sin.html

A trig question: Who first knew that sin' = cos?

The standard proof which occurs in nearly all modern calculus texts that the derivative of the sine is the cosine involves using the addition formula for the sine together with the facts, which are usually established geometrically, that and approach 1 and 0 respectively as t approaches 0. When you start looking back through old calculus books, the proofs are much different. Here is one taken verbatim from Professor Olney's General Geometry and Calculus (New York, 1886)

Thm: If y = sin x, then dy = cos(x) dx

Proof: Let x take an infinitesimal increment dx, and let dy represent the contemporaneous infinitesimal increment of y. Then the consecutive state of the function is y + dy = sin(x + dx) = sin(x) cos(dx) + cos(x) sin(dx). In the limit cos(dx) = 1 and sin(dx) = dx. So y + dy = sin(x) + cos(x) dx. Subtract y = sin(x) and arrive at the conclusion dy = cos(x) dx. QED

Now Professor Olney remarks in a footnote that 'the student may be inclined to say that in the limit sin(dx) = 0, and though there is no error in that it is equally true that in the limit sin(dx) = dx and we do not wish dx to vanish from the equation'.

This kind of argument is almost universally eschewed by mathematician today (and actually before Olney's time) as so much gibberish because the term infinitesimal increment is not well-defined. It does have some elements of a correct argument however(the addition formula is used), so I would give him partial credit. It makes for a 'plausibility argument' or a 'first approximation' of a proof that sin' = cos.

When we search for results about the sine function from Newton's time (1660s and 70s), we see a lot of power series. These were treated as merely polynomials of infinite degree with not much attention paid to intervals of convergence.

Newton's work

Newton obtained the series for sine and cosine in his unpublished paper "Treatise on the methods of series and fluxions" (1671).

> sin(x) =series(sin(x),x,12);

> cos(x) = series(cos(x),x,12);

The manner in which he did this is interesting.

First, he got a power series for y = arcsin(x) by

[Maple Plot]

In the diagram, we know that the area of the pieslice is half the angle y/2, but that is also

[Maple Math] .

Hence arcsin(x) = [Maple Math] . Next, Newton obtained a power series for and integrated it term by term using Cavalier's integral formula .

> sqrt(1-t^2) =series(sqrt(1-t^2),t,12);

> Int(sqrt(1-t^2),t=0..x) =int(subs(O=0,series(sqrt(1-t^2),t,12)),t=0..x)*`...`;

[Maple Math]

> arcsin(x) = 2*(x-1/6*x^3-1/40*x^5-1/112*x^7-5/1152*x^9-7/2816*x^11)-
x*(1-1/2*x^2-1/8*x^4-1/16*x^6-5/128*x^8-7/256*x^10);

> eqn :=subs(arcsin(x)=y, expand(%));

>

Next, Newton had neat method for inverting a power series (that is, solving the equation for x in terms of y).

He would set x = 0 + a[1]*y + a[2]*y^2 + a[3]*y^3 + a[4]*y^4 + `...`, plug it into eqn, expand and successively obtain the coefficients of the series. Note, he knew that a[0] = 0, since arcsin(0) = 0.

> eqn2 :=subs(x = 0 + a[1]*y + a[2]*y^2 + a[3]*y^3 + a[4]*y^4,eqn);

>

> eqn3 :=collect(expand(eqn2),y):

> eqn4 := collect(eqn3 - (y=y),y):

> coeff(rhs(eqn4),y,1)=0;
coeff(rhs(eqn4),y,2)=0;
coeff(rhs(eqn4),y,3)=0;
coeff(rhs(eqn4),y,4)=0;

>

By inspection then, [Maple Math] , etc, so

[Maple Math]

To get the power series for the cosine, Newton substituted the series just obtained into the series for [Maple Math] .

Of course, you realize that no attention was paid at this time to the determination of the interval of convergence of these series or to where they actually represented the function in question.

In any case, Newton had the power series for sin and the cos

> sin(x) = series(sin(x),x,12);
cos(x) = series(cos(x),x,12);

He had no problem with integrating series term by term, so he could tell by inspection that sin is an antiderivative of cos. Hence he must have know that the derivative of the sin is the cos, but he probably did not think to ask.

Leibnitz's work

Leibnitz in the 1670's reinvented calculus and then some. He introduced the notion of the infinitesimal increment and the differential, and the notation we use in calculus today. He drew 'differential triangles' on a curve y = f(x) and used that to write down a differential equation for the function. For example, he obtained the second order equation for the sin function

[Maple Math]

in this manner.

[Maple Plot]

In the diagram, the hypotenuse and base of the large yellow right triangle are 1 and [Maple Math] respectively. The large yellow right triangle and the small blue differential triangle are similar so . (Right here he could have seen that = cos(t) satisfies this equation.) Square both sides and get = . Leibnitz then considers the differential of t as a constant and take the differential of both sides to get [Maple Math] . Cancelling 2*dy from both sides and dividing by dt^2, we get the equation expressing the fact that the second derivative of the sine is -sine. Leibnitz then proceeded to get the power series for the sine by a 'method of undetermined coefficients' similar to Newtons. Except he plug the series into the differential equation and used the equations resulting from that to get the coefficients of the series for sine.

So we can say that Leibnitz knew the second derivative of the sin, even if he didn't think to ask for the first derivative.

Problem. Carry out Leibnitz's method to get the first 2 nonzero coefficients of the series for sine.