The number of distinct solution of `cos(x)/(4)=cos(x)` in `x in [0,24 pi]` is

Options

A. 23
B. 24
C. 25
D. 20

Answer

To find the number of distinct solutions of the equation \(\frac{\cos(x)}{4} = \cos(x)\) in the interval \(x \in [0, 24\pi]\), we can follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the equation:
\[ \frac{\cos(x)}{4} = \cos(x) \]
Multiply both sides by 4 to eliminate the fraction:
\[ \cos(x) = 4\cos(x) \]

### Step 2: Simplifying the Equation
Now, we can simplify this equation:
\[ \cos(x) - 4\cos(x) = 0 \]
This simplifies to:
\[ -3\cos(x) = 0 \]
Dividing both sides by -3 gives:
\[ \cos(x) = 0 \]

### Step 3: Finding General Solutions
The cosine function equals zero at odd multiples of \(\frac{\pi}{2}\):
\[ x = \frac{\pi}{2} + n\pi \quad \text{where } n \text{ is an integer} \]

### Step 4: Finding Solutions in the Given Interval
We need to find the values of \(n\) such that \(x\) lies within the interval \([0, 24\pi]\):
\[ 0 \leq \frac{\pi}{2} + n\pi \leq 24\pi \]
Subtracting \(\frac{\pi}{2}\) from all parts:
\[ -\frac{\pi}{2} \leq n\pi \leq 24\pi - \frac{\pi}{2} \]
Dividing the entire inequality by \(\pi\):
\[ -\frac{1}{2} \leq n \leq 24 - \frac{1}{2} \]
This simplifies to:
\[ 0 \leq n \leq 23.5 \]
Since \(n\) must be an integer, the possible values for \(n\) are \(0, 1, 2, \ldots, 23\).

### Step 5: Counting Distinct Solutions
The integer values of \(n\) from \(0\) to \(23\) give us a total of:
\[ 23 - 0 + 1 = 24 \text{ distinct solutions.} \]

### Conclusion
Thus, the number of distinct solutions of the equation \(\frac{\cos(x)}{4} = \cos(x)\) in the interval \(x \in [0, 24\pi]\) is **24**.

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