Problem 38 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. The equation \(\sin ^{4} x-2 \cos ^{2} x+a^{2}=0\) is solvable for (a) \(-\sqrt{3} \leq a \leq \sqrt{3}\) (b) \(-\sqrt{2} \leq a \leq \sqrt{2}\) (c) \(-1 \leq a \leq 1\) (d) None

Step-by-Step Solution

Answer

Therefore, none of the given range of a values in options (a), (b), or (c) match with our found range of a; \(-2 \leq a \leq 2\). So, option (d) None is the correct answer.

Step 1: Review the equation

The given equation is \(\sin ^{4} x-2 \cos ^{2} x+a^{2}=0\).

Step 2: Use trigonometric identity

Use the identity \(1 - cos^2x = sin^2x\) to rewrite the first term in the equation in terms of \(\cos ^{2}x\). So, the equation becomes \( (1-\cos ^{2}x)^{2}-2 \cos ^{2} x+a^{2}=0\), which simplifies to \(1-2\cos ^{2} x+\cos ^{4}x-2 \cos ^{2} x+a^{2}=0\). This further simplifies to \(1-4\cos ^{2} x+\cos ^{4}x+a^{2}=0\).

Step 3: Organizing the equation

Rearrange the terms to set the equation to zero and group the cosines together. The equation becomes \(\cos ^{4}x -4 \cos ^{2} x +1+a^{2}=0\).

Step 4: Notice a pattern

This is a form of a quadratic equation \(z^2 -4z + b = 0\), where \(z = \cos^2x\) and \(b = 1 + a^2\). Solve for \(z\) treating \(b\) as constant.

Step 5: Solve for z

By using the quadratic formula \(z = \frac{-(-4) \pm \sqrt{(-4)^2 - 4*1*b}}{2*1}\), replace \(b\) with \(1 + a^2\), you get \(z = \frac{4 \pm \sqrt{16 - 4(1 + a^2)}}{2}\). The discriminant of quadratic ( the part inside square root ) must be greater than or equal to zero to provide real solution. So, we get \(16 - 4(1 + a^2) \geq 0\).

Step 6: Find the range of a

Solve for \(a\) from the inequality \(16 - 4(1 + a^2) \geq 0\). This simplifies to \(4 - a^2 \geq 0\), then you get \(-\sqrt{4} \leq a \leq \sqrt{4}\), which simplifies to \(-2 \leq a \leq 2\).

Key Concepts

Quadratic Trigonometric Equations

When facing a \textbf{quadratic trigonometric equation}, the approach is largely similar to solving a standard quadratic equation, except that the variable is a trigonometric function. In our exercise, this becomes evident as we are dealing with \( \sin^4x \) and \( \cos^2x \).

To solve such equations, one might need to utilize a combination of algebraic manipulation and trigonometric identities. In the exercise, rearranging the initial equation allows it to take the form of a traditional quadratic, \( z^2 - 4z + b = 0 \) where \( z \) represents \( \cos^2x \). This reformation is crucial as it allows us to tackle the problem using well-known algebraic methods, such as factoring or applying the quadratic formula.

Trigonometric Identities

In trigonometry, \textbf{trigonometric identities} are equations that express one trigonometric function in terms of another. These identities are the backbone of simplifying and solving complex trigonometric equations. In our step-by-step solution, the identity \( 1 - \cos^2x = \sin^2x \) is used to transform \( \sin^4x \) into an expression involving \( \cos^2x \) alone.

Why is this beneficial? Using these identities reduces the number of variables in the equation, making it more straightforward to solve. From the perspective of a student, being fluent in trigonometric identities is akin to having a Swiss Army knife for trigonometric problems; it equips you with the necessary tools to adapt and reframe problems for simpler resolution.

Common Trigonometric Identities

• Pythagorean Identities, like \( \sin^2x + \cos^2x = 1 \)
• Angle Sum and Difference Identities
• Double Angle and Half-Angle Identities
• Product-to-Sum and Sum-to-Product Identities

Memorizing these identities can significantly aid in your problem-solving arsenal as it opens up various paths to tackle equations that initially seem daunting.

Quadratic Formula

The \textbf{quadratic formula} is a powerful tool that provides the solution to any quadratic equation of the form \( ax^2 + bx + c = 0 \). Through this universal formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), one can find the roots of the quadratic equation by simply plugging in the values of \(a\), \(b\), and \(c\).

In our trigonometric context, this formula is used after rewriting the equation as a quadratic with \( \cos^2x \) as the variable. The discriminant, \( \sqrt{b^2 - 4ac} \), tells us if the roots are real or complex. If the discriminant is negative, there are no real solutions. In the exercise, ensuring it is non-negative further provides constraints to the variable \( a \), leading to the solvability conditions for the original equation.

Importance of the Discriminant

The sign of the discriminant is crucial:

• A positive discriminant indicates two distinct real roots.
• A zero discriminant indicates one real repeated root.
• A negative discriminant signals no real roots, only complex ones.

Hence, understanding the role of the discriminant not only allows for the solving of the equation but also informs us about the nature of its solutions. This is particularly useful in trigonometric equations where the solution set can drastically change based on the coefficient values.