of the angles at the base is to the tiuigcnt of half their difference. ELEMENTARY PRINCIPLES OP SPHERICAL TRIGONOMETRY. 1. A...
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A Popular Course of Pure and Mixed Mathematics ...: With Tables of ...

Peter Nicholson - Mathematics - 1825 - 1046 pages
...the proportion AC + CB : AC— CB:: tang (B+C) : tang-i (B—C) it follows that in any triangle the sum of any two sides is to their difference, as the tangent of half the sum of the two angles opposite these sides, is to the tangent of half the difference of these same angles. Let...
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The New American Practical Navigator: Being an Epitome of Navigation ...

Nathaniel Bowditch - Nautical astronomy - 1826 - 732 pages
...any triangle (supposing any side to be the basr, and calling the other two the sides) the sum of the sides is to their difference, as the tangent of half the sum of the angles at the base is to the tangent of half the difference of the tame angles. Thus, in the triangle ABC, if we call...
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