# Talk:Differentiation of trigonometric functions

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## Inverses

There should probably also be the inverse trig functions because I can't find another page that has proofs for them. --Bobianite (talk) 01:32, 17 April 2008 (UTC)

Just wondering, since all the proofs for inverse trig functions are almost identical (as I am finding out while I do them), does anyone have a good way to put all of them into one single generalized proof? I tried creating a generalized explanation, but what about a proof? --Bobianite (talk) 01:53, 27 April 2008 (UTC)

The proofs are not identical as far as I can tell. Usually you prove the derivatives for the trig functions and then use those results, with implicit differentiation, to compute the derivatives of the inverse trig functions. Paul Laroque (talk) 01:42, 21 December 2009 (UTC)

Nowhere on the page are the derivatives of arcsec, arccsc, or arccot discussed, proved, or indeed, given. 173.71.106.60 (talk) 18:22, 29 December 2009 (UTC)

## Other things

I think the "a" needs to be included, as in the derivative of cos(ax) is -asin(ax)... right? Well yeah, the "a"s need to be included, to help people like me (Bonzai273 (talk) 16:22, 25 May 2008 (UTC))

In the proof of derivative of sine function, the first principle should be sin'(x)= lim(h->0) (sin(x+h)-sin(x))/h. But it is written cosine something multiplied to sine something. I don't understand how it comes. — Preceding unsigned comment added by 134.160.173.54 (talk) 03:00, 7 July 2011 (UTC)

## Unproven trigonometric limits

The proofs omit proofs that ${\displaystyle \lim _{\theta \rightarrow 0}{\frac {\sin \theta }{\theta }}=1}$ and ${\displaystyle \lim _{\theta \rightarrow 0}{\frac {1-\cos \theta }{\theta }}=0}$. Are they posted anywhere on Wikipedia? --Moly 18:14, 16 October 2008 (UTC)

They are easily obtained by looking at the Taylor series for ${\displaystyle \cos \theta }$ and ${\displaystyle \sin \theta }$. Paul Laroque (talk) 01:35, 21 December 2009 (UTC)
And how do you get the Taylor series unless you already know the derivatives?98.109.232.157 (talk) 06:12, 1 September 2014 (UTC)
The proof supplied seems to be what Ehrenpreis was famous for saying was a logical circle. How do you prove, exactly, that the area is proportional to the subtended angle unless you use calculus to calculate the area? How do you associate a real number to the subtended angle (unless it is a real number, you cannot divide by it so you cannot put an angle in the denominator) without using calculus? (You have to prove the circumference is rectifiable and use calculus to calculate the arclength...integral calculus, which comes after differential calculus, so this is a logical circle.) 98.109.232.157 (talk) 06:10, 1 September 2014 (UTC)
Why does the proof use areas, rather than arc lengths? The fact that the arc length of the circle segment from A to B is squeezed in between the length of the line segments AB and AC is no more obscure than the fact about the areas. In fact, using arc length would seem to be much more direct, as it directly uses the definition of the real number associated to an angle. In fact, the proof of the rectifiability of the circular arc by Archimedes uses exactly the same line segments. This would potentially address the issues mentioned above "circular reasoning". --345Kai (talk) 04:17, 8 September 2014 (UTC)

## Deletion of image leaves article nonsensical

Read through this article, you'll see that a section refers to an image that is no longer there.76.114.70.10 (talk) 19:25, 21 April 2009 (UTC)

## Proof of derivative of the sine cosine function

I cannot understand how you(?) got to the last identity in the article, the one with the multiplication of the two limits. you spend a lot of time proving the sin(x)/x limit, but no time at all explaining the more complicated part.

I would love to know what's going on there.

ManuRoitman (talk) 14:36, 12 July 2011 (UTC)

## Inverse Trig Functions Omit Proof of Differentiability

Unless I'm missing something, the derivation for the derivatives of inverse trig formulas is incomplete. Working with the arcsine function, for example, the proof shows that if ${\displaystyle {d \over dx}\arcsin x}$ exists, it must equal ${\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}}$. Of course the arcsine function is differentiable, but there are plenty of other right inverses for sine on [-1,1] which are not even continuous, let alone differentiable, and nothing in the argument excludes these cases so long as cosine is always positive in the range of arcsine. Of course it follows very easily from the inverse function theorem that for any point p in (-π/2,π/2) sine has a differentiable inverse on a neighborhood of p, which must then be continuous and hence differs from arcsine by a constant (easily proven with LUB property), which of course wouldn't make a difference for differentiability. Hence arcsine is differentiable on (-π/2,π/2) and so the formula given is correct. The somewhat more advanced implicit function theorem can also be used to roughly the same effect as the inverse function theorem. The proof is important and not difficult, but it's certainly above the level intended for the article, so instead of adding it, I wonder if anyone can give a more elementary proof that arcsine is differentiable or a way to mention the issue without making it too complicated. 129.15.139.200 (talk) 22:23, 3 March 2012 (UTC)

## Proofs of derivative of the sine and cosine functions - simplify explanation for sine function proof

I notice that for the proof for the limit of sin(θ)/θ as θ → 0, the variable u is introduced to denote the "chosen unit of measurement". It is not at all clear why this is done. It complicates the explanation and, as it turns out, the u's drop out once you get to the triple inequality stage of the proof. Therefore, I suggest that the references to u be dropped as they are not needed for the proof and may only confuse the reader. I will do the amendments, but will await any feedback before doing so. --Chewings72 (talk) 05:55, 22 April 2012 (UTC)

## An Easier Proof of the Limit of sin(θ)/θ

If point D was on line OA making OA perpendicular to BD, Why not use the length of line BD, arc AB, and line AC? — Preceding unsigned comment added by 24.151.68.8 (talk) 03:17, 18 October 2015 (UTC)

How can you prove that line BD is shorter than arc AB then? Just by visualizing it?--Raycheng200 (talk) 02:46, 23 June 2016 (UTC)

## Better way of proving cos'(x)=-sin(x) using sin'(x)=cos(x)

sin'(x)=cos(x)
sin'(x+π/2)=cos(x+π/2)
cos'(x)=-sin(x)
??? — Preceding unsigned comment added by DarwinKKim (talkcontribs) 03:23, 18 October 2015 (UTC)