Why Do We Use Radians in Calculus?

This video starts with a very simple argument for why the choice of unit for angles makes a difference in calculus. Then a detailed proof that the derivative of the sine function is the cosine function is provided. An exercise at the end gives you feedback on your mastery of this topic.

[harrymills]

Yes. It all goes back to h = angle = arc length, when you use radians, in the derivation of the proofs of the derivatives of sine and cosine. In degrees the arc length corresponding to angle h is s = Pi*x/180.

I always just mumbled something about "radians and arc length..." when students asked. *sigh*

[JohnSmith-rftx]

Nice demonstration. I had one question though. At 17:57, you're dividing through by h*cos(h). And you mention that as h is small and positive, this is fine and the inequality signs aren't affected. But the actual limit we're interested in is h->0, not h->0+. I.e. we want the whole limit, not just approaching zero from the positive side. So, it's important that we also look at h->0- isn't it? My impression is that since you're using a geometric argument in the proof, and a "negative angle" doesn't really have meaning in that context, maybe this isn't actually an issue. Is that right? If you allow for "negative angles", I think you might have to start over with the areas and have the relevant areas of the sectors including the absolute value of h instead of just the naked h. I'm sure the end result is the same, but a mention of why we can just ignore the small, negative h case or how it is dealt with would be better than no mention.

[anilmishrassblintercollege]

Excellent, sir why we use radian in inverse circular function

[ankurgoyal]

I have a question in calculus and that i never the got answer and searched online also but couldnt find satisfactory result. What does limit at infinity actually mean please help me when can find limit at a certain point this make sense but when limit at infinity what does this actually please tell.

[lvpsitv-larongvolleyballpa]

Can i ask? Do limits must be in radian mode? like sin 0.6?

[Pgan]

It would mean more if you can demonstrate why using "degrees" would go bonkers compared to using "rad" in a real world application eg Sniper Bullet leaving the barrel.