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Topic: Trigometry-Degrees to a decimal (Read 1700 times) |
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Mugwump101
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Trigometry-Degrees to a decimal
« on: Apr 18th, 2004, 11:06am » |
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Is there a way to convert the degrees in an angel to a decimal by hand? If so, can you tell me how? Just out of curiousity, and I could use it in class.
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towr
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Re: Trigometry-Degrees to a decimal
« Reply #1 on: Apr 18th, 2004, 11:48am » |
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I don't think I understand what you want.. Could you give an example?
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Icarus
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Re: Trigometry-Degrees to a decimal
« Reply #2 on: Apr 18th, 2004, 3:11pm » |
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Yes - are you wanting to convert minutes and seconds to decimal degrees, or are you trying to convert degrees into radians? Or something else?
For these two:
Degrees, minutes, seconds [to] decimal degrees
d = D + (M + S/60)/60 = D + M/60 + S/3600
(The first form is usually easier when calculating by hand.)
Decimal degrees [to] Radians
R = d([pi]/180) = d(0.017453...)
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Sir Col
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Re: Trigometry-Degrees to a decimal
« Reply #3 on: Apr 18th, 2004, 3:44pm » |
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Judging by the title of the thread, I wonder if she is looking for a way of "converting degrees into sine/cosine/tangent"?
If this is the case, Mugwump101, then there is no easy way. You will need to either use tables, in which you look up the angle and read off the corresponding sin, cos, or tan value; or, you will need to use a calculator (which is essentially the same as using tables). You enter the angle and the calculator "looks up" the sine/cosine/tangent of the angle.
If you're set on trying to do this by hand then you could draw a circle with a unit radius. If you measure, say 40o, in an anticlockwise direction from the x-axis around the circle to make point P, join this point to the origin, O. Now draw a vertical line down to meet the x-axis. You will have made a right angle triangle. The height of the triangle is the sine of the angle, and the base length of the triangle is the cosine of the angle.
If you want more information on this, I have a FAQ relating to this on my website:
http://mathschallenge.net/index.php?section=faq&ref=library/geometry /trigonometric_identities
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| « Last Edit: Apr 18th, 2004, 3:48pm by Sir Col » |
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Mugwump101
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Re: Trigometry-Degrees to a decimal
« Reply #4 on: Apr 18th, 2004, 6:54pm » |
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on Apr 18th, 2004, 3:44pm, Sir Col wrote:Judging by the title of the thread, I wonder if she is looking for a way of "converting degrees into sine/cosine/tangent"?
If this is the case, Mugwump101, then there is no easy way. You will need to either use tables, in which you look up the angle and read off the corresponding sin, cos, or tan value; or, you will need to use a calculator (which is essentially the same as using tables). You enter the angle and the calculator "looks up" the sine/cosine/tangent of the angle.
If you're set on trying to do this by hand then you could draw a circle with a unit radius. If you measure, say 40o, in an anticlockwise direction from the x-axis around the circle to make point P, join this point to the origin, O. Now draw a vertical line down to meet the x-axis. You will have made a right angle triangle. The height of the triangle is the sine of the angle, and the base length of the triangle is the cosine of the angle.
If you want more information on this, I have a FAQ relating to this on my website:
http://mathschallenge.net/index.php?section=faq&ref=library/geometry /trigonometric_identities
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Yeah this is what I mean, thanks! I'll check out the website, and thanks for your information, and everyone else too.
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Icarus
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Re: Trigometry-Degrees to a decimal
« Reply #5 on: Apr 18th, 2004, 8:15pm » |
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on Apr 18th, 2004, 3:44pm, Sir Col wrote:| you will need to use a calculator (which is essentially the same as using tables). You enter the angle and the calculator "looks up" the sine/cosine/tangent of the angle. |
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I'm sure you're just hiding the complexities with this statement, but I still wouldn't say that the calculator "looks up" the values of sine, cosine, and tangent. It calculates them each time from the formulas.
Mugwump101 - To calculate trigonometric functions directly, the first thing you need to do is convert the angle from degrees into radians, using the formula I gave above. It also helps to use trig identities to make the angle you are needing to calculate for as small as possible - the smaller it is, the faster you will get a reasonable number. By using the following identities, you can get the angle to be [le] [pi]/4 = 45o:
cos(x + [pi]) = -cos(x); sin(x+[pi]) = -sin(x); cos([pi]/2 - x) =sin(x); sin([pi]/2 - x) = cos(x).
Some other identities can allow you to get it even smaller. Once you have done this, then you can calulate cos(x) & sin(x) using the following formulas:
cos(x) = 1 - x2/2! + x4/4! - x6/6! + ...
sin(x) = x - x3/3! + x5/5! - x7/7! + ...
Both of these sums would have to go on forever for equality to be reached. But well before then, the terms get so small that the remainder of the sums are negligible. By making x as small as you can first, you can make it so you only need 4 or 5 terms each to get good accuracy.
There is also a formula for tangent, but it is not as easy to remember, nor does it converge as fast as these. Still, it might be easier to use than calculating both of these and taking their ratio.
However, Sir Col is right. Before calculators, people created tables of values for these functions. The easiest way to find the value of the functions was to look up the nearest table entries, and use linear interpolation to estimate your value. This was a much faster method than calculating them from the formulas. With calculators, we can get much greater accuracy with no effort on our part at all.
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rmsgrey
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Re: Trigometry-Degrees to a decimal
« Reply #6 on: Apr 19th, 2004, 4:40am » |
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I can't vouch for modern calculators, but certainly it is not uncommon in real-time 3D graphics calculations (games) to use look-up tables for trig functions (roughly one value per degree and linear interpolation) as a time saving exercise...
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Icarus
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Re: Trigometry-Degrees to a decimal
« Reply #7 on: Apr 20th, 2004, 3:48pm » |
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That may be - it's a trade-off of memory vs computation time. It used to be that memory was the determining factor. Nowadays, memory is more abundant and the computations are more complex, so time becomes the critical factor.
Still, my guess is most calculators calculate it out each time. Calculators are not engaged in such intensive graphics manipulation, and generally are required to answer with more precision than needed for 3D graphics modeling. My 26 year old hand calculator that I still use for most simple calculations gives the answer to 10 decimal places. For these graphics apps, 4 or 5 is probably more than enough to get smooth action.
Every programming language I have explored that deep into also calculated it from scratch each time. Though some may not now: I haven't explored into the mathematical methods behind a language for many years.
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"Pi goes on and on and on ...
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I wonder: Which is larger
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Sir Col
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Re: Trigometry-Degrees to a decimal
« Reply #8 on: Apr 20th, 2004, 4:34pm » |
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You might be interested in this article found at the Math Forum's Dr. Math archives.
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Mugwump101
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Re: Trigometry-Degrees to a decimal
« Reply #9 on: Apr 21st, 2004, 9:16pm » |
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Thanks again. I've tell you my results when I follow it through.
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