Should we re-define Sine?

Question

Sine is usually defined as the ratio of the opposite side to an angle to the hypotenuse in a right angle triangle. Another common definition is based on the unit circle. However I think these geometrical definitions could lead to confusion and misconception. I’ve been wondering about this for a while so now I’m bringing some reasons for this argue and want to know whether this taught is right or not.

Here’s the confusion which this definition causes: When taking oscillatory motion lessons, students are told about the Differential Equations. The very first equation they learn is $$\frac{d^2y}{dx^2}+ky=0$$

Then the solution for that is written as $y=A\sin(\sqrt{k}x+\phi)$ and questions begin to raise: “Where did that sine come from?!”. The typical answer would be: “See, if you take the second derivative and put it in, it satisfies the equation”. “Yes, that’s true but… where is the circle? Where is the angle? There’s just an object connected to a spring”. And so on, many students have problems.

This confusion occurs because, when you first defined the sine function for them, there was a circle, and the argument was an angle and so on. It’s no wonder if those less curious students accept this phenomenon (that an object connected to a spring moves like sine function) as accident or something like that and pass. To solve this, I’m suggesting a re-definition. We can define sine as The answer to $y’’+ay=0$ considering $y(0)=0$ and $y’(0)=1$. It can be proved (with a little effort) that the opposite/hypotenuse ratio also obeys the same differential equation so by [the new] definition, this ratio would be equal to sine of the angle. This should resolve the issues discussed above. Another interesting subject is pi which is related to this discussion. I think I can convince you that pi could be quite confusing:

One day, looking at the formulas in some book or other, I discovered a formula for the frequency of a resonant circuit. There was a mystery about this number that I didn't understand as a youth, but this was a great thing, and the result as that I looked for pi everywhere.

[??Something missing here] which was f = 1/2 pi LC, where L is the inductance and C [is] the capacitance of the [capacitor, and there was also a pi. But where is the] circle? You laugh, but I was very serious then. Pi was a thing with circles, and here is pi coming out of an electric circuit. Where was the circle? Do those of you who laughed know how that comes about? I have to love the thing. I have to look for it. I have to think about it. And then I realized, of course, that the coils are made in circles. About a half year later, I found another book which gave the inductance of round coils and square coils, and there were other pi's in those formulas. I began to think about it again, and I realized that the pi did not come from the circular coils. I understand it better now; but in my heart I still don't know where that circle is, where that pi comes from.

Richard Feynman – “What is science?”

Another example from the famous article:

THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."

Naturally, we are inclined to smile about the simplicity of the classmate's approach. Nevertheless, when I heard this story, I had to admit to an eerie feeling because, surely, the reaction of the classmate betrayed only plain common sense. I was even more confused when, not many days later, someone came to me and expressed his bewilderment [1 The remark to be quoted was made by F. Werner when he was a student in Princeton.] with the fact that we make a rather narrow selection when choosing the data on which we test our theories. "How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?" It has to be admitted that we have no definite evidence that there is no such theory. The preceding two stories illustrate the two main points which are the subjects of the present discourse. The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, …

Eugene Wigner – “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”

So after all these stories from big minds, we can see that curious minds have difficulty relating pi to equations which don’t even involve a circle. But if we instead define pi as half of sine function period (newly defined one), then usage of pi in absence of circles and triangles would not be a surprise.

At the end of this long post, I want to ask for opinions. I wonder if all what I said above makes sense or not? Math is a precise area and definitions are everything, so I think what I’m asking here is an important question.

Edit: A few friends here have marked this as "Opinion Based". Thanks for reading my question, but I really don't see how can a definition be "Opinion Based", in a precise field like math. So the only way I can imagine is that the two be "equivalent", which is what I'm somehow concluding from the answers, but still looking for a clear proof for that point which I don't see here.

Answer 1

The geometric meaning of $\pi$ and $\sin$ is really not that far from those other examples. In the case of the second-order differential equation $y'' + y = 0$, this naturally transforms to the first-order system $$ \eqalign{y' = v\cr v' = -y\cr}$$ Now notice that $y^2 + v^2$ is an invariant for this system: $$\dfrac{d}{dt} (y^2 + v^2) = 2 y v - 2 v y = 0$$ Therefore the trajectories of the system are circles in the $y-v$ plane, and that leads you to $\sin$ and $\cos$.

In the case of the normal distribution, the normalizing factor $1/\sqrt{2\pi}$ comes from the improper integral

$$ J = \int_{-\infty}^\infty e^{-x^2/2}\; dx = \sqrt{2\pi}$$

The standard trick for this is to go to a double integral

$$ J^2 = \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-x^2/2} e^{-y^2/2}\; dx\; dy = \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(x^2+y^2)/2} \; dx\; dy$$

and go to polar coordinates: here come the circles again!

$$ J^2 = \int_0^\infty \int_0^{2\pi} e^{-r^2/2} r \; d\theta \; dr = 2\pi$$

Yes it's a trick, but I might say that the fundamental reason this works is that the joint density of two independent standard normal random variables is invariant under rotations. So again, geometry is there behind the scenes.

Answer 2

That $y''=-y$ is satisfied by sine and cosine comes geometrically from the fact that something rotating around the unit circle at unit speed (something at $(\cos t,\sin t)$ by definition) has acceleration vector pointing towards the center of the circle, which is the negative of the position vector. By this reasoning, I don't think it's too strange that sine appears when solving similar differential equations.

That said, if you have the theorems about differential equations, then it's certainly fine to use an initial value problem to (re)define sine.

Answer 3

You begin your question with:

Sine is usually defined as the ratio of the opposite side to an angle to the hypotenuse in a right angle triangle. Another common definition is based on the unit circle.

This very beginning is the problem with your question, because sine and cosine are defined as you say only in middle and high school. Mathematics students are taught that the series $\sum \limits _{n \ge 0} (-1)^n \frac {x^{2n+1}} {(2n+1)!}$ and $\sum \limits _{n \ge 0} (-1)^n \frac {x^{2n}} {(2n)!}$ converge on $\Bbb R$ absolutely and uniformly. Their respective sums are called, by definition, $\sin x$ and $\cos x$. $\pi$ is defined as the smallest non-zero root of $\sin$. Alternatively, one may show that $\sin$ and $\cos$ as defined above are periodic, with the same period; one then defines $\pi$ as half of this period.

There is nothing to redefine, then. You have just discovered what was already known - but in a slightly more complicated context (because series are conceptually simpler than differential equations - this, in turn, because series are just sequences and sequences are conceptually more elementary than differential calculus). So why do we use the definitions of $\sin$ and $\cos$ from Euclidean geometry? Well, does it seem reasonable to you to introduce the above notions about series to people in middle school?