Roger Stafford wrote:
> "sudhir singla" <sudhir.414@gmail.com> wrote in message
> <hob39o$n85$1@fred.mathworks.com>...
>> hey ya all,
>> i have the following equations. how can i optimally find solution for
>> the angles a1 and a2 ?
>>
>> cos(a1)+ cos(a2) =m (where m is a constant that varies from 0.4 to 1)
>> cos(3a1)+ cos(3a2) =0
>> cos(5a1)+ cos(5a2) =0
>> cos(7a1)+ cos(7a2) =0
>> cos(9a1)+ cos(9a2) =0
> If x and y are defined as x = (a1+a2)/2 and y = (a1a2)/2, then because
> of the trigonometric identity
>
> cos(A)+cos(B) = 2*cos((A+B)/2)*cos((AB)/2) ,
>
> your equations can be replaced by the equivalent equations
>
> 2*cos(x)*cos(y) = m
> 2*cos(3*x)*cos(3*y) = 0
> 2*cos(5*x)*cos(5*y) = 0
> 2*cos(7*x)*cos(7*y) = 0
> 2*cos(9*x)*cos(9*y) = 0
>
> In each of these last four equations one of the cosines must equal zero
> to yield a zero product, which means that its argument must be an odd
> integral multiple of pi/2 and this is a stringent condition indeed.
The first and third equations together have solutions involving +/Pi/10
and +/3*Pi/10, independent of m. There are also solutions involving
+/arccos(1/4*m*(102*5^(1/2))^(1/2)+1/20*m*(102*5^(1/2))^(1/2)*5^(1/2))
which does not simplify to any particularly nice multiple of Pi.
> According to my (somewhat hasty) calculations, it is actually impossible
> to satisfy all four equations without causing m to also equal 0. If I
> am correct in this, it means that your plot can have only the single
> value of zero for m, and this does not lie in the range 0.4 to 1.
5 equations, actually, but Yes, solving them simultaneously requires m=0
. However, based upon the questions about multiple lines on the same
plot, I am suspecting that the original poster wanted to solve the first
equation in combination with one of the other equations at a time.
Solving the first and last equations together and exploring all the
roots of the quartic, leads to some pretty odd plots! I'm still
scratching my head about where some of the discontinuities originate
from  that is, having trouble figuring out why they are where they are.
