Science:Math Exam Resources/Courses/MATH110/December 2016/Question 07 (a)

MATH110 December 2016

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Question 07 (a)
Consider the function $f(x)={\frac {\sin(x)}{1-\cos(x)}}$. (a) Find the domain of ${\textstyle f}$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
At which points does the denominator of the function vanish?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We need to find those points where ${\textstyle f}$ is defined. Since ${\textstyle \sin(x)}$ and ${\textstyle \cos(x)}$ are defined for all real number ${\textstyle x}$, we only need to exclude the points where the denominator vanishes, namely $$1-\cos(x)=0$$ $$\cos(x)=1.$$ The solution is $$x=2k\pi$$ where ${\textstyle k}$ is any integer. Therefore, the domain of ${\textstyle f}$ is all real numbers except integer multiples of ${\textstyle 2\pi }$, or ${\textstyle \mathbb {R} \setminus \{\dots ,-4\pi ,-2\pi ,0,2\pi ,4\pi ,\dots \}}$. Answer: the domain of ${\textstyle f}$ is ${\textstyle {\color {blue}\mathbb {R} \setminus \{\dots ,-4\pi ,-2\pi ,0,2\pi ,4\pi ,\dots \}}}$.