- #1

- 1,455

- 1,769

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter BWV
- Start date

In summary: 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.

- #1

- 1,455

- 1,769

Science news on Phys.org

- #2

- 26,514

- 18,113

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

- #3

Mentor

- 14,690

- 9,022

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.

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.

Last edited:

- #4

Staff Emeritus

Science Advisor

Gold Member

- 5,583

- 1,511

It's probably not worth it just for that

- #5

Mentor

- 14,690

- 9,022

- #6

- 469

- 192

- #7

- 2,617

- 1,795

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

- #8

Mentor

- 14,690

- 9,022

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.

- #9

- 26,514

- 18,113

jedishrfu said: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.

- #10

Mentor

- 14,690

- 9,022

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.

- #11

- 26,514

- 18,113

$$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}

$$

- #12

Science Advisor

Gold Member

- 6,829

- 10,202

- #13

- 469

- 192

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 said:

- #14

Science Advisor

Gold Member

- 6,829

- 10,202

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

- #15

Mentor

- 14,690

- 9,022

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.

- #16

Science Advisor

Gold Member

- 6,829

- 10,202

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.jedishrfu said:

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.

- #17

- 469

- 192

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

- #18

Mentor

- 14,690

- 9,022

- #19

- 469

- 192

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.WWGD said: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.

- #20

Gold Member

- 1,083

- 1,687

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 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.

Last edited:

- #21

- 14,018

- 7,552

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.

Nice, but are you rotating the vector or the basis?PeroK said:A counter-clockwise rotation by α is represented by the matrix ##~\dots##

- #22

- 26,514

- 18,113

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##:kuruman said:Nice, but are you rotating the vector or the basis?

$$

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.

- #23

- 14,018

- 7,552

It is important to learn complex exponential trigonometric derivations in precalculus because they are fundamental concepts that are used in higher level math courses such as calculus and linear algebra. They also have applications in fields such as physics, engineering, and economics.

Complex exponential trigonometric derivations have many real-world applications, such as modeling the behavior of electrical circuits, analyzing the growth and decay of populations, and understanding the behavior of waves and oscillations in physics. They are also used in signal processing and digital communication systems.

While there are no shortcuts or tricks for memorizing complex exponential trigonometric derivations, practicing and understanding the underlying concepts can make them easier to remember. It is also helpful to break down the derivations into smaller steps and practice them regularly to build familiarity.

It is not recommended to skip learning complex exponential trigonometric derivations in precalculus. These concepts are essential for understanding more advanced topics in math and science, and skipping them may hinder your understanding of future concepts. It is important to have a strong foundation in precalculus in order to succeed in higher level math courses.

One way to make learning complex exponential trigonometric derivations more interesting and engaging is to look for real-world examples and applications. This can help you see the relevance and practical use of these concepts. You can also try to break down the derivations into smaller steps and work on solving practice problems to build your understanding and retention of the material.

Share:

- Replies
- 18

- Views
- 3K

- Replies
- 5

- Views
- 3K

- Replies
- 32

- Views
- 2K

- Replies
- 3

- Views
- 1K

- Replies
- 39

- Views
- 4K

- Replies
- 8

- Views
- 2K

- Replies
- 9

- Views
- 1K

- Replies
- 4

- Views
- 1K

- Replies
- 19

- Views
- 4K

- Replies
- 160

- Views
- 14K