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This is a beautiful problem from ISI MStat 2019 PSA problem 12 based on finding the domain of the function. We provide sequential hints so that you can try.
Domain of a function- ISI MStat Year 2019 PSA Question 12
What is the set of numbers \(x\) in \( (0,2 \pi)\) such that \(\log \log (\sin x+\cos x)\) is well-defined?
- \( [\frac{\pi}{8},\frac{3 \pi}{8}] \)
- \( (0,\frac{\pi}{2}) \)
- \( (0,\frac{ \pi}{4}] \)
- \( (0,\pi) \cup (\frac{3 \pi}{2}, 2 \pi) \)
Key Concepts
Domain
Basic inequality
Trigonometry
Check the Answer
But try the problem first…
Answer: is \( (0,\frac{\pi}{2}) \)
Source
Suggested Reading
ISI MStat 2019 PSA Problem 12
Pre-college Mathematics
Try with Hints
First hint
\(logx\) is defined for \( x \in (0,\infty)\).
Second Hint
\(sinx+cosx > 0\).
\(log(sinx+cosx) > 0 \Rightarrow sinx + cosx > 1\)
\( sin(x+\frac{\pi}{4}) > \frac{1}{\sqrt{2}}\)
For \(y\) in \( (0,2 \pi)\) , \(siny > \frac{1}{\sqrt{2}} \iff \frac{\pi}{4} < y < \frac{3\pi}{4 } \)
Final Step
Hence we have \( 0< x < \frac{\pi}{2 } \) .
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