Math Is Fun Forum

mikau

Member

Registered: 2005-08-22

Posts: 1,504

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 (2009-12-07 20:23:22)

---

A logarithm is just a misspelled algorithm.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

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 (2009-12-08 07:20:26)

---

A logarithm is just a misspelled algorithm.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

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 (2009-12-08 07:46:40)

---

A logarithm is just a misspelled algorithm.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

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.

devian18

Member

Registered: 2009-02-24

Posts: 12

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

i have no clue about it