Question

If \(\cos 4x = \cos 3x\), find the general solution for \(x\).

• A. \( x = 2n\pi \) or \( x = \frac{2n\pi}{7} \), where \( n \in \mathbb{Z} \)
• B. \( x = n\pi \) or \( x = \frac{n\pi}{7} \), where \( n \in \mathbb{Z} \)
• C. \( x = \frac{2n\pi}{7} \) only
• D. \( x = 2n\pi \) only

Hint:

For \( \cos A = \cos B \), avoid dividing by terms that might be zero.
Always use the general form \( A = 2n\pi \pm B \) to ensure all possible roots are captured in the general solution.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In trigonometry, the general solution provides all possible values of an angle that satisfy a given trigonometric equation.
For the cosine function, the general solution connects two angles that have the same cosine value.
Step 2: Key Formula or Approach:
The formula for the general solution of the equation \( \cos \theta = \cos \alpha \) is:
\[ \theta = 2n\pi \pm \alpha \]
where \( n \in \mathbb{Z} \).
Step 3: Detailed Explanation:
We are given the trigonometric equation:
\[ \cos 4x = \cos 3x \]
Comparing this with the standard form \( \cos \theta = \cos \alpha \), we can identify \( \theta = 4x \) and \( \alpha = 3x \).
Substituting these expressions into the general solution formula, we obtain:
\[ 4x = 2n\pi \pm 3x \]
This equation splits into two separate cases based on the plus and minus signs.
Case 1: Considering the positive sign, we have:
\[ 4x = 2n\pi + 3x \]
Subtracting \( 3x \) from both sides yields:
\[ 4x - 3x = 2n\pi \]
\[ x = 2n\pi \]
Case 2: Considering the negative sign, we get:
\[ 4x = 2n\pi - 3x \]
Adding \( 3x \) to both sides gives:
\[ 4x + 3x = 2n\pi \]
\[ 7x = 2n\pi \]
Dividing by 7, we find:
\[ x = \frac{2n\pi}{7} \]
Combining both cases, the possible solutions are \( x = 2n\pi \) or \( x = \frac{2n\pi}{7} \), where \( n \in \mathbb{Z} \).
Step 4: Final Answer:
The correct option is (A).