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Numerade Educator

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Problem 5 Medium Difficulty

Find the domain of the function.

$ f(x) = \dfrac{\cos x}{1 - \sin x} $

Answer

The denominator cannot equal $0, \operatorname{so~} 1-\sin x \neq 0 \Leftrightarrow \sin x \neq 1 \Leftrightarrow x \neq \frac{\pi}{2}+2 n \pi .$ Thus, the domain of
\[
f(x)=\frac{\cos x}{1-\sin x} \text { is }\left\{x | x \neq \frac{x}{2}+2 n \pi, n \text { an integer }\right\}
\].

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Video Transcript

let's find the domain of this function. So if we were just looking at a coastline function, it's domain would be all real numbers. And if we were just looking at a sine function, it's domain would be all real numbers. However, here we have a denominator with one minus sign of X, and we know we can't divide by zero. That's undefined. So we need to figure out what would make one minus sign of X equal to zero and then exclude those numbers from the domain. So let's solve this equation for X. So if this is true, then sign of X would be one. And we want to figure out what angles have assigned value of one X is the inverse sign of one. So I'm going to think about my unit circle and on the unit circle. The X coordinate of the point is the co sign and the Y coordinate of the point is the sign. So up at pi over two radiance. We have assigned value of one because we have the 10.1 and nowhere else on the unit circle. Do we have that? However, we could go all the way around a circle again and back to that, so we would be adding to pie. So the original one is pi over two. And if we add two pi to that, we get five pi over two. And if we had to part of that, we get nine pi over two, etcetera. There will be infinitely many. So the way we can describe that is we can say that X is high over to plus two pi times in where two Pi represents going all the way around the circle and end represents any integer so you could go all the way around the circle once all the way around the serval twice and so on. So these are the numbers that we can't have in the domain. So we need to say what is in the domain, and that's pretty tricky. So instead of saying what is in the domain, we're going to say what is not so The domain is the set of X values, so that X is a real number, and X is not equal too high or too plus two pion