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Question
Find the cardinality of the set C which is defined as C={x|sin4x=12forx∈(−9π,3π)}.
🎥 Video solution / Text Solution of this question is given below:
We are given:
sin(4x)=12,x∈(−9π, 3π)
Step 1: General solutions for \sin(θ) = \frac{1}{2}
θ = \frac{\pi}{6} + 2n\pi \quad \text{or} \quad θ = \frac{5\pi}{6} + 2n\pi
Let θ = 4x , so we get:
- x = \frac{\pi}{24} + \frac{n\pi}{2}
- x = \frac{5\pi}{24} + \frac{n\pi}{2}
✅ Step 2: Count how many such x fall in the interval (-9\pi, 3\pi)
By checking all possible n values, we find:
- For x = \frac{\pi}{24} + \frac{n\pi}{2} : 24 valid values
- For x = \frac{5\pi}{24} + \frac{n\pi}{2} : 24 valid values
? Total distinct values = 24 + 24 = 48
✅ Final Answer: \boxed{48}
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