Constructing Trigonometric Identities: A Geometric Approach for Solving ODEs

[e^(i Pi)+1=0]

Can't figure out how they're constructed. I'm guessing it would be geometric in nature. It would be nice to be able to derive at least a few of the ones I inevitably forget through disuse.

Math level: ordinary differential equations.

 

[Simon Bridge]

The trig IDs are geometric - yes.
Look for the trig function definitions that use the unit circle:
http://www.haverford.edu/physics/MathAppendices/Trig_Ident.pdf

 

[tim9000]

I hope this is relevant but can anyone tell me what the calculator does when you press inverse sin cos or tan when you take a value? Say you did inverse cos 1/2, it gives you 60 degrees. You can see it on a triangle if you drew cos .. = A/H = 1/2 But as an equation I can't seem to rearrange it to get that...Does this make sense? Or am I just not using my head

Edit:
A thought, is the calculator just doing some algorithm to get the answer close to the value, not actually finding the value? If so does anyone know what the algorithm is? Like guessing a value that fits.

 

[tim9000]

Also forgot to mention, for the thread question I used this page once:

http://math.stackexchange.com/questions/126894/how-to-derive-a-double-angle-identity

 

[Simon Bridge]

@tim9000: depends on the calculator - old pocket calculators just used lookup tables and I think that's what most of the math software does these days. You can still buy the dead-tree version of these tables.

Re: main question - I found an old draft I wrote ages ago:
http://www.scribd.com/doc/149266846/Math-Bits
... its got typos and stuff but should be readable. In particular, see fig.3 on p6.
Trig functions are defined in terms of the ratios on the triangle so stuff like $\cos\theta = A/H$ is a definition ... the LHS is a special way of writing the RHS.

 

[tim9000]

Thanks!
Interesting link

 

[Jheavner724]

Also, you can derive them from simply trying to solve certain trig problems without using them. Although this approach would not function as a proper derivation, and you would have to try a lot of problems.

 

[tim9000]

Simon,
It's funny, lately I've watched quite a few Richard Stallman videos and so it's a coincidence to see your article on Free and open source; also your doc on convincing people to choose freedom over confinement. I think Ted Neslon would agree with you about temporary fixs' becoming perminant.

I'm thinking about making a cluster computer out of 4 raspberry PIs but I've only ever programmed in basic, C and C++ so I don't know if its a good idea.

 

[lurflurf]

What do you mean lookup tables?

Yes most calculators give approximate answers two common methods are equal error polynomials series and cordic. For rough answers by hand I use multiples of 15° and 18° to get within 3°.

 

[tim9000]

What do you mean lookup tables?

I imagine he means like a data sheet for the properties of a materials...but not that, along those lines.

Ok, I'll look into "equal error polynomials series and cordic" cheers.

 

[Simon Bridge]

Yeah - I mean a table of data and an algorithm used to look up the answers on them.
http://www.pearsoned.co.nz/student/primary/product-listing-page/product-details-page/?isbn=9780868636313

The forward calculation is easy - so you do it for lots of angles and put them in a table. The inverse problem is then a matter of just running a finger down the entries in the table. You exploit the properties of the trig functions to make the tables smaller. With computers, you can store very big tables and still look up entries very quickly.

Note:
... cluster computing is fun, but a topic for another thread.
... Software freedom is important to scientific and academic work - also a topic for another thread ;)

 

[tim9000]

Yeah - I mean a table of data and an algorithm used to look up the answers on them.
http://www.pearsoned.co.nz/student/primary/product-listing-page/product-details-page/?isbn=9780868636313

Note:
... cluster computing is fun, but a topic for another thread.
... Software freedom is important to scientific and academic work - also a topic for another thread ;)

If it's published in NZ why is it called 'Statistical and MATH tables'? :P obviously not concentrating on the domestic market (unless they speak more like americans). If I end up starting a thread about either, that I think is of interest I'll send you a link.
Some would say intellectual property is a contradiction in terms, they may be correct.

 

[Simon Bridge]

heh heh - it's the Eton stat and math tables - Eton is in the UK ;D
Actually - wee cultural lesson here: the subject gets abbreviated to "maths" but objects associated with the subject are "math-" whatever. NZ relies on exports so tends to be very globally oriented. Except when it comes to rugby.

Just noticing that OP has yet to respond .. feedback time.