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y=tan(theta) has an asymptote when theta= pi/2 because 1/0 is undefined, and the Taylor Series for tan approaching this point just goes on indefinitely I'm guessing (I'm in gr 11 so I'm new to all this).

When reciprocating tan, how and why are we allowed to reciprocate 1/0. I'm sure there's rules math rules behind reciprocating fractions when dividing. Can someone maybe offer

1) a proof of why reciprocating a fraction when dividing, turning it into multiplication works.

2) If it isn't already clear from the answer to 1), could you explain what allows us to reciprocate 1/0 to 0/1 when converting tan to cot.

when reciprocating something don't we also set restrictions to the denominator. but even then, is tan(theta) the initial function we are setting restrictions on or is cot(theta) the initial function?

What I think might be the answer is that cot is only the inverse of tan when tan or cot has the restriction that theta cannot equal n*(pi/2) where {nEZ}. And in cases where theta is n*(pi/2) where {nEZ}, cot and tan are not the reciprocal of each other.

My teacher insists on using circular arguments, so im stuck

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    $\begingroup$ $\cot\theta = 1/\tan\theta$. $\endgroup$ – amd Apr 22 '19 at 21:58
  • $\begingroup$ Please use MathJax to format your posts. They'll be a lot easier to read. $\endgroup$ – saulspatz Apr 22 '19 at 22:32
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The best way to look at this is geometrically. Picture a unit circle in the $x,y$ plane centered at the origin and let point $A$ be at $(1,0)$. Now draw point $P$ on the circle and let $P$ wander along the circle in such a way that at any given time, angle $AOP$ is $\theta$.

Drop perpendiculars to the $x$ and $y$ axes from $p$; let their (signed) lengths be $X$ and $Y$.

In this geometric picture, the function $\tan \theta$ can be viewed as the ratio $Y/X$ provided $P$ is not on the $y$ axis. We don't try to define $\tan \theta$ when $P$ is on the $y$ axis, because we want to stay away from dividing by zero.

Similarly, the function $\cot \theta$ can be viewed as the ratio $X/Y$ provided $P$ is not on the $x$ axis. Again, we avoid division by zero. Then the fundamental relation holds everywhere that cotangent and tangent can both be defined: $$ \cot \theta = \frac1{\tan \theta} $$ Now we look at how a plot of $\tan \theta$ on a graph, with $\theta$ as the left-right coordinate and $\tan \theta$ as the vertical coordinate. As we approach the "forbidden" point the value gets greater and greater without limit; we say it "approaches infinity" and we note that it gets closer and closer to the vertical line $x = 90^\circ$.

It is not a coincidence that at than same value of $\theta$, $\cot \theta = 0$, because of necessity it will get arbitrarily small as $\tan \theta$ gets arbitrarily large. But we can't, without a good deal more care than you would be able to give at your level, blithely say that "$1/\infty = 0$" or "$1/0 = +\infty$". That works out fine in this case but it would be a bad habit to get into because it can lead to manipulations which give wrong answers in other cases.


ADDED AFTER OP ADDED TO THE QUESTION:

Sorry, I guess this is a "circular" argument as well...

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  • $\begingroup$ (+1) IMO, just because your idea relies on the circle, doesn't necessarily mean it's bad. It provides an intuition from the definition of the functions - and besides, maybe OP's teacher is just not explaining it well, you know? $\endgroup$ – Eevee Trainer Apr 22 '19 at 22:36
  • $\begingroup$ I acknowledge that it approaches infinity, but at the asymptotic line itself, the value is undefined, in terms proving this with variables, your proof would only work if your denominator is never equal to zero. Since the value is undefined, shouldn't there be a discontinuity or something in the line for cot, where the asymptote is for tan?... It took me a solid 10 min to finally catch onto the pun $\endgroup$ – Allan Henriques Apr 22 '19 at 22:37

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