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.
Solving any triangle means calculating its main elements: $a$, $b$, $c$, $\hat{A}$, $\hat{B}$ and $\hat{C}$. To do this, it is necessary to provide three information about the triangle, at least one of which is the measurement of a segment linked to it (side, height, median, etc.). The story The first European treatise dedicated entirely to solving triangles, both plane and spherical, was De triangulis omnimodis (“On triangles of all types”), written around 1464 by the German astronomer Regiomontanus and published only in 1533. He organized the solving cases in a very similar way to what we will do here, and the book was the gateway to trigonometry in Renaissance Europe.
We will deduce formulas that allow us to calculate the notable segments of a triangle — heights, medians, internal angle bisectors, radii of the inscribed and circumscribed circles — knowing only the measurements of the sides. In all of them, the semiperimeter $p = \frac{a + b + c}{2}$ appears. The story The formula for the area on three sides is named after Heron of Alexandria (1st century), who demonstrated it in the Metric, a manual of practical measurements. The Persian mathematician al-Biruni, however, attributed it to Archimedes, two and a half centuries earlier. Heron’s demonstration is purely geometric, with the circumference inscribed; the one we will do here is algebraic, starting from the law of cosines.
Four theorems complete the trigonometric properties of triangles: one decomposes each side into the projections of the other two, two give the area and the last, the law of tangents, relates sums and differences of sides to sums and differences of angles. The story The law of tangents was published by the Danish Thomas Fincke in 1583, in the same book in which he named the tangent (see chapter VIII), and François Viète also used it at the same time. It became very practical in the era of logarithm tables: as it only involves quotients and products, it allows you to solve a triangle knowing two sides and the angle between them without adding squares, which the law of cosines would require. With logarithms, each multiplication or division became an addition or subtraction of numbers taken from the table.
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).