Worth learning complex exponential trig derivations in precalc?

[BWV]

This is a pedagogical /time management / bandwidth / tradeoff question, no argument that learning the complex exponential derivation is valuable, but is it a good strategy for preparing for first year Calculus? my 16YO son is taking AP precalc and AP calc next year and doing well, but struggled with the recent trig unit - If you went back to high school and had to relearn precalc and Calc I, would you learn your trig identities the normal way, or cut to the complex exponential derivation and skip memorizing identities? wondering if it would help or just distract him. complex numbers and exponentials are in the precalc syllabus, but not complex exponentials (at least to my knowledge)

 

[PeroK]

I can't remember things like that anymore, so I have a sheet of paper with them all written down.

 

[jedishrfu]

Some are good to know, others can be derived when needed.

as an example: $sin(x)^2 + cos(x)^2 = 1$ is good to know as are the definitions for tan, cot, sec, csc.

From those basics, these two identities can be derived:

$tan(x)^2 + 1 = sec(x)^2$ and $1 + cot(x)^2 = csc(x)^2$

Others to know are $sin(\alpha + \beta)$ and it’s cosine cousin. From those, half angle, double angle, and subtracting angles can be derived and are a good exercise in handling trig definitions and identities.

https://en.wikipedia.org/wiki/Trigonometric_functions

Reading the wiki article reminds me of the breadth of trig and why it is an important and useful topic to know for its own sake. It pops up everywhere in physics and engineering sometimes in the strangest of places. One would be remiss not to study and appreciate it.

 

[Office_Shredder]

I think the complex representation is most useful for remembering the double angle/half angle formulas, which are not trivially justified through normal geometry as far as I know (yes, I know you can draw a very clever picture and cleverly label it to get the answer)

It's probably not worth it just for that

 

[jedishrfu]

Yes Eulers identity is a remarkable equation and can be used to derive a lot of trig stuff and simplify some integrals if an identity is spotted within.

 

[valenumr]

There are applications where the Euler identities are extremely useful. A particular example is in numerical simulation, as computing an exponential derivate is quite straightforward.

 

[sysprog]

I skipped trig and went straight to calculus -- maybe not my best decision ever.

 

[jedishrfu]

During my review of old math topics, I decided to do a proof of the sine of the sum of two angles usually done geometrically identifying line segments that add up to the answer. It was tough going and I had to look up the proof finally.

Later my coworker mentioned you could do it via the Euler identity. It's true it becomes very trivial. My only beef was, I felt I was using a higher-order identity to prove a lower-order one.

 

[PeroK]

Later my coworker mentioned you could do it via the Euler identity. It's true it becomes very trivial. My only beef was, I felt I was using a higher-order identity to prove a lower-order one.

Using rotation matrices is another easy option. And, that is more elementary.

 

[jedishrfu]

I've not heard of those, I'll have to check them out. Thanks @PeroK !

EDIT: okay I think I know what you mean but I felt they were too advanced too. Basically, I wanted to do it the way the ancients may have done it with the math they knew.

But, yes they are a good way to do this and I didn't think to try them.

 

[PeroK]

A counter-clockwise rotation by $\alpha$ is represented by the matrix:
$$R_\alpha =
\begin{bmatrix}
\cos \alpha & -\sin \alpha\\
\sin \alpha & \cos \alpha
\end{bmatrix}
$$Now, if we take the vector $(\cos \beta, \sin \beta)$ and apply a rotation by $\alpha$, then we should get the vector $(\cos (\alpha + \beta), \sin(\alpha + \beta))$. Hence:
$$
\begin{bmatrix}
\cos (\alpha +\beta)\\
\sin (\alpha + \beta)
\end{bmatrix} =
\begin{bmatrix}
\cos \alpha & -\sin \alpha\\
\sin \alpha & \cos \alpha
\end{bmatrix}
\begin{bmatrix}
\cos \beta\\
\sin \beta
\end{bmatrix} =
\begin{bmatrix}
\cos \alpha \cos \beta - \sin \alpha \sin \beta\\
\sin \alpha \cos \beta + \cos \alpha \sin \beta
\end{bmatrix}
$$

 

[WWGD]

Complex Exponential beyond some basic identities becomes way too difficult and full of traps quickly: Branches, Branch Cuts, using simple-connectedness in existence results. Maybe just keep it somewhat basic at Taylor Series and basic identities.

 

[valenumr]

Complex Exponential beyond some basic identities becomes way too difficult and full of traps quickly: Branches, Branch Cuts, using simple-connectedness in existence results. Maybe just keep it somewhat basic at Taylor Series and basic identities.

I wouldn't agree. For example, in AC circuit analysis, one can just do everything in exponentials without having to think about "algebraic substitution rules", or even trigonometry in general (well, until the final answer). It is very straight forward, no memory by rote required. Wikipedia has some good examples of solving trigonometric integrals and identities using the complex exponential forms as well. I personally find it much easier to think about, and less error prone. But I guess it depends on the problem.

 

[WWGD]

I wouldn't agree. For example, in AC circuit analysis, one can just do everything in exponentials without having to think about "algebraic substitution rules", or even trigonometry in general (well, until the final answer). It is very straight forward, no memory by rote required. Wikipedia has some good examples of solving trigonometric integrals and identities using the complex exponential forms as well. I personally find it much easier to think about, and less error prone. But I guess it depends on the problem.

I have no problem with algebraic manipulations of it, just not going deeper into the theory, which requires knowledge of branches, branch cuts, etc.

 

[jedishrfu]

This is always a problem of math cutting off discussion when it starts to go deeper than the course intentions.

Sometimes this means students never learn about deeper issues unless they are laser focused on places where the instructor skips over something and quiz the teacher or do self investigation. For the vast majority of students we are led along a path and we follow it not deviating left or right.

While we learn the subject, we are blind to the deeper truths.

 

[WWGD]

This is always a problem of math cutting off discussion when it starts to go deeper than the course intentions.

Sometimes this means students never learn about deeper issues unless they are laser focused on places where the instructor skips over something and quiz the teacher or do self investigation. For the vast majority of students we are led along a path and we follow it not deviating left or right.

While we learn the subject, we are blind to the deeper truths.

I think there's a viable middle of the road, where you're presented with the material in greater depth, but not required to really understsnd it. I know it doesn't seem to maje sense, but this will give you a first exposure, after which the second one will seem easier, and so on. The material will ruminate in the back of your mind and maybe click. Because not presenting the more complex material early, will delay getting to understand it. I used to be a bottom-up type that refused to ccontinue until I understood everything. But that's not really very effective. Now, I understand whatever bit I can, andcut often happens that it clicks unexpectedly.

 

[valenumr]

I have no problem with algebraic manipulations of it, just not going deeper into the theory, which requires knowledge of branches, branch cuts, etc.

Im not sure I would even call it theory, but more on the level of axioms. It is just identities that can be used to simplify problems in some cases. It also makes my brain hurt less to only have to think about two equations, vs a bunch of trig identities.

 

[jedishrfu]

I think of it as an ever expanding spiral Of math knowledge revision areas but from a slightly different perspective.

 

[valenumr]

I think there's a viable middle of the road, where you're presented with the material in greater depth, but not required to really understsnd it. I know it doesn't seem to maje sense, but this will give you a first exposure, after which the second one will seem easier, and so on. The material will ruminate in the back of your mind and maybe click. Because not presenting the more complex material early, will delay getting to understand it. I used to be a bottom-up type that refused to ccontinue until I understood everything. But that's not really very effective. Now, I understand whatever bit I can, andcut often happens that it clicks unexpectedly.

I don't know. I feel pretty hindered by my exposure to physics education early on. Just wrong concepts being dumbed down. I mean, classical Newtonian stuff is fine, if not rusty, but trying to learn really modern physics with the installed ideas is tough.

 

[Klystron]

Along with matrices and set theory, I found studying basic transcendental functions very useful for understanding physics and electronics even before formally learning calculus. One can insert or emphasize characteristics of these functions into many STEM subjects where they aid understanding and problem solving. This view implies an affirmative answer to the OP.

The relations between transcendental functions such as sine/cosine, sinh/cosh, exp/log, etc., plus basic identities provide alternate paths to solutions and numerical methods that help a student throughout their career. Students trained primarily in algebraic methods seem to have more difficulty grasping advanced concepts such as complex numbers, non-Euclidean geometries, logarithmic scales, etc., than those also versed in transcendent mathematics.

The most familiar transcendental functions are the logarithm, the exponential (with any non-trivial base), the trigonometric, and the hyperbolic functions, and the inverses of all of these. Less familiar are the special functions of analysis, such as the gamma, elliptic, and zeta functions, all of which are transcendental.

 

[kuruman]

I learned the basic trig identities including the sine of the sum of angles in high school. I memorized them just like I memorized the multiplication table in second grade. Then, also in high school, I saw complex numbers and then I saw complex exponentials. When I saw he derivation of the identity for $\sin(\alpha+\beta)$, my reaction was something like "Isn't that interesting?"

To @BWV: If he were my son, I would stress the importance of memorizing the identities as a means of recognizing what he is looking at when doing algebra involving trig functions. Then I would teach him the rudiments of complex exponentials as a backup in case his memory fails him or in case there is an identity that he has not encountered before, e.g. writing $\sin^3\!\theta$ in terms of first order powers of sines.

A counter-clockwise rotation by α is represented by the matrix $~\dots$

Nice, but are you rotating the vector or the basis?

 

[PeroK]

Nice, but are you rotating the vector or the basis?

That's rotating the vector counter-clockwise and expressing its components in the original basis. E.g. take $\alpha = \frac \pi 2$ and the vector $u = (1, 0)$ is rotated to the new vector $v$:
$$
R_\alpha u =
\begin{bmatrix}
\frac 1 {\sqrt 2} & -\frac 1 {\sqrt 2}\\
\frac 1 {\sqrt 2} & \frac 1 {\sqrt 2}
\end{bmatrix}
\begin{bmatrix}
1\\
0
\end{bmatrix} =
\begin{bmatrix}
\frac 1 {\sqrt 2}\\
\frac 1 {\sqrt 2}
\end{bmatrix} = v
$$That's how I remember which sine is negative!

A change of basis, on the other hand, would rotate the axes and change the components of the vector $u$ to be expressed in the new basis. You can equally derive the same formulas by considering things that way.

 

[kuruman]

Remembering the sine's sign is easy, negative above the diagonal just like the Pauli matrix $\sigma_{\!y}~$. It's what's being rotated that I have to verify by doing a sample rotation like you did.