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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

## 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...

## 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,...

## 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 = 9, = 0.954243...

## 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)...

## A treatise on navigation, and nautical astronomy

Edward Riddle - Nautical astronomy - 1824 - 572 pages
...= cot B, and tan DAC = cot C. PROPOSITION VI. The slim of any two sides of a triangle is to tlieir difference, as the tangent of half the sum of the angles opposite to those sides is to the tangent of half their difference. Let А В С be any plane triangle. Then AB...

## A Popular Course of Pure and Mixed Mathematics ...: With Tables of ...

Peter Nicholson - Mathematics - 1825 - 372 pages
...R 42. From the proportion AC + CB : AC— CB:: tangí (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...

## An Introduction to the Theory and Practice of Plain and Spherical ...

Thomas Keith - Navigation - 1826 - 442 pages
...of triangles are to each other as the chords of double their opposite angles. PROPOSITION IV. (E) 1. In any plane triangle, the sum of any two sides is...their difference, as the tangent of half the sum of their ^opposite angles, is to the tangent of half their difference. Let ABC be any triangle; make BE...

## The New American Practical Navigator: Being an Epitome of Navigation ...

Nathaniel Bowditch - Nautical astronomy - 1826 - 710 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,...