Mathematica returns empty set when I expect two different values

• To: mathgroup at smc.vnet.net
• Subject: [mg32329] Mathematica returns empty set when I expect two different values
• From: "spiraltooth" <spiraltooth at netzero.net>
• Date: Tue, 15 Jan 2002 02:30:06 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```The following is a Mathematica file in which a spiral is described in
polar coordinates. I am trying to find a point Ttheta,theta on the
spiral given another point Tt,t . These two points are separated by the
straight line distance Mtheta. Both Ttheta and Mtheta are functions of
theta and are the unknowns. Thank
you for helping me.
Alan Notis

In(1)=     S=6Pi
Out(1)=   6Pi
In(2)=     MR=.7
Out(2)=   .7
In(3)=     t=Pi
Out(3)=   Pi
In(4)=     Itheta=MR+(theta / S)
Out(4)=   0.7+(theta / 6Pi)
In(5)=     It=MR+(t / S)
Out(5)=   0.8666667
In(6)=     CM=.025
Out(6)=   .025
In(7)=     CX=.275
Out(7)=   .275
In(8)=     Mtheta=CX-(((CX-CM)theta) / S)
Out(8)=   0.275-0.0132629theta
In(9)=     Ttheta=Itheta+Mtheta   (*  This describes a radius Ttheta
given theta as a point on a spiral from ~-.1Pi to ~6.1Pi  *)
Out(9)=   0.975+0.0397887theta
In(10)=    Mt=CX-(((CX-CM)t) / S)
Out(10)=  0.233333
In(11)=    Tt=It+Mtheta   (*   This describes a radius Tt given t as
a point on the same spiral as Ttheta,theta; Again t is demonstrated at
Pi here but in reality will vary from 0 to 6Pi   *)
Out(11)=  1.14167-0.0132629theta
In(12)=
alpha=ArcCos[((Ttheta^2)+(Tt^2)-(Mtheta^2))/(2(Ttheta)(Tt))]
Out(12)=
ArcCos[(-(0.275-0.0132629theta)^2+(1.14167-0.0132629theta)^2+(0.975+0.0397887theta)^2) / 2(1.14167-0.0132629theta)(0.975+0.0397887theta)]
In(13)=    Solve[alpha==ArcCos[(Ttheta^2+Tt^2-Mtheta^2) /
(2(Ttheta(Tt)))] , theta ]     (*   Here I'm Using the law of cosines to
find theta where alpha is the angle between t and theta . I'm expecting
two values for theta one where theta < t and one where theta > t .
Mathematica seems unable to do this as it returns the null set.  *)
Out(13)=  {{}}
In(14)=    Solve[2X^2 + 14X +12 == 0 , X]     (*    But
Mathematica of course has no problem solving an equation in a quadratic
form. Do I need to load a certain package? Am I doing something
mathematically wrong? Do I need to use ( and learn ) vector calculus?
*)
Out(14)=  {{X -> -6},{X -> -1}}

```

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