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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 ... - Σελίδα 117
των Dennis M'Curdy - 1846 - 138 σελίδες
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The New American Practical Navigator: Being an Epitome of Navigation ...

Nathaniel Bowditch - 1826 - 617 σελίδες
...AE : : DH : AF or HB ; that is, AD, the sum of the legs AC and CB, is to AE their difference, as DH the tangent of half the sum of the angles at the base (the radius being AH) is to ИВ the tangent of half the difference of the same angles, (to the same...

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

Nathaniel Bowditch - 1826 - 617 σελίδες
...triangle (supposing any aide to be the base, and calling the other two the tide*) the sum of the sida is to their difference, as the tangent of half the sum of tht ongfcs at the base is to the tangent of half the difference of the tame angla. Thus, in the triangle...

An Elementary Treatise on Plane and Spherical Trigonometry: And on the ...

Silvestre François Lacroix - 1826 - 165 σελίδες
...^r;» ^'otn which tang i (a' -f- 6') sin a' + sin 6' we infer, that the sum of the sines of two arcs is to their difference, as the tangent of half the sum of these arcs is to the tangent of half their difference, is obtained immediately by a very elegant geometrical...

The Elements of Euclid: The Errors by which Theon, Or Others, Have Long ...

Robert Simson - 1827 - 513 σελίδες
...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 two sides AB,...

An Analytical Treatise on Plane and Spherical Trigonometry, and the Analysis ...

Dionysius Lardner - 1828 - 317 σελίδες
...plane triangle are as the sines of the opposite angles. (73.) The sum of two sides of a plane triangle is to their difference as the tangent of half the sum of the opposite angles to the tangent of half their difference. •* ^74.) Formulae for the sine, cosine,...

The Southern Review, Τόμος 3

1829
...first of these cases is shewn to depend on the theorem, that, " the sum of two sidi\s of a triangle is to their difference, as the tangent of half the sum of the opposite angles to the tangent of half their difference." This half difference added to half the sum,...

Elements of Surveying: With the Necessary Tables

Charles Davies - 1830 - 300 σελίδες
...should obtain, THEOREM. 44. In any plane triangle, the sum of tfte two sides containing either angle, is to their difference, as the tangent of half the sum of the other two angles, to the tangent of half their difference. Let ABC (PI. I. Fig. 3) be a triangle ;...

A concise system of mathematics

Alexander Ingram - 1830 - 120 σελίδες
...sura. PROP. XXXIX. In any triangle ABC, of which the sides are unequal, the sum of the sides AC + AB is to their difference as the tangent of half the sum of the opposite angles B and C, to the tangent of half their difference. CA + AB : CA — AB : : tan. £ (B...

The Principles of Plane Trigonometry, Mensuration, Navigation and Surveying

Jeremiah Day - 1831 - 370 σελίδες
...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 <if half their difference. Therefore, R :tan(ACH-45°)::tan|(ACB +...

The Principles of Plane Trigonometry, Mensuration, Navigation and Surveying

Jeremiah Day - 1831 - 370 σελίδες
...therefore, from the preceding proposition, (Alg. 389.) that the sum of any two sides of a triangle, is to their difference ; as the tangent of half the sum of the opposite angles, to the tangent of half their difference. This is the second theorem applied to the...




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