Is it possible to gain intuition into these trig compound angle formulas - and in general, final year high school math?

Question

Does anyone have any insight into the trig sum and difference formulas? The formulas in themselves are very elegant, but I don't really like the proofs that have been given, even the geometric proofs. I feel that none of them address the following points:

cos(A+B) = cosAcosB - sinAsinB

• If you keep one angle constant in this formula, say angle A, and change angle B, why will it trace out the exact cosine curve, just displaced by angle A? This doesn't seem obvious at all from the formula.

• Also, the geometric proof is nice, but it doesn't show (to me) why adding angle A to B is the same as adding angle B to A

• Earlier today I asked if trigonometry could exist in one dimension, and I think the answer is yes

• In another question on this site, someone brought up the idea of matrix multiplication - now I know the mechanics of it, but have literally no idea why it works

So, if anyone had any extra insight into this weird formula, that would really be greatly appreciated!! I realise that this question may seem very strange, maybe even stupid, but hopefully you see where I am sort of coming from. Thanks.

I have been scouring the net for quite a while now but haven't gained much intuition. I do have a sort of obsession with really trying to understand all the formulas that they teach me at high school (final year coming up), and in a way it has delayed my progression in the subject.

So I guess another question would be - is it even possible to gain a deep intuition into high school mathematics, so for example, to feel as natural manipulating these trig equations as multiplication, or manipulating logarithms for example.

Answer 1

I believe you will only be satisfied when you understand $\cos$ and $\sin$ as related to $\exp$ via:

$\cos(z) = \frac{1}{2} ( e^{iz} + e^{-iz} )$ for any $z \in \mathbb{C}$

$\sin(z) = \frac{1}{2i} ( e^{iz} - e^{-iz} )$ for any $z \in \mathbb{C}$

I still think that the best way to define them all is via their power series which arises naturally from solving the first and second order ordinary linear differential equations, which in turn arise from naturally occurring phenomena. Of course, to do so would require various concepts such as limits and differentiability, but it is in my opinion very well motivated. After that, you can proceed as in https://math.stackexchange.com/a/802678/21820, which gives all the fundamental properties, and periodicity especially arises as the path of $\exp(it)$ along the unit circle for real $t$.

Now with these it is immediately clear that the special characteristics of all the common trigonometric identities arise from the characteristics of the exponential function, in particular that $\exp(x+y) = \exp(x) \exp(y)$. You can try proving all of them by just converting them to the equivalent identities for the exponential function. For example here is the particular formula you asked about:

$\cos(a+b) = \frac{1}{2} ( e^{i(a+b)} + e^{-i(a+b)} ) = \frac{1}{2} ( e^{ia} e^{ib} + e^{-ia} e^{-ib} )$

$\cos(a) \cos(b) - \sin(a) \sin(b) = \frac{1}{4} ( e^{ia} + e^{-ia} ) ( e^{ib} + e^{-ib} ) + \frac{1}{4} ( e^{ia} - e^{-ia} ) ( e^{ib} - e^{-ib} )$

$ = \frac{1}{2} ( e^{ia} e^{ib} + e^{-ia} e^{-ib} )$

And if you only need real $a,b$:

$\cos(a+b) = Re(e^{i(a+b)}) = Re(e^{ia}e^{ib}) = Re(e^{ia}) Re(e^{ib}) - Im(e^{ia}) Im(e^{ib})$

$ = \cos(a) \cos(b) - \sin(a) \sin(b)$ where the third equality can be checked directly from the multiplication of complex numbers.

Concerning your side question about matrix multiplication, left-multiplication of a matrix with a vector corresponds to some linear transformation of a vector. To get the desired property that the product of two matrices corresponds to the (non-commutative) composition of the corresponding linear transformations, we necessarily have to define the matrix multiplication exactly the way you have been taught. You should derive it for yourself to see that it is indeed the only way.

Answer 2

When you are at a certain mathematical level you have at your disposal and can use all the material that has been presented to you up to this point. You will observe some patterns, like the addition formulas for the trig functions. In many cases you have a full understanding of what's going on, and in other cases you can parse the proof of such patterns step for step, but you feel uneasy.

Another example: You understand the construction for bisecting an angle, and you hope that the teacher will show you a method for trisecting an angle next week. But, as we all know, there is no such method. The insight why this is so only comes after you have climbed to a higher level of mathematical knowledge and understanding.

Same thing with the trigonometric functions: That they behave as they do is a miracle to you now, but it can be understood from a (moderately) higher mathematical viewpoint. Once you have reached this point it will be obvious to you that all the formulas (and other properties) valid for the trig functions stem from one basic fact: the addition theorem ("law of exponents") for the exponential function, valid not only for real, but also for complex exponents: $$e^{z_1+z_2}=e^{z_1}\cdot e^{z_2}\ .$$