Saturday, 16 April 2016

calculus - Is the proof of the derivative of sin(x) circular?




The proof of ddxsin(x) goes something like this:



limh0sin(x+h)sin(x)h=limh0sin(x)cos(h)+cos(x)sin(h)sin(x)h=limh0sin(x)(cos(h)1)+cos(x)sin(h)h=limh0sin(x)(cos(h)1)h+cos(x)limh0sin(h)h=0+cos(x)×1=cos(x)




My doubt about this is that it uses the limit of sin(h)h and cos(h)1h in the proof. However, proving these two limits uses L'Hopital's rule, which uses the derivative of sin(x) and cos(x) to prove the limits. This causes a circular argument because we're using the derivative of sin(x) to prove the derivative of sin(x). Is there a way to prove these two limits without using L'Hopital's rule or just looking at the graph, or is there a way to find ddxsin(x) without using these two limits?



There is a nice way to prove the limits using geometry here, but I'm wondering if there's a way to do it without using this, either.



Edit: The way I'm defining sine is by the unit circle definition, not the Taylor series one.


Answer




Here is a different approach that uses the integral definition of the arcsine function. We will deduce the limit of interest, limh0sin(h)h=1, without appeal to geometry or differential calculus. (Note that cos(h)1=2sin2(h/2))




Instead, we only rely on elementary analysis of continuous functions and their inverses along with simple properties of the Riemann integral. To that end, we now proceed.







We define the sine function, sin(x), as the inverse function of the function f(x) given by




f(x)=x011t2dt





for |x|<1.




NOTE: It can be shown that the sine function defined as the inverse of f(x) given in (1) has all of the familiar properties that characterize the circular function sin(x).




It is straightforward to show that since 11t2 is positive and continuous for t(1,1), f(x) is continuous and strictly increasing for x(1,1) with limx0f(x)=f(0)=0.



Therefore, since f is continuous and strictly increasing, its inverse function, sin(x), exists and is also continuous and strictly increasing with limx0sin(x)=sin(0)=0.







From (1), we have the bounds (SEE HERE)




1f(x)x11x2




for x(1,1), whence applying the squeeze theorem to (2) yields




limx0f(x)x=1






Finally, let y=f(x) so that x=sin(y). As x0, y0 and we can write (3) as



limy0ysin(y)=1



from which we have





limy0sin(y)y=1




as was to be shown!








NOTE:



We can deduce the following set of useful inequalities from (2). We let x=sin(θ) and restrict x so that x[0,1). In addition, we define new functions, cos(θ)=1sin2(θ) and tan(θ)=sin(θ)/cos(θ).



Then, we have from (2)



ycos(y)sin(y)ytan(y)



which are the familiar inequalities often introduced in an introductory geometry or trigonometry course.




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