Law of sines 624 Knowing that $\operatorname{sen}(180^{\circ} - x) = \operatorname{sen}x$ and $\cos(180^{\circ} - x) = -\cos x$, determine: a) $\operatorname{sen}120^{\circ}$; b) $\cos 150^{\circ}$; c) $\cos 135^{\circ}$; d) $\operatorname{sen}150^{\circ}$; e) $\operatorname{sen}135^{\circ}$; f) $\cos 120^{\circ}$. Ver respostaa) $\dfrac{\sqrt{3}}{2}$ · b) $-\dfrac{\sqrt{3}}{2}$ · c) $-\dfrac{\sqrt{2}}{2}$ · d) $\dfrac{1}{2}$ · e) $\dfrac{\sqrt{2}}{2}$ · f) $-\dfrac{1}{2}$ 625 Determine the value of $x$ in the following cases: Ver respostaa) $6\sqrt{2}$ · b) $6\sqrt{6}$
Medians, heights and angle bisectors can only be calculated from the three sides. All formulas come out of metric relations — and a single relation, Stewart’s, solves them all at once. Who discovered The median formula is known as Apollonius’ theorem: Papo of Alexandria attributes it to Apollonius of Perga (3rd century BC). The height formula uses the area of the triangle as a function of the sides, the Heron’s formula, demonstrated by Heron of Alexandria in the 1st century, in the Metric (the Arabs attributed it to Archimedes). The relation that holds for any segment connecting a vertex to the opposite side was published by the Scotsman Matthew Stewart in 1746, in the book Some General Theorems of Considerable Use in the Higher Parts of Mathematics.
The Pythagorean theorem has a correction term when the angle is not right. This term is the law of cosines. Who discovered The metric relations of this topic are in The Elements of Euclid (c. 300 BC), in area language: the proposition II.12 deals with the side opposite to an obtuse angle and II.13, opposite an acute angle. The cosine form, $a^2 = b^2 + c^2 - 2bc\cos A$, appeared much later, along with the trigonometric tables. The Persian astronomer Jamshid al-Kashi presented it in 1427, in The Key to Arithmetic; in France, the law of cosines is still called théorème d’Al-Kashi. François Viète wrote it with cosines in 1593.
In any triangle, each side divided by the sine of the opposite angle always gives the same number: the diameter of the circumscribed circle. Who discovered The law of sines was born in astronomy, for triangles drawn on the celestial sphere. The spherical version was obtained in the 10th century by astronomers from the Islamic world — including Abu al-Wafa and Abu Nasr Mansur. The form for flat triangles appears systematically in the Treatise on the quadrilateral (1260) by the Persian Nasir al-Din al-Tusi, who made trigonometry a discipline independent of astronomy. In Europe, it spread with the book On triangles of all kinds, by Regiomontanus, written in 1464 and printed in 1533.