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ISI MStat 2019 PSA Problem 12 | Domain of a function

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

Answer: is $(0,\frac{\pi}{2})$

ISI MStat 2019 PSA Problem 12

Pre-college Mathematics

Try with Hints

$logx$ is defined for $x \in (0,\infty)$.

$sinx+cosx > 0$.
$log(sinx+cosx) > 0 \Rightarrow sinx + cosx > 1$
$sin(x+\frac{\pi}{4}) > \frac{1}{\sqrt{2}}$
For $y$ in $(0,2 \pi)$ , $siny > \frac{1}{\sqrt{2}} \iff \frac{\pi}{4} < y < \frac{3\pi}{4 }$

Hence we have $0< x < \frac{\pi}{2 }$ .