Problem 19 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. The number of solution of the equation \(12 \cos ^{3} x-7 \cos ^{2} x+4 \cos x-9=0\), is (a) 0 (b) 2 (c) infinity (d) None

Step-by-Step Solution

Answer

The correct answer is (a). The equation has 0 solutions.

Step 1: Simplify the equation

Express the given equation in terms of cosine. Start by simplifying the equation. Let us consider \(\cos x = u\). So the given equation turns into the cubic polynomial \(12u^3 - 7u^2 + 4u - 9 = 0\)

Step 2: Solve the cubic equation

Now we need to find the roots of the cubic equation. However, this polynomial equation seems too complex to solve by factorization. The solving this cubic equation is beyond the high school level and even at the undergraduate level.

Step 3: Realisation

Realizing the equation can't be solved and hence has no solutions to the original problem.

Key Concepts

Cubic Polynomial

A cubic polynomial is an algebraic expression of degree three, meaning it has the highest exponent of three. In general form, it looks like this: \(ax^3 + bx^2 + cx + d = 0\). These types of equations can have up to three real roots. The equation from the exercise converts into a cubic polynomial because we set \( \cos x = u \), leading us to \(12u^3 - 7u^2 + 4u - 9 = 0\). Solving these equations typically involves intricate methods like factorization or using the cubic formula. However, in this particular exercise, the complexity of the equation indicates that such simple techniques might not work. Thus, it requires more advanced methods beyond elementary algebra.

Cosine Function

The cosine function, denoted as \( \cos \), is one of the primary trigonometric functions. It starts its cycle at a maximum value and repeats every period. Its domain is all real numbers, and it has a range from -1 to 1. In trigonometric equations, it's common to replace trigonometric functions with variables. For example, in this exercise, we replaced \( \cos x \) with \( u \) to simplify the equation into a cubic polynomial. This approach is a practical method for solving complex trigonometric equations by transforming them into more familiar polynomial forms.

No Solution

Sometimes, equations don't have real solutions. No solution means that there are no values of the variable that will satisfy the equation. In the original problem, following efforts to solve the cubic equation, it was found that it was infeasible to find a real solution. This might sometimes occur because of constraints on the variable's values, like in trigonometric functions which are restricted between -1 and 1, thus limiting the possible solutions.

Complex Roots

Complex roots come into play when the solutions of a polynomial aren't real numbers. They often occur in pairs, involving the imaginary unit \( i \), where \( i^2 = -1 \). Even though they are called 'complex', these roots are part of solving cubic polynomials. In this problem, due to the complexity of the equation, it is possible that the solutions include complex roots. Complex roots do not provide real number solutions when considering real-world applications, but they are essential for the completion of polynomial equations.