Question

Find the cardinality of the set \( C \) which is defined as \( C = \{ x \mid \sin(4x) = \frac{1}{2} \text{ for } x \in (-9\pi, 3\pi) \} \):

• A. 24
• B. 48
• C. 36
• D. 12

Hint:

When solving trigonometric equations like \( \sin(4x) = \frac{1}{2} \), remember to find the general solution and then restrict it to the desired interval.

Answer 1

The Correct Option is B

Step 1: Solve \( \sin(4x) = \frac{1}{2} \). 
We know that: \[ \sin(4x) = \frac{1}{2} \quad \text{has solutions at} \quad 4x = n\pi + (-1)^n \frac{\pi}{6}, \quad n \in \mathbb{Z}. \] Thus, the general solutions for \( x \) are: \[ x = \frac{n\pi}{4} + (-1)^n \frac{\pi}{24}, \quad n \in \mathbb{Z}. \] Step 2: Find the range of solutions for \( x \) in \( (-9\pi, 3\pi) \).
The interval given is \( (-9\pi, 3\pi) \), so we will determine how many solutions fit into this range by considering the form of the general solution for \( x \). 
Step 3: Count the solutions.
By evaluating the number of integer values of \( n \) that satisfy \( x \in (-9\pi, 3\pi) \), we find that there are 48 distinct values for \( x \).