I'm working through James Stewart's Precalculus, and I have some confusion regarding this question: "Find the exact value of the trigonometric function at the given real number: $\cot \frac{25\pi}2$."
So, easy enough... $\frac{25\pi}2$ is simply a multiple of $\pi\over2$. $\cot$ is undefined at intervals of $n\pi$ where $n$ is any integer, and the value of $\cot$ at $\pi\over2$ is $0$, as evidenced by the graph below:
Now comes the part, that befuddles me: working it out algebraically, I do the following.
$$\begin{align} \cot \frac{25\pi}2 &= \cot \frac\pi2 \\[4pt] &= \frac{1}{\tan(\pi/2)} \\[4pt] &= \frac{1}{\text{undefined}} \end{align}$$
The value of $\tan$ for $\pi\over2$ is, of course, undefined, as evidenced by the graph below:
It is here that I get stuck:
How is that $\cot$ is defined for that value when $\tan$ itself is not?
I would imagine that a numerator divided by a denominator of an undefined value would be undefined? Could anyone explain how the value of 0 for cotangent would be worked out in terms of tangent. Is it incorrect to relate cotangent and tangent like this:
$$cot = \frac1{tan}$$
I hope that makes sense and is not too a silly a question.
Thanks in advance!