Problem 246 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ 2 \sin ^{2} x-4 \sin x \cos x+9 \cos ^{2} x>0 $$

Step-by-Step Solution

Answer

The analytic solution to this inequality involves solving a quadratic equation in cosine, which may require the use of trigonometric or numerical techniques. Alternatively, the solution can be estimated using graphical techniques by observing the intervals where the graph of the function is above the x-axis.

Step 1: Convert into quadratic inequality

First, let's use the double angle identities to convert the inequality into a quadratic form. Replace sin^2x and cos^2x with (1-cos(2x))/2 and (1+cos(2x))/2 respectively :\[ f(x) = 2 (1-\cos(2x))/2 - 4 \sin x \cos x+ 9 (1+\cos(2x))/2 \] Next, simplify the sine term using the identity \(\sin (2x) = 2\sin(x)\cos(x)\), so, we get:\[ f(x) = (1-\cos(2x)) - \sin(2x) + (1+\cos^2(2x)) > 0 \] Now, we have a quadratic inequality of the form \(Ax^2 + Bx + C > 0\) where \(x = \cos(2x)\), \(A = 9\), \(B = -4\), and \(C = 2\).

Step 2: Solve the quadratic inequality

This problem is now a quadratic inequality. Set \(f(x) = 0\) and solve for \(x\):\[ 0 = (1 - x) - \sin(2\arccos(x)) + (1+9x^2) = 10x^2 - x - \sin(2\arccos(x)) > 0 \]Using the substitution \(y=\arccos(x)\), \[ 0 = 10\cos^2(y) - \cos(y) - \sin(2y)\] This can be solved using trigonometric techniques or by numerical methods.

Step 3: Graphical interpretation

To complete the solution, graph the function and the inequality to visualize the solutions. An online graphing calculator may be useful for this purpose. The solutions will be the intervals where f(x) > 0. This completes the solution for the inequality.

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where the variable appears in a quadratic form, such as \(Ax^2 + Bx + C > 0\). These inequalities can often be found in problems dealing with parabolas or curves, where the solution requires determining when the quadratic expression is positive, negative, or zero.
Understanding the roots of the equation or where the quadratic crosses the x-axis helps in determining the solution. In this scenario, once our trigonometric inequality is converted to a quadratic one, the task becomes finding intervals where the inequality holds true.

• Set the quadratic expression to zero and find its roots.
• Determine the intervals using test points.
• Decide if the inequality holds in each interval.

By following these steps, we isolate regions where our inequality stays true, leading to a comprehensive solution.

Double Angle Identities

Double angle identities are crucial tools in trigonometry used to simplify expressions involving angles. These identities help transform complex trigonometric expressions by relating different trigonometric functions of double angles to single-angle functions.
For instance:

• \(\sin(2x) = 2\sin(x)\cos(x)\) helps to simplify terms involving a product of sine and cosine.
• \(\cos(2x) = \cos^2(x) - \sin^2(x)\) is another identity useful for reducing expressions with double angles.

In our exercise, these double angle identities allow us to transform terms such as \(\sin^2(x)\) and \(\cos^2(x)\), simplifying the original inequality into a format more easily managed as a quadratic. This makes analysis and solving more straightforward.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. These identities serve as essential tools for simplifying trigonometric expressions and solving various problems involving angles and trigonometry.
Commonly used trigonometric identities include:

• Pythagorean Identity: \(\sin^2(x) + \cos^2(x) = 1\).
• Angle Sum Identities: Such as \(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\).

In solving trigonometric inequalities as in our example, these identities enable us to express complex multiple-angle terms in single angles, simplifying the computation.
Once simplified, the inequality transforms into a more recognizable algebraic form, enabling further analysis and eventual resolution.

Graphical Interpretation

Graphical interpretation is a powerful method to analyze equations or inequalities visually to identify solutions or characteristics such as the roots, extrema, and intervals where a function lies above or below a given line.
By plotting the function derived from our trigonometric inequality, students can visualize the intervals where the curve is above the x-axis to find the solution set for the inequality \(f(x) > 0\).
Steps for graphical interpretation include:

• Sketch or use graphing tools to plot the equation.
• Identify x-intercepts by finding roots where the graph crosses the x-axis.
• Determine regions above or below the x-axis.

Through this approach, students gain a better intuitive grasp of the inequality's behavior, confirming solutions obtained through algebraic methods.