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Exercises from chapter X — Any triangles

Before calculating, ask: what was given? Two sides and the angle between them, or the three sides, require the law of cosines; one side and the opposite angle, the law of sines. In this chapter the printed template has several errors; when the answer here differs from him, the difference is explained. Law of cosines 360 Two sides of a triangle measure 8 m and 12 m and form an angle of $120^\circ$ between them. Calculate the third side. See the resolutionCalling $b = 8$ and $c = 12$ the given sides and $\hat{A} = 120^\circ$ the angle between them, the law of cosines gives $$a^2 = b^2 + c^2 - 2bc\cos\hat{A} = 64 + 144 - 2 \cdot 8 \cdot 12 \cdot \left(-\frac{1}{2}\right) = 64 + 144 + 96 = 304,$$then $a = \sqrt{304} = 4\sqrt{19}$ m.

October 24, 2026 Read
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Law of cosines and law of sines

The two laws in this topic are the main tools for working with triangles that are not right angles. Both are demonstrated by reducing the problem to right triangles: plotting a height, in the first, and a diameter, in the second. The story In geometric form, the law of cosines is already found in Euclid’s The Elements (book II, propositions 12 and 13), which compare the square on the opposite side to an obtuse or acute angle with the sum of the squares of the other two — without talking about cosine, of course. In trigonometric form, it appears in Miftah al-hisab (“The Key to Arithmetic”, 1427) of the Persian astronomer Jamshid al-Kashi, and that is why the French call it Al-Kashi’s theorem. The law of sines for flat triangles was enunciated and demonstrated in the 13th century by Nasir al-Din al-Tusi, in his Treatise on the quadrilateral, the first work to treat trigonometry as a discipline independent of astronomy (see chapter III).

October 24, 2026 Read
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Exercises from chapter 15 — Any triangles

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) $6\sqrt{6}$

October 9, 2026 Read
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Calculation of notable lines

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.

October 9, 2026 Read
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Metric relations and cosine theorem

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.

October 9, 2026 Read
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Sines theorem

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.

October 9, 2026 Read