The phrase was popularized in the Western world by François Viète in the 16th century. At the beginning of the 19th century, modern algebraic notation made it possible to write the law of cosine in its current symbolic form. If a = b, that is, if the triangle is isosceles and the two sides at the angle are γ equal, the law of cosine simplifies considerably. Because a2 + b2 = 2a2 = 2ab, the law of cosine This proof requires a slight modification if b < a cos(γ). In this case, the right triangle to which Pytaggoras` theorem is applied moves outside the triangle ABC. This has no effect on the calculation except by replacing the set b − a cos(γ) with a cos(γ) − b. Since this quantity is only included in the calculation by its square, the rest of the proof is not affected. However, this problem only occurs when β is blunt and can be avoided by reflecting the triangle around the γ bisector. If we remember the Pythagorean identity, we get the law of cosine: the law of cosine refers to the relationship between the lengths of the sides of a triangle with respect to the cosine of its angle. It is also known as the cosine rule. If ABC is a triangle, then according to the statement of the law of cosine: With more trigonometry, the law of cosine can be derived using the Pythagorean theorem only once.
In fact, using the right triangle on the left side of Fig. 6 It can be shown that: c 2 = ( b − a cos γ ) 2 + ( a sin γ ) 2 = b 2 − 2 a b cos γ + a 2 cos 2 γ + a 2 sin 2 γ = b 2 + a 2 − 2 a b cos γ , , {displaystyle {begin{aligned}quad c^{2}&=(b-acos gamma )^{2}+(asin gamma )^{2}&=b^{2}-2abcos gamma +a^{2}cos ^{2}gamma +a^{2}sin ^{2}gamma &= b^{2}+a^{2}-2abcos gamma ,end{aligned}}} This is the thesis of Euclid 12 of Book 2 of the Elements. [5] To convert it to the modern form of the law of cosine, note that the law of cosine is useful for calculating the third side of a triangle when two sides and their closed angle are known, and for calculating the angles of a triangle when all three sides are known. The theorem is used in triangulation to solve a triangle or circle, i.e. (see Figure 3): It is important to solve more problems based on the formula of the law of cosine by changing the values of pages a, b & c and the overlap law of the cosine calculator given above. In trigonometry, the law of cosine (also known as the cosine formula, cosine rule, or al-Kashi`s theorem[1]) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as shown in Figure 1, the law of cosine states that we have just seen how to find an angle if we know three sides. It took a few steps, so it`s easier to use the «direct» formula (which is just a rearrangement of the formula c2=a2+b2−2ab cos(C)). It can be in one of the following forms: This formula can be converted into a law of cosine by stating that CH = (CB) cos(π − γ) = −(CB) cos γ.
Thesis 13 contains a completely analogous statement for pointed triangles. If the angle is γ small and the adjacent sides, a and b, are of similar length, the right side of the standard form of the law of cosine is subject to catastrophic suspension in numerical approximations. In situations where this is an important concern, a mathematically equivalent version of the law of cosine, similar to Haversinformel, may prove useful: Euclid`s elements paved the way for the discovery of the law of cosine. In the 15th century, Jamshīd al-Kāshī, a Persian mathematician and astronomer, provided the first explicit statement of the law of cosine in a form suitable for triangulation. He provided accurate trigonometric tables and expressed the theorem in a form suitable for modern use. Since the 1990s, the cosine law in France has always been called the Al-Kashi Theorem. [1] [3] [4] Although the concept of cosine was not yet developed in its time, Euclid`s elements of the 3rd century BC contain an early geometric theorem that almost corresponds to the law of cosine. The cases of blunt triangles and pointed triangles (corresponding to the two cases of negative or positive cosine) are treated separately in theses 12 and 13 of Book 2. Since trigonometric functions and algebra (especially negative numbers) were missing in Euclid`s time, the statement has a more geometric aftertaste: versions similar to the law of cosine for the Euclidean plane also apply on a unit sphere and in a hyperbolic plane. In spherical geometry, a triangle is defined by three points u, v and w on the unit sphere and the arcs of great circles connecting these points. If these large circles form angles A, B and C with opposite sides a, b, c, then the spherical law of cosine states that the following two relations are valid: The law of cosine is a tool for solving triangles.
That said, with some information about the triangle, we can find more. In this case, the tool is useful if you know two sides and their angle included. From there, you can use the law of cosine to find the third side. It works on any triangle, not just right-angled triangles. In hyperbolic geometry, a pair of equations is collectively called the law of hyperbolic cosine. The first is an advantage of this proof is that it does not require the consideration of different cases if the triangle is acute, straight or blunt.
