Question

If the sides \(a, b, c\) of a triangle \(A B C\), satisfy the equation \(x^{3}-11 x^{2}+38 x-40=0\), then \(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}\) is (a) \(\frac{16}{9}\) (b) 1 (c) \(\frac{9}{16}\) (d) \(\frac{7}{16}\)

Short answer

Answer: (c) \(\frac{9}{16}\)

Step-by-step solution

Use the Cosine Rule in terms of the triangle sides

Cosine rule for a triangle states that the square of a side is equal to the sum of squares of the other two sides, minus twice the product of the other two sides, times the cosine of the angle between them. Formally, it looks like \(a^{2} = b^{2} + c^{2} - 2bc \cos A\) \(b^{2} = a^{2} + c^{2} - 2ac \cos B\) \(c^{2} = a^{2} + b^{2} - 2ab \cos C\) Now, we need to rewrite the cosine of the angles in terms of the sides.

Rearrange the formulas for the cosine of each angle

Rearrange the cosine rule equations as follows: \(\cos A = \frac{b^{2} + c^{2} - a^{2}}{2bc} \) \(\cos B = \frac{a^{2} + c^{2} - b^{2}}{2ac} \) \(\cos C = \frac{a^{2} + b^{2} - c^{2}}{2ab} \) Now, we can use these relationships to find the sum \(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}\).

Plug in the cosine values into the required sum and simplify

Substitute the values for \(\cos A\), \(\cos B\), and \(\cos C\) that we found earlier into the expression \(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}\). We get: \(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c} = \frac{\frac{b^{2} + c^{2} - a^{2}}{2bc}}{a} + \frac{\frac{a^{2} + c^{2} - b^{2}}{2ac}}{b} + \frac{\frac{a^{2} + b^{2} - c^{2}}{2ab}}{c}\) Now, let's simplify this expression by taking out a common factor: \(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c} = \frac{1}{2}\left(\frac{b^2+ c^2 - a^2}{abc} + \frac{a^2 + c^2 - b^2}{abc} + \frac{a^2 + b^2 - c^2}{abc}\right)\) Now, notice that the sum of the first two terms in each fraction is the same. This sum is equal to \((a^2+b^2+c^2)\). So, we can rewrite the expression as: \(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c} = \frac{1}{2}\left(\frac{a^2+b^2+c^2}{abc}\right)\)

Compare the expression with the given options

Now we need to find which option matches with the expression \(\frac{1}{2}\left(\frac{a^2+b^2+c^2}{abc}\right)\). Since our expression is a ratio, we just need to find the expression for the sum of the cubes of roots, and the product of roots from the given equation. We are given \(x^{3}-11x^{2}+38x-40=0\). By Vieta's formulas, we know that the roots of the polynomial have a sum equal to the second coefficient, and the product equal to the constant term which is – the last coefficient. So, we have: Sum of roots (\(a+b+c\)) = \(11\) Product of roots (abc) = \(40\) Now, we need to find the value of \((a^2+b^2+c^2)\). Recall that \(a+b+c=11\) and \((a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ac)\). We know that the value of \(ab+bc+ac\) can be found by considering the third term in the polynomial, which is \(38\). Now, we have: \(11^2 = a^2+b^2+c^2+2(38)\) \(a^2+b^2+c^2 = 121-76 = 45\) Now, let's plug these values into the expression \(\frac{1}{2}\left(\frac{a^2+b^2+c^2}{abc}\right)\). We get: \(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c} = \frac{1}{2}\left(\frac{45}{40}\right) = \frac{9}{16}\) Therefore, the correct answer is (c) \(\frac{9}{16}\).

Key concepts

Vieta's Formulas

Vieta's formulas, a critical piece within polynomial equation solutions, allow us to find relationships among the roots of a polynomial without actually solving the equation. This mighty tool says that, for a polynomial of the form

\[x^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0\],

the sum of the roots taken one at a time is \(-a_{n-1}\), the sum of the products of the roots taken two at a time is \(a_{n-2}\), and so on, with each succeeding coefficient given a sign based on its position. For a cubic polynomial \(x^3 + a_2x^2 + a_1x + a_0 = 0\), Vieta's formulas tell us particularly that:

• The sum of the roots \(r_1, r_2, r_3\) is \(-a_2\).
• The sum of the products of the roots taken two at a time is \(a_1\).
• The product of the roots is \(-a_0\).

These algebraic shortcuts help solve an array of problems in mathematics, especially when dealing with quadratic or cubic equations. They play a significant role in problems connecting algebra with geometry, like relating the roots of polynomial equations to the elements of a geometric figure such as a triangle.

Trigonometry in Triangles

Trigonometry unfolds within the corners of every triangle, carving the relationship between the angles and lengths of sides. The Cosine Rule – or the Law of Cosines – is a key trigonometric concept used to delve into these relationships. It relates the lengths of the sides of a triangle to the cosine of one of its angles. In a triangle \(ABC\), with sides \(a, b, c\), opposite to angles \(A, B, C\) respectively, the rule is expressed as:

• \(a^2 = b^2 + c^2 - 2bc\cos A\)
• \(b^2 = a^2 + c^2 - 2ac\cos B\)
• \(c^2 = a^2 + b^2 - 2ab\cos C\)

These formulas not only permit the calculation of an unknown angle when we know the lengths of all three sides, but they also allow us to determine an unknown side given two sides and the included angle. Through appropriate manipulation, we can isolate the cosine term to find the measurement of the angle. In educational contexts, understanding the Cosine Rule is essential for solving various geometric problems, notably those involving non-right-angled triangles.

Polynomial Equations

Polynomial equations are like treasure maps that lead to numerous mathematical treasures, from simple curves to complex shapes encoding the fundamentals of algebra. A polynomial equation is constructed as a function that involves variables raised to non-negative integer powers – also known as degrees – and their coefficients.A general polynomial of degree \(n\) is written as \(P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_0\) are constants. Solving polynomial equations often means finding the values of \(x\) that satisfy the equation. There are various methods to handle these equations, including factoring, using the quadratic formula, synthetic division, and graphing.Polynomial equations are vital in finding dimensions, predicting outcomes, and solving real-world problems across disciplines such as physics, engineering, and economics. A profound grasp of polynomial behavior and patterns is a valuable asset in a student's toolkit, empowering them to tackle more elaborate mathematical situations.