       Re: Solve Equation

• To: mathgroup at smc.vnet.net
• Subject: [mg20846] [mg20846] Re: [mg20813] Solve Equation
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Wed, 17 Nov 1999 03:41:05 -0500 (EST)
• References: <199911142314.SAA02155@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Mecit Yaman wrote:
>
> i am trying to solve 3 equations on 3 variables. Mathematica is complaining
> about trigonometric functions, sin and cosine.
>
> L= 5+4Cos[a]
> L=13-12 Cos[b]
> L=10 - 6Cos[a+b]
>
> I tried to change
> Cos[a]  -> x
> Cos[b] -> y
> Cos[a+b] - > (x y - Sqrt[1-x x]Sqrt[1- y y])
>
> But this time it only gives me a trivial solution.(L=0)
> I am trying to get the solution.
>
> L=7, b=Pi/3 , a= Pi/3*5

I rather hope Solve will not give that as a result.

exprs = {5+4*Cos[a]-L, 13-12*Cos[b]-L, 10-6*Cos[a+b]-L};

In:= exprs /. {L->7, b->Pi/3 , a->Pi/3*5}
Out= {0, 0, -3}

The following will get you a pair of fairly concise solutions.

SetOptions[Roots, Cubics->False]
soln = Solve[exprs==0, {L,a,b}];

In:= FullSimplify[{L,a,b} /. soln]
2     3
Out= {{Root[-1300 + 441 #1 - 42 #1  + #1  & , 3],
2        3
>     ArcCos[Root[-5 + 96 #1 - 108 #1  + 16 #1  & , 3]],
2        3
>     ArcCos[Root[13 - 48 #1 + 12 #1  + 48 #1  & , 1]]},
2     3
>    {Root[-1300 + 441 #1 - 42 #1  + #1  & , 2],
2        3
>     ArcCos[Root[-5 + 96 #1 - 108 #1  + 16 #1  & , 2]],
2        3
>     ArcCos[Root[13 - 48 #1 + 12 #1  + 48 #1  & , 2]]}}

In:= N[%]
Out= {{27.8341, 0. + 2.42735 I, 3.14159 - 0.674426 I},
>    {8.94378, 0.167852, 1.22599}}

Daniel Lichtblau
Wolfram Research

```

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