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  •  » intuitive way to calculate the sine and cosine of an angle

#1 2009-12-08 19:18:39

mikau
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intuitive way to calculate the sine and cosine of an angle

Here is an amusing method for finding the sine of an angle that makes intuitive sense:

suppose we  want to find the sine of 1 radian, about 57 degrees.

First we choose a power of 2 that is large relative to the angle. I'll choose 32.

1/32 is fairly small and so sin(1/32)  ≈ 1/32, and cos(1/32)  ≈ 1.

Now that we know sin(1/32) and cos(1/32) we can use 5 applications of the double angle identities: sin(2x) = 2sin(x)cos(x) and cos(2x)=cos^2(x) - sin^2(x)  to get the final sine and cosine!
I worte a simple program to do it for me. Have a look:

sin(1/32): 0.03125 cos: 1
sin(2/32): 0.0625 cos: 0.999023
sin(4/32): 0.124878 cos: 0.994142
sin(8/32): 0.248293 cos: 0.972723
sin(16/32): 0.48304 cos: 0.884541
final (32/32): sin: 0.854537 cos: 0.549085

results by windows calculator:
sin: 0.8414709848078965066525023216303, cos: 0.54030230586813971740093660744298

Hey! Thats pretty close! Of course! you can make it more accurate by choosing a larger power of 2. Choosing 2^10 for instance yields: sin: 0.841881 cos: 0.540566

Its hardly the fastest way to do it, but its easy to understand why it works and I think that makes it fun!
Originally I envisioned this as a series of line reflections. But I eventually realized I was essentially using the double angle formulas. Maybe I'll post a diagram later.

Last edited by mikau (2009-12-08 19:23:22)


A logarithm is just a misspelled algorithm.
 

#2 2009-12-09 05:54:55

bobbym
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Re: intuitive way to calculate the sine and cosine of an angle

Hi mikau;

We can represent your algorithm as a coupled set of difference equations:

a[n+1]=2a[n]b[n]
b[n+1]=b[n]^2 - a[n]^2   with b[0]=1 and a[0] = 1 / 65536

Using that I calculated .8414774 and .54030642


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#3 2009-12-09 06:13:54

mikau
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Re: intuitive way to calculate the sine and cosine of an angle

Nice! And of course we can say a[0]=x/2^M  for the general case! B)

(edit) also i think it should say that a[M]=sin(x) and b[M]=cos(x), otherwise the above difference equation just describes a sequence, and we don't know where to stop it.

Last edited by mikau (2009-12-09 06:20:26)


A logarithm is just a misspelled algorithm.
 

#4 2009-12-09 06:28:45

bobbym
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Re: intuitive way to calculate the sine and cosine of an angle

Hi mikau;

We can describe it as a sequence, a finite one. For, a[0] = 1 / 2^n then we are interested in the nth term of the sequence. For instance a[0] = 1 / 2^20 then a[20], b[20] are our answers.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#5 2009-12-09 06:46:24

mikau
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Re: intuitive way to calculate the sine and cosine of an angle

That works too! :]

Originally, I thought of it this way: start with the point P= (1,0) and reflect it about the line L through the origin and Q= (1,x/2^n), to produce a point R
now set P equal to Q, and Q equal to R, and repeat. In a way its kind of like rolling a cone of length 1 and circumference x/2^n, through 2^n revolutions until the large end falls onto the point (sin(x),cos(x))

Last edited by mikau (2009-12-09 06:46:40)


A logarithm is just a misspelled algorithm.
 

#6 2009-12-09 09:06:16

bobbym
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Re: intuitive way to calculate the sine and cosine of an angle

Hi mikau;

That idea reminds me of the old Cordic method to compute trig values.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
 

#7 2009-12-09 09:53:31

mikau
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Re: intuitive way to calculate the sine and cosine of an angle

you said hello 3 times! ;]

Cordic method huh? I'll look it up


A logarithm is just a misspelled algorithm.
 

#8 2009-12-10 04:24:59

devian18
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Re: intuitive way to calculate the sine and cosine of an angle

i have no clue about it

 
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