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Thread: trig under the hood

  1. #1 trig under the hood 
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    hey guys,

    any chance you could recommend any reading (book/internet) on exactly why and how the trig ratios (sine, cos etc) work...

    i've read on wikipedia about something called a taylor series, but honestly that's a little out of my league!

    i'm really interested in understanding exactly how this works...

    thanks!


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    Before one of those math wizards comes on here and baffles you completely, I am going to give you my dumbed down version of trigonometry.

    Picture a triangle of any shape. It doesn't have to be a right triangle. Imagine that triangle scaled up or scaled down. Zoom in or zoom out. It should be obvious that the angles stay the same, and the three sides stay in the same proportion. We could completely define that family of similar triangles by listing the ratios of each side to the other. Or we could list the three angles. Any triangle with the same 3 angles will have the same ratio of their corresponding sides. We could make up a table of triangles and their angles and the ratios of the sides.

    Instead of trying to make tables for all combinations of 3 angles we decide to stick to right triangles. Then our table only has to have one angle because the other two will be the right angle and 90 degrees minus the angle in our table (since all triangles have 180 degrees). The three sides can be defined as the side opposite the angle in our table, the adjacent side and the hypotenuse, which is the long side. For each angle we can measure the ratios of these sides. We call these ratios the sine (opposite over hypotenuse), cosine (adjacent over hypotenuse) and tangent (opposite over adjacent).

    This is exactly the kind of table I had when I learned trig, back in the days before we had calculators.

    It turns out these ratios are very handy, which you will find out by working out some problems in a textbook.


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    hey harold, thanks for the very helpful reply!

    cool, i do understand about the trig tables and ratios etc, but there are an infinite number of angles right? so generating a table to accommodate all the possible angles would be impossible... my question is how do you actually work out the ratio for an arbitrary angle, i.e. as you would using a calculator/computer?

    could you give examples for sin, cos and tan if possible? thanks!
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  5. #4  
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    for example, taylor series (how does this work?):

    http://upload.wikimedia.org/math/0/0...bb9f19d69f.png

    is there a simpler way?
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    Quote Originally Posted by rgba
    hey harold, thanks for the very helpful reply!

    cool, i do understand about the trig tables and ratios etc, but there are an infinite number of angles right? so generating a table to accommodate all the possible angles would be impossible... my question is how do you actually work out the ratio for an arbitrary angle, i.e. as you would using a calculator/computer?

    could you give examples for sin, cos and tan if possible? thanks!
    We used to interpolate the tables. I thought you wanted to stay away from the taylor series and such. I'll probably have to step aside and let the real mathematicians weigh in on that one.
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    cool, so just interpolation... thanks for the great help! i'm still interested in hearing how the taylor series works though (a *really* dumbed down version though )
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    Quote Originally Posted by rgba
    for example, taylor series (how does this work?):

    http://upload.wikimedia.org/math/0/0...bb9f19d69f.png

    is there a simpler way?
    As to how it works, I cannot answer directly, but one of the reasons it works is that, when you use the mathematicians measure of angle, the radian, you can treat it as an 'ordinary' number in equations and therefore create equations which treat these ratios as sums of infinite series, as former maths heroes like Euler, Gauss et al discovered back in the 18th and 19th centuries.

    While the particular series to which you linked looks complicated, if you start to use it you will discover that each further term in the series becomes a smaller and smaller number (it converges rapidly, in calculus speak), so that you could well get a suitably accurate answer after calculating just a few initial terms (the first 5 or first 10 terms should be enough for just about any calculation you needed to make).

    To understand the theory of how these equations were derived however, besides taking me further than my feeble maths knowledge, you would have to have a decent basic understanding of the differential and integral calculus and how, therefore, mathematicians become comfortable with creating sums of infinite series and manipulating them. This tends to be a bit beyond what we might call high school algebra, and will require some study.

    We do have some frightfully knowledgable and jelpful professional mathematicians on this forum, but please only ask them questions after ensuring they appreciate your current levels of mathematical understanding so that they can, if they respond to your query, tailor the response to what you can appreciate and understand.
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    Forum Radioactive Isotope MagiMaster's Avatar
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    There was a previous topic on Sin, Cos and Tan here that talked about the taylor series of those functions (though it also deals with complex numbers).
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  10. #9  
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    hey guys, thanks very much for the replies! sure, i do understand that the taylor series works by converging on the result, and i do know a *little* basic calculus, but will be doing more studying on calculus once i'm finished with linear algebra, but i know pretty much all i need to know for now... although i can never accept a formula for what it is, but need to know exactly why and how it works! guess i'll just have to be patient

    cheers guys, very helpful! much appreciated
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    Quote Originally Posted by rgba
    cool, so just interpolation... thanks for the great help! i'm still interested in hearing how the taylor series works though (a *really* dumbed down version though )
    I believe it is just a ratio between circles created from one common angle to two other angles in a triangle.

    Here is how it was taught at one time in the U.S.

    Harold is like a Rocket scientist to me. So this is some really dumb stuff.



    I believe there is a slight error on this page. At the bottom of the page. Where they say draw SR parallel to MS I believe they mean draw SR parallel to ML.










    Sincerely,


    William McCormick
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  12. #11  
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    Here are a few more.

    Sincerely,


    William McCormick
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  13. #12  
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    hi william,

    thanks a million for posting those images! apologies for the late reply, i've been so busy lately, i've been meaning to reply for a while now but it keeps escaping my head!

    thanks
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  14. #13  
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    Quote Originally Posted by rgba
    hi william,

    thanks a million for posting those images! apologies for the late reply, i've been so busy lately, i've been meaning to reply for a while now but it keeps escaping my head!

    thanks





    I had meant to post these, I just ran off to do something else. I am glad you enjoyed the blast from the past.

    I believe there are some more pages about it. If I get a chance I will try to post them too.

    Personally I would have converted everything to inches in that rafter exercise.

    And I would have taken half the ridge board off the run first. They show three quarters of a foot taken off for half the ridge board thickness. I suspect that they meant three quarters of an inch. But I would have taken the three quarters of an inch off the run.



    Sincerely,


    William McCormick
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