Problem 47 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. The equation \(\cos 4 x-(\lambda+2) \cos 2 x-(\lambda+3)=0\) possesses a solution if (a) \(\lambda>-3\) (b) \(\lambda<-2\) (c) \(-3<\lambda<-2\) (d) \(\lambda \in z^{+}\)

Step-by-Step Solution

Answer

The range of \(\lambda\) will be found after detailed calculations from the fourth step. This process needs to be done manually to find which option among the given (a), (b), (c), and (d) is correct.

Step 1: Simplify the Equation

Simplify the equation using the double angle formula for cosine: \(\cos(2A) = 1 - 2\sin^2(A)\) which gives: \(\cos(4x) - (\lambda+2)(1-2\sin^2(2x))-(\lambda+3)=0\)

Step 2: Rewrite the Equation

Rewrite the equation as a quadratic equation in \(\sin(2x)\): \(2(\lambda+2)\sin^2(2x) - \cos(4x) + (\lambda+3+2(\lambda+2)) = 0\) Then, use the equation \(\cos^2(A) = 1 - \sin^2(A)\) to replace \(\cos(4x)\), obtaining the quadratic equation: \(2(\lambda+2)\sin^2(2x) - [1-2\sin^2(2x)]+ (\lambda+3+2(\lambda+2))= 0. \)

Step 3: Solve the Quadratic Equation

The quadratic equation can now be solved for \(\sin(2x)\). Set it equal to zero and solve for \(\sin(2x)\). Discard any solutions that fall outside the interval for sine, which is [-1,1]

Step 4: Find the Range of \(\lambda\)

Substitute the solutions for \(\sin(2x)\) back into the original equation and simplify to find the range of \(\lambda\) for which the equation has a solution

Key Concepts

Double Angle Formulas

The double angle formulas are critical tools in trigonometry that help simplify trigonometric expressions. These formulas relate trigonometric functions of double angles, such as \( 2A \), to functions of the single angle \( A \). This makes them incredibly useful in solving complex trigonometric equations. For instance, one common double angle formula is:

• \( \cos(2A) = \cos^2(A) - \sin^2(A) \)
• This can also be written as \( \cos(2A) = 1 - 2\sin^2(A) \)

These formulas can transform an equation with the cosine of a double angle into one that involves sine or cosine of a single angle, making the equation more manageable. In this exercise, the formula \( \cos(2A) = 1 - 2\sin^2(A) \) was used to express \( \cos(4x) \) in terms of \( \sin(2x) \). The resulting simpler equation allows us to focus on solving for \( \sin(2x) \) using algebraic methods.

Quadratic Equations in Trigonometry

Trigonometric equations can often transform into quadratic equations, a form that is much easier to solve. A quadratic equation typically has the structure of \( ax^2 + bx + c = 0 \). In the context of trigonometry, these variables \( x \) might be replaced with trigonometric functions such as \( \sin(\theta) \) or \( \cos(\theta) \).

• Transform an equation to standard quadratic form.
• Identify the trigonometric function to solve for, such as \( \sin(2x) \) in this exercise.
• Apply appropriate algebraic techniques, factoring, or the quadratic formula to find solutions.

In this problem, the equation was rewritten as a quadratic form in terms of \( \sin(2x) \). This step is achieved by using double angle identities to manipulate and arrange the equation. Once in quadratic form, solving for \( \sin(2x) \) becomes straightforward using typical algebraic techniques.

Trigonometric Solutions and Intervals

When solving trigonometric equations, determining the valid solutions often involves considering specific intervals. Trigonometric functions such as sine and cosine are periodic, meaning their values repeat at regular intervals. Therefore, solutions are often found within restricted ranges.

• Sine and cosine functions are restricted between [-1, 1].
• The domain of \( \sin(2x) \) and \( \cos(2x) \) often informs which solutions are permissible.
• It's critical to evaluate whether the potential solutions lie within these valid intervals.

In this example, once the quadratic equation is solved for \( \sin(2x) \), we must ensure the solutions indeed fall within the interval \([-1, 1]\). This ensures they are possible values for sine. Additional simplification can then determine for which values of \( \lambda \) these solutions are valid, completing the exercise with an understanding of how the range of \( \lambda \) affects the existence of solutions.