June 8, 2020

ISI MStat 2019 PSA Problem 12 | Domain of a function

Content

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

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 \( 0< x < \frac{\pi}{2 } \) .