December 21st, 2013, 05:27 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, 06:05 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 Math Focus: Elementary mathematics and beyond  Re: How to prove this question? 
December 21st, 2013, 07:03 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, 08:49 AM  #4  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 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, 04:45 AM  #5  
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: How to prove this question? Quote:
Quote:
 
December 23rd, 2013, 04:58 AM  #6  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 Math Focus: Elementary mathematics and beyond  Re: How to prove this question? Quote:
 

Tags 
prove, question 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
one prove question  450081592  Applied Math  1  September 15th, 2010 03:35 AM 
one prove question  450081592  Calculus  0  September 14th, 2010 05:26 PM 
one prove question  450081592  Linear Algebra  1  February 25th, 2010 02:16 AM 
prove question  igalep132  Calculus  8  February 7th, 2008 04:36 AM 
How to prove this question?  jiasyuen  Algebra  0  December 31st, 1969 04:00 PM 