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Re: Re: Simple problem for math wiz
*To*: mathgroup at christensen.cybernetics.net
*Subject*: [mg1745] Re: [mg1708] Re: Simple problem for math wiz
*From*: brucec (Bruce Carpenter)
*Date*: Mon, 24 Jul 1995 00:44:07 -0400
In article <DBKwIr.2qv at wri.com>, nomail at me.net (REVEAL) wrote:
> Anyone here know how to find the solution set to: sin^2x+sinx-1=0 ? the
>trick, that is?
> (limited to trig & algebra)
>
If you write Sin[x] in terms of its complex exponential:
In[1]:=
f[x_] := (E^(I x) - E^(-I x))/(2 I)
Then the eqaution becomes:
In[2]:=
Expand[f[x]^2 + f[x] - 1] == 0
Out[2]=
-2 I x 2 I x
1 I -I x I I x E E
-(-) + - E - - E - ------- - ------ == 0
2 2 2 4 4
Multiply through by -4 E^(2 I x), (which is never zero):
In[3]:=
Expand[-4 E^(2 I x) (f[x]^2 + f[x] - 1) ] == 0
Out[3]=
I x 2 I x 3 I x 4 I x
1 - 2 I E + 2 E + 2 I E + E == 0
This yields a quartic equation in powers of E^(I x), which can be solved
exactly:
In[4]:=
soln = Solve[y^4 + 2 I y^3 + 2 y^2 - 2 I y + 1 == 0]
Out[4]=
-I I Sqrt[-2 - 2 Sqrt[5]]
{{y -> -- - - Sqrt[5] - --------------------},
2 2 2
-I I Sqrt[-2 - 2 Sqrt[5]]
{y -> -- - - Sqrt[5] + --------------------},
2 2 2
-I I Sqrt[-2 + 2 Sqrt[5]]
{y -> -- + - Sqrt[5] - --------------------},
2 2 2
-I I Sqrt[-2 + 2 Sqrt[5]]
{y -> -- + - Sqrt[5] + --------------------}}
2 2 2
The principal real solutions are now easily picked out:
In[5]:=
ans = Chop[N[-I Log[y] /. soln]]
Out[5]=
{-1.5708 - 1.06128 I, -1.5708 + 1.06128 I, 2.47535,
0.666239}
A check on the results:
In[6]:=
Sin[x]^2 + Sin[x] - 1 /. {{x->ans[[3]]},{x->ans[[4]]}}
Out[6]=
-16 -16
{-1.11022 10 , 1.11022 10 }
Of course you get infinitely many solutions because the Log is actually
infinitely valued--just take any solution + 2 k Pi for k an integer. Eg,
In[7]:=
ans[[3]]-8Pi//N
ans[[4]] + 12Pi//N
Sin[x]^2 + Sin[x] - 1 /. {{x->ans[[3]]-8Pi},
{x->ans[[4]]}}
Out[7]=
-22.6574
Out[8]=
38.3654
Out[9]=
-16 -16
{-1.11022 10 , 1.11022 10 }
Is this what you meant by being limited to trig and algebra?
Cheers,
---------------------------------------------
Bruce Carpenter
Wolfram Research, Inc.
100 Trade Center Drive, Champaign, IL 61820
email: brucec at wri.com
---------------------------------------------
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