THE cosine law is trigonometric relation used to relate sides and angles on one triangle any, that is, that triangle which does not necessarily have a right angle. Note the following triangle ABC with the measures highlighted:
THE lawFromcosines can be given by one of the following expressions:
The2 = b2 + c2 – 2·b·c·cosα
B2 = the2 + c2 – 2·a·c·cosβ
ç2 = b2 + the2 – 2·b·a·cosθ
Observation: It is not necessary to memorize these three formulas. Just know that the lawFromcosines can always be built. Note, in the first expression, that α is the angle opposite the side whose measure is given by The. We start the formula with the square on the opposite side of the angle that will be used in the calculations. It will be equal to the sum of the squares of the other two sides, minus twice the product of the two sides that are not opposite this angle by the cosine of α.
In this way, the three formulas above can be reduced to:
The2 = b2 + c2 – 2·b·c·cosα
As long as we know that “The" is the measurement on the opposite side of "α", and that "b" and "c" are the measurements of the other two sides of the
triangle.Demonstration
Given the triangle Any ABC, with the measures highlighted in the following figure:
Consider the triangles ABD and BCD formed by the height BD of triangle ABC. Using the Pythagorean theorem in ABD, we will have:
ç2 = x2 + h2
H2 = c2 – x2
Using the same theorem for the triangle BCD, we will have:
The2 = y2 + h2
H2 = the2 - y2
Knowing that there is2 = c2 – x2, we will have:
ç2 – x2 = the2 - y2
ç2 – x2 + y2 = the2
The2 = c2 – x2 + y2
Note in the picture of triangle where b = x + y, where y = b – x. Substituting this value in the result obtained before, we will have:
The2 = c2 – x2 + y2
The2 = c2 – x2 + (b - x)2
The2 = c2 – x2 + b2 – 2bx + x2
The2 = c2 + b2 – 2bx
Still looking at the figure, notice that:
cosα = x
ç
c·cosα = x
x = c·cosα
Substituting this result in the previous expression, we will have:
The2 = c2 + b2 – 2bx
The2 = c2 + b2 – 2b·c·cosα
This is exactly the first of the three expressions presented above. The other two can be obtained analogously to this one.
Example - At the triangle then calculate the measure of x.
Solution:
Using the lawFromcosines, note that x is the measurement of the side opposite the 60° angle. Therefore, the first "number" to appear in the solution should be it:
x2 = 102 + 102 – 2·10·10·cos60°
x2 = 100 + 100 – 2·100·cos60°
x2 = 200 - 200·cos60°
x2 = 200 – 200·1
2
x2 = 200 – 100
x2 = 100
x = ± √100
x = ± 10
As there are no negative lengths, the result should only be the positive value, ie x = 10 cm.
by Luiz Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-lei-dos-cossenos.htm
