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I have the following equations -

T = (2*cos[α]*sin[β])/sin[α + β];

sin[α]/sin[β] = sqrt[1 - V/E];

And I want to eliminate β from the two equations so as to obtain an expression for T. How should I proceed?

Kuba
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Indeterminate
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  • Never use upper-case letters (T, V, E, ...) to start the names of variables as these may conflict with internal names in Mathematica. Sin is the proper usage, not sin, Sqrt not sqrt, ... etc. – David G. Stork Jan 22 '18 at 19:45
  • Thanks David, I am new to this software. – Indeterminate Jan 22 '18 at 19:45
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    Also note all built in functions all start with caps (Sin,Sqrt, etc). You canSolve` the second expression for beta and sub into the first. – george2079 Jan 22 '18 at 19:49

3 Answers3

3

Or can just use eliminate. First get into equation form without the upper case variables. E is a particularly bad idea unless you mean the built in constant E.

eq1 = t == (2*Cos[α]*Sin[β])/Sin[α + β]//TrigExpand;
eq2 = Sin[α]/Sin[β] == Sqrt[1 - v/e];

Eliminate[{eq1, eq2}, β] // Simplify
(*t^2*v == 4*e*(t - 1)*Cos[α]^2*)
Kuba
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Bill Watts
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Solve[Sin[α]/Sin[β] = Sqrt[1 - V/E], β]

(*

π - ArcSin[(Sqrt[E] Sin[α])/Sqrt[E - V]]

*)

T /. β -> π - ArcSin[(Sqrt[E] Sin[α])/Sqrt[E - V]]

$-\frac{2 \sqrt{e} \sin (\alpha ) \cos (\alpha ) \csc \left(\alpha -\sin ^{-1}\left(\frac{\sqrt{e} \sin (\alpha )}{\sqrt{e-V}}\right)\right)}{\sqrt{e-V}}$

Kuba
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David G. Stork
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1

note the result here simplifies nicely using TrigExpand

(2*Cos[α]*Sin[β])/Sin[α + β] /. 
   Solve[Sin[α]/Sin[β] == Sqrt[1 - v/e] , β] // 
  Simplify

(* conditional expression *) 

result = Simplify[%, Assumptions -> C[1] ∈ Integers] // 
   TrigExpand // Simplify

enter image description here

and simpler still if you add Assumptions -> 0 < v < e (if true of course)

enter image description here

Plot[Evaluate[result /. {v -> .1, e -> .2}], {α, -2 Pi, 2 Pi}]

enter image description here

Kuba
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george2079
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  • Thanks for the answer. What does this mean C[1] [Element] Integers ? – Indeterminate Jan 23 '18 at 16:26
  • the first conditional expression is in terms of unspecified integer constants ( due to the inverse trig function). When you insert that back into the expression for t those constants simplify out (due to the forward trig functions). Unfortunately we are left with a ConditionalExpression that still says C[1] must be an integer even though there is no C[1] in the expression, so my last simplify just gets rid of that. – george2079 Jan 23 '18 at 16:58
  • My assumptions expression doesn't work and I get this - https://imgur.com/jgWCW75 – Indeterminate Jan 23 '18 at 17:05
  • looks like you have a letter o instead of zero in there. – george2079 Jan 23 '18 at 17:25