Problem 29 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve: \(\sin 3 x \cdot \cos x+\sin ^{2} x \cos ^{2} x\) \(+\sin x \cdot \cos ^{3} x=1,0 \leq x \leq 2 \pi\)

Step-by-Step Solution

Answer

The solutions for the equation are x=0, π/2, π, 3π/2, or 2π (or any integer multiple of these solutions, if the interval for x wasn't restricted.)

Step 1: Simplification

Utilize trigonometric identities to simplify the given equation. Starting with the first term, the formula \(\sin 2x = 2 \sin x \cos x\) can be a hint to find a useful substitution. If you substitute \(3x = 2x + x\), and apply \(\sin (a + b)\) identity, you end up with \(2 \sin 2x \cos x\).

Step 2: Reduction to Quadratic Form

Substitute y = \(\sin x\) which also implies \(\cos x = \sqrt{1-y^2}\), after replacing these values, the equation reduces to a quadratic form 'ay^2 + by + c = 0'. Solve this quadratic equation for y.

Step 3: Solve for y

In order to find the roots of this equation, we may use the quadratic formula, \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). You may also have multiple roots, since this is based on \(0 \leq x \leq 2\pi\). Check all the roots and substitute back y with \(\sin x\) in order to get x.

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools that help simplify complex expressions involving trigonometric functions. They provide relationships between functions like sine, cosine, and tangent. In our exercise, these identities make the problem much more manageable.

One identity we use is the angle addition formula:

• \( \sin(a + b) = \sin a \cdot \cos b + \cos a \cdot \sin b \)

By applying this identity to \(3x = 2x + x\), the equation simplifies into more familiar terms. Another key identity is \(\sin 2x = 2 \sin x \cos x\), which helps to condense expressions like \(\sin 3x \cdot \cos x\) by re-expressing them in terms of double angles.

The transformation from complex trigonometric expressions to simpler forms is essential, especially when solving equations over specified intervals like \([0, 2\pi]\). Recognize and leverage these identities to break down daunting tasks into solvable pieces.

Quadratic Equations

Quadratic equations come into play when you convert trigonometric expressions into a form that resembles \(ay^2 + by + c = 0\). This conversion is crucial for solving real-world equations that involve squares of sine or cosine.

In the given problem, substituting \(y = \sin x\) allows us to rewrite the original trigonometric equation in a quadratic form. This substitution makes it easier to handle the equation and use standard techniques, like the quadratic formula:

• \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula finds the roots of the quadratic equation, paving the way to discover the values of \(x\).

Quadratic equations often involve multiple roots. It’s important to check which roots satisfy the original equation, particularly within the domain \([0, 2\pi]\). This ensures that all possible solutions are considered, keeping with the constraints of the problem.

Angle Transformations

Angle transformations involve changing or manipulating angles to simplify or solve trigonometric equations. They are especially useful when breaking down complex expressions into more straightforward parts.

Consider \(\sin(3x)\) as an example. By expressing \(3x\) in terms of \(2x\) and \(x\), we use the previously mentioned identities to rethink the angles. This angle manipulation can lead to significant simplifications in the equation.

• Transformations often utilize addition or subtraction formulas which can breakdown challenging expressions into familiar trigonometric relationships.
• Recognizing these transformations allows deeper understanding of periodic functions and their properties over specific intervals.

By transforming angles intelligently, students can draw out hidden patterns or solutions in trigonometric expressions, ensuring a thorough and efficient approach to solving such equations.