[Guest]

For what integral numbers a, b and c, where 0 < a < b < c < 90, is the following equation satisfied, and why?

cos²aº + cos²bº + cos²cº = 1.

Edited November 7, 2009 by jerbil

[Guest]

I would think your "why" question means there aren't any solutions

Edited November 9, 2009 by ljb

[Guest]

Not quick enough to edit

I checked them all and couldn't find any that worked. If it was <= 90 on the limit, it would work.

[Guest]

I'm really having no luck solving this.

Just something to note: If you use a computer to find solutions it may miss some or create false solutions if you simply check if the sum equals one. This is due to rounding errors given that the computer can only store values to a limited accuracy.

[Guest]

36, 60, 72

Here's my python script: with indenting signified by ~'s

#!/usr/bin/python

import math

for a in range(0,88):

~for b in range(a+1,89):

~~for c in range(b+1,90):

~~~z = math.cos(a*math.pi/180)**2;

~~~z += math.cos(b*math.pi/180)**2;

~~~z += math.cos(c*math.pi/180)**2;

~~~if math.fabs(z-1) < 0.000001:

~~~~print a, b, c;

[Guest]

Sorry, I meant to edit, not repost

36, 60, 72

Here's my python script: with indenting signified by ~'s

#!/usr/bin/python

import math

for a in range(0,88):

~for b in range(a+1,89):

~~for c in range(b+1,90):

~~~z = math.cos(a*math.pi/180)**2;

~~~z += math.cos(b*math.pi/180)**2;

~~~z += math.cos(c*math.pi/180)**2;

~~~if math.fabs(z-1) < 0.000001:

~~~~print a, b, c;

--------------------------

Note that though this was found without checking equality,

I was able to verify equality symbolically on my calculator (not using floating point calculations with round off errors)

Edited November 10, 2009 by mmiguel1

[Guest]

cos^2(x) = (1/2)(1+cos(2x))

Using this formula,

we get (1/2)(3 + cos(2a) + cos(2b) + cos(2c) ) = 1

or

cos(2a) + cos(2b) + cos(2c) = -1

letting a,b,c = 36, 60, 72

We get

Cos(72) + Cos(120) + Cos(144) = -1

This is the same as

Cos(2 pi/5) + Cos(2 pi/3) + Cos(4 pi/5) = -1

From a table,

Cos(2 pi/5) = (1/4)( Sqrt[5] - 1 )

Cos(2 pi/3) = -1/2

Cos(4 pi/5) = (1/4)(-1 - Sqrt[5] )

Summing these together gives -1

Meaning that the equation is satisfied and a,b,c = 36, 60, 72 satisfies the original relation.

[Guest]

Nicely done, solvers. I append my own ideas, which also explains why I entitled the problem the way I did.

Explanation of Problem.doc

[Guest]

Very nice solution.

Other close calls:

20, 71, 84

31, 61, 80

39, 54, 77

39, 55, 75

42, 53, 73

These are all off by an error of the order 10^(-5)