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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 $$0< x < \frac{\pi}{2 }$$ .

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