Problem 26 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve: \(\cos x \cos 2 x \cos 4 x=\frac{1}{4}\) \(0 \leq x \leq \pi\)

Step-by-Step Solution

Answer

The solutions to the equation \(\cos x \cos 2x \cos 4x = \frac{1}{4}\) in the interval \(0 ≤ x ≤ \pi\) are \(x = \frac{\pi}{21}, \frac{2 \pi}{21}, \frac{8\pi}{21},\) and \(\frac{10\pi}{21}.\)

Step 1: Use of Product-to-Sum Identity

Rewrite the equation \(\cos x \cos 2x \cos 4x = \frac{1}{4}\) using product-to-sum identities. The identity \(\cos a \cos b = \frac{1}{2} [\cos(a+b)+\cos(a-b)]\) can be used twice. Firstly applying it to \(\cos x \cos 2x\), then to \(\cos(3x)\cos(4x)\).

Step 2: Further Simplification

Keep simplifying the equation. Multiply it by 2 to get \(2 \cos x \cos 2x \cos 4x = \frac{1}{2}\), which can be rewritten as \(\cos 7x = \frac{1}{2}\). This simple equation can be solved using known values of cosine.

Step 3: Solve for x

Finally, we can use the fact that \(\cos(\pi/3) = \frac{1}{2}\) and that cosine is also positive in the fourth quadrant (\(2 \pi - \pi/3\)). Hence the solutions are all \(x = \frac{\pi}{21}\) (from first solution) and \(x = \frac{2 \pi}{21}\) (from second solution) plus any multiples of \(2\pi/7\) (from periodicity of cosine), which lie in the desired interval \(0 ≤ x ≤ \pi\).

Key Concepts

Product-to-Sum Identity

Trigonometric identities are tools that allow us to simplify complex expressions. Among these, the product-to-sum identities convert products of trigonometric functions into sums. This can make equations much easier to handle. Specifically, the identity for cosine is:

• \( \cos a \cos b = \frac{1}{2} [\cos(a+b) + \cos(a-b)] \)

To apply this identity to a problem involving multiple cosines, it helps to manage the terms step by step. For instance, with the equation \( \cos x \cos 2x \cos 4x \), you'd use the identity twice:

• First, convert \( \cos x \cos 2x \) into a sum using the identity.
• Then, in the resulting expression, apply the identity again to combine terms that include \( \cos 4x \).

This approach greatly simplifies the equation, often reducing it to a manageable form that can be solved algebraically.

Cosine Equation Solutions

Solving trigonometric equations like \( \cos 7x = \frac{1}{2} \) involves understanding when the cosine function equals specific values. The cosine function is known to equal \( \frac{1}{2} \) at certain key angles:

• The first is \( x = \frac{\pi}{3} \).
• In the unit circle, cosine is also positive in the fourth quadrant, meaning \( \cos(2\pi - \frac{\pi}{3}) = \frac{1}{2} \), leading to solutions around \( \frac{5\pi}{3} \).

However, in equations with a more complex argument like \( 7x \), these solutions transform:

• We solve for \( x \) by dividing these known angles by 7, such as \( x = \frac{\pi}{21} \).
• Solutions repeat periodically; for \( \cos 7x \), periodicity results in expressions like \( x = \frac{\pi}{21} + \frac{2k\pi}{7} \) for integer \( k \).

By checking each solution within the given interval \( 0 \leq x \leq \pi \), we find the valid values for \( x \).

Periodicity of Trigonometric Functions

Periodicity is a core property of trigonometric functions, meaning these functions repeat their values in regular intervals. This characteristic is crucial for solving equations as it gives us an infinite set of solutions.For cosine, the function repeats every \( 2\pi \) radians. However, when dealing with transformations, the apparent period might differ. In the equation \( \cos 7x \) = \( \frac{1}{2} \), the factor 7 affects the period:

• The new period is \( \frac{2\pi}{7} \).
• This means every solution should account for this change, repeating every \( \frac{2\pi}{7} \) units.

When solving equations, we look for specific solutions within an interval, but thanks to periodicity, a simple adjustment lets us determine all possible solutions. For instance, starting from \( x = \frac{\pi}{21} \), adding periods gives more solutions: \( x = \frac{\pi}{21} + m \times \frac{2\pi}{7} \) for integers \( m \). Periodicity ensures that once you find one solution, you can easily predict others by simply adding the function's period.