December 21st, 2013, 06:27 PM  #1 
Senior Member Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36  How to prove this question?
Prove that d(sin(x))/dx=cos(x),

December 21st, 2013, 07:05 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,898 Thanks: 1093 Math Focus: Elementary mathematics and beyond  Re: How to prove this question? 
December 21st, 2013, 08:03 PM  #3 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: How to prove this question?
Using what definition of "sin(x)" and cos(x)? greg1313 showed how to do it if you are given some definiton, such as the definition in terms of unit circles. that lead immediately to sin(a+ b)= sin(a)cos(b)cos(a)sin(b) and that sine and cosine are continuous at 0. But there are logical difficulties with defining "arc length" for a unit circle before you have defined sine and cosine so some texts use "y= sin(x) is the function satisfying y''+ y= 0 with initial conditions y(0)= 0, y'(0)= 1 and y= cos(x) is the function satisfying y''+ y= 0 with initial conditions y(0)= 1, y'(0)= 0. Using that definition, if y= (sin(x))', then y'= (sin(x))''= sin(x) (because (sin(x))''+ sin(x)= 0) so y''=  sin'(x)= y so that y''+ y= 0. Further y(0)= sin'(0)= 1 and sin''(0)= y(0)= 0. Since y= (sin(x))' satisfies y''+ y= 0, y(0)= 1, y'(0)= 0, y'= (sin(x))'= cos(x). Yet another perfectly valid definition of sin(x) and cos(x) is and We show that sin(x) converges for all x so converges uniformly in any closed and bounded interval. That means that, in particular, sin(x) is "term by term differentiable" at any x. So 
December 22nd, 2013, 09:49 AM  #4  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,898 Thanks: 1093 Math Focus: Elementary mathematics and beyond  Re: How to prove this question? Quote:
From my understanding of the Taylor series for sine, it's necessary to take the derivative of the sine function to arrive at that series.  
December 23rd, 2013, 05:45 AM  #5  
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: How to prove this question? Quote:
Quote:
 
December 23rd, 2013, 05:58 AM  #6  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,898 Thanks: 1093 Math Focus: Elementary mathematics and beyond  Re: How to prove this question? Quote:
 

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