Math Is Fun Forum

kappa_am

Member

Registered: 2013-12-20

Posts: 63

60-coordination

Hi All,
I have a question about decomposing a vector on a 60°-coordination.  In 90°-cordination, it is easy and we are familiar with that. In X-axis direction is |V| cosθ and in Y-axis direction is |V|sinθ. What about 60°-coordination? is there any equation by which we find compositions of a vector in 60°-coordination? For more information, I have shown a 60°-coordination below.

Thank you

Last edited by kappa_am (2018-06-23 06:17:59)

Grantingriver

Member

Registered: 2016-02-01

Posts: 129

Re: 60-coordination

My approach to this problem is to generalize the definition of trigonometric functions by taking the angle

opposite to the hypotenuse to be

. In this case, by using the sine laws, it can be show that:

and hence in your situation the decomposition will be:

Notice that:

The above formulae can be used with any oblique Cartesian coordinate system. I hope that will answer your question.

kappa_am

Member

Registered: 2013-12-20

Posts: 63

Re: 60-coordination

It helps a lot. Thank you very much. could you please introduce me a reference about this subject. I'd like to go deep into that.

Grantingriver

Member

Registered: 2016-02-01

Posts: 129

Re: 60-coordination

Dear kappa_me, to what extent do you what to go deeper. The subject is very broad, we can talk about plain Trigonometry,  spherical Trigonometry, hyperbolic Trigonometry and a mixture of them. The formulae which I have given you is my generalization of  the traditional definition of the Trigonometric functions (which is a paraphrasing of the sines law). If you wonder why they are work, the reason is very simple: The nomenclature of any science is an equivalence class and the definitions into each vicar of the nomenclature aggregate into definitional clusters which have infinitely many vicars that could be used to represent them. If you would like the derivation of these formulae I could provided it to you, also if you have questions about the "plain Trigonometry" (since you are asking about it), please go head and ask them. Finally, if you want a reference about the plain Trigonometry with a lot of solved questions I would recommend "Schaum's Outline of Trigonometry, 5th Edition".

By the way, what is your profession?

kappa_am

Member

Registered: 2013-12-20

Posts: 63

Re: 60-coordination

Dear Grantingriver,
Actually, I have Ph. D in electrical engineer, and like to use this theory in the switching of a converter. I think plain and spherical trigonometry can help me, because in future I may extend this work to 3D switching. I know basics of trigonometry like solving trigonometry equations, some formulas of equivalence, etc all the subject we studied about trigonometry in university. However, generalized trigonometry was totally new to me.  From physical facts, I extracted following equations after posting this thread.
D=√3*v*sin(θ)
H=√3*v*cos(θ+30)
There may be a 3/2 or 2/3 coefficient difference between these and your formulation but it is ok. It depends on voltage vector definition. I really astonished how math matches physical phenomena. thank you again for making me familiar with this subject.
Could you please send me the derivation of these formulas, I also will read the book you refer me to.

thank you again for taking your invaluable time and helping me.

Last edited by kappa_am (2018-06-27 09:15:39)

Grantingriver

Member

Registered: 2016-02-01

Posts: 129

Re: 60-coordination

Hi kappa_am, sorry for delay. In fact sciences are human inventions, so we can modify, change or create any concepts to be suitable to answer our needs (always keep in your mind that you can solve any problem, but you have to figure out how to do it). The derivations are very simple. Suppose that we have an oblique triangle with sides a, b and c which are opposite to the angles

and

, respectively, and suppose further that we have taken the angle

to be opposite to the hypotenuse (c in this case), then from the sines law we have:

Now define the generalized Trigonometric functions (suggested by their traditional definitions) as follows:

But from the sines law we have:

and hence:

Which are the required results.

Note: The fundamental identity can be generalized using the cosine law and the other identities have also their generalized versions, but they are very sophisticated (ugly). Please do not hesitate to ask, if you have any question. It was a great time to meet you.

kappa_am

Member

Registered: 2013-12-20

Posts: 63

Re: 60-coordination

Completely understood.

Thank you

kappa_am

Member

Registered: 2013-12-20

Posts: 63

Re: 60-coordination

Hi,
Could you please tell me what is operant 60° counterclockwise rotation in this coordination? for example, J is operant of 90° rotation in 90° frame which is exp(jπ/2). I convert 60° rotation in 90° frame to this frame by using the equation we talked about. but I did not get correct answers.

I would be very thankful if you give me hint on this subject.

Grantingriver

Member

Registered: 2016-02-01

Posts: 129

Re: 60-coordination

Hi kappa_am, Sorry I just saw the post. What exactly you are trying to do? Please give me more information.