Usually we take, $$\tan90^\circ=+\infty$$
But how fair is it? From the tan curve we get $\tan90^\circ$ is inditerminate. Can someone explain please?
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7We don't say that. In a loose sense, the tangent of the right angle is "infinite", but no sign is implied. – Jul 13 '18 at 12:22
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5I don’t say that ... I say it’s undefined ! In a limit procession sense that would make sense – Tolaso Jul 13 '18 at 12:22
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1See Extended real number line. – Mauro ALLEGRANZA Jul 13 '18 at 12:25
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1There was a post to this not long ago that should answer your question...Find the possible values of $\tan x$ – bjcolby15 Jul 13 '18 at 12:49
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1FYI: https://math.stackexchange.com/questions/2424005/what-does-infinity-in-complex-analysis-even-mean/2424111 – Darío A. Gutiérrez Jul 13 '18 at 13:08
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The tangent of $90^{\circ}$ is undefined. $\tan 90^{\circ}=\infty$ is a shorthand way of saying that as $\theta$ gets very close to $90^{\circ},\ \tan \theta$ becomes arbitrarily large; it isn't meant to be taken literally. You'll see a more detailed discussion of this phenomenon when you take calculus.
saulspatz
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+1 ... though I'm not sure a typical calculus course would include a discussion of that particular abuse of notation. – hmakholm left over Monica Jul 13 '18 at 12:28
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@HenningMakholm Right. By "this" I meant "as θ gets very close to 90∘, tanθ becomes arbitrarily large," but I see that I wasn't very clear. I'll see if I can come up with a better way to say it. – saulspatz Jul 13 '18 at 12:32
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This also bothers me because it goes to $-\infty$ from the other side. – Randall Jul 13 '18 at 13:49
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@Randall Good point. I didn't even notice the + sign in the OP's post. – saulspatz Jul 13 '18 at 13:56
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The query "tangent special values" on Google images shows numerous tables of values. Most of them say "not defined", "undefined" or "-" for $\tan 90°$. I saw one with $\pm\infty$, one with "Infinity", and another with $\infty$.
I bet that $+\infty$ is even more scarce and IMO would be a true mistake. So this "usually" is excessive.
By default, Wolfram Alpha considers complex numbers and reports $\tilde\infty$, the complex infinity.
Interestingly, Microsoft Mathematics returns "Indeterminate" in the real mode, and $\tilde\infty$ in the complex one, while the Calculator just says "Invalid input".
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If you've ever seen this, it was abbreviating a left-hand limit; the right-hand limit is if course $-\infty$. Depending on the context, one might only care about a one-sided limit.
J.G.
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IMO, $\tan 90°=+\infty$ is wrong because the positive sign is enforced, whereas this only makes sense for the left-side limit.
$\tan 90°=\pm\infty$ and, more loosely $\tan 90°=\infty$ are more acceptable as they leave room for the indeterminate sign.
In any case, we admit that $\tan90°$ is short for $\lim_{\theta\to90°}\tan\theta$.