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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

The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh ...

Euclid - Geometry - 1810 - 518 pages
...of half their difference. • Let ABC be a plane triangle, the sum of any two sides, AB, AC will be to their difference as the tangent of half the sum of -;' the angles at the base ABC, ACB to the tangent of half their difference. About A as a centre, with AB the greater side for...

An Introduction to the Theory and Practice of Plane and Spherical ...

Thomas Keith - Navigation - 1810 - 482 pages
...xxvi.) (H) In any spherical triangle, I. If the perpendicular fall within the triangle. The co-tangent of half the sum of the angles at the base, Is to tangent of half their difference ; As the tangent of half the •vertical angle, Is to the tangent...

The Modern Preceptor ; Or, a General Course of Education, Volume 1

John Dougall - 1810
...to EF; AD, however, is the sum of the sides, and AE their difference; while DC was shown above to be the tangent of half the sum of the angles at the base, and EF is the tangent of rulf their difference ; the proposition is, therefore, demonstrated. PROP....

Institutes of Natural Philosophy: Theoretical and Practical

William Enfield - Astronomy - 1811 - 428 pages
...side MR. In the triangle SRM, the sides RS, RM, being thus found, the sum of the two sides RS, RM, is to their difference, as the tangent of half the sum of the angles at the base RSM, RMS, is to the tangent of half their difference. To half the sum add half the difference, and...

A Treatise of Plane and Spherical Trigonometry: In Theory and Practice ...

Francis Nichols - Plane trigonometry - 1811 - 128 pages
...of the angles at A and B, may be found by Cor. 32. 1. PROP VI. 61. In any 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. Let ABC be the proposed triangle, whose...

The Theory and Practice of Surveying: Containing All the Instructions ...

Robert Gibson - Surveying - 1811 - 592 pages
...In any Jilane triangle ABC, the sum of the two given sides AB and BC, including a given angle ABC, is to their difference, as the tangent of half the sum of the two unknown angles A and C is to Che tangent of half their difference. Produce AB and make HB=BC, and...

A Course of Mathematics: In Three Volumes : Composed for the Use of the ...

Charles Hutton - Mathematics - 1811 - 422 pages
...k readily converted into a very nsefnl proportion, viz, The sum of the sines of two arcs or angles, is to their difference, as the tangent of half the sum of those arcs ' or angles, is to the tangent of half their difference. 2f . Operating with the third and...

A Course of Mathematics: For the Use of Academies as Well as ..., Volume 2

Charles Hutton - Mathematics - 1812 - 632 pages
...readily converted into a very useful proportion, viz, The sum of the sines of tiuo arcs or angles, is to their difference, as the tangent of half the sum of those arcs or angles, is to the tangent of half their difference. 26. Operating with the third and...