Re: Inverse function solution

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• Subject: [mg132654] Re: Inverse function solution
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Tue, 29 Apr 2014 01:33:51 -0400 (EDT)
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• References: <20140428014444.9FBD16A4E@smc.vnet.net>

```In Mathematica 9.0.1, I'm getting:

Solve[{x == Cos[u], y == Cos[u + v]}, {u, v}, Reals]
{{u -> ConditionalExpression[-ArcCos[x] +
2 \[Pi] C[1], (C[1] | C[2]) \[Element] Integers && -1 <= x <=
1 && -1 <= y <= 1],
v -> ConditionalExpression[
ArcCos[x] - ArcCos[y] - 2 \[Pi] C[1] +
2 \[Pi] C[2], (C[1] | C[2]) \[Element] Integers && -1 <= x <=
1 && -1 <= y <= 1]}, {u ->
ConditionalExpression[-ArcCos[x] +
2 \[Pi] C[1], (C[1] | C[2]) \[Element] Integers && -1 <= x <=
1 && -1 <= y <= 1],
v -> ConditionalExpression[
ArcCos[x] + ArcCos[y] - 2 \[Pi] C[1] +
2 \[Pi] C[2], (C[1] | C[2]) \[Element] Integers && -1 <= x <=
1 && -1 <= y <= 1]}, {u ->
ConditionalExpression[
ArcCos[x] +
2 \[Pi] C[1], (C[1] | C[2]) \[Element] Integers && -1 <= x <=
1 && -1 <= y <= 1],
v -> ConditionalExpression[-ArcCos[x] - ArcCos[y] - 2 \[Pi] C[1] +
2 \[Pi] C[2], (C[1] | C[2]) \[Element] Integers && -1 <= x <=
1 && -1 <= y <= 1]}, {u ->
ConditionalExpression[
ArcCos[x] +
2 \[Pi] C[1], (C[1] | C[2]) \[Element] Integers && -1 <= x <=
1 && -1 <= y <= 1],
v -> ConditionalExpression[-ArcCos[x] + ArcCos[y] - 2 \[Pi] C[1] +
2 \[Pi] C[2], (C[1] | C[2]) \[Element] Integers && -1 <= x <=
1 && -1 <= y <= 1]}}

Or, a bit simpler:

Reduce[{x == Cos[u], y == Cos[u + v]}, {u, v}, Reals]
(C[1] | C[2]) \[Element] Integers && -1 <= x <= 1 && -1 <= y <=
1 && ((u == -ArcCos[x] +
2 \[Pi] C[1] && (v ==
ArcCos[x] - ArcCos[y] - 2 \[Pi] C[1] + 2 \[Pi] C[2] ||
v == ArcCos[x] + ArcCos[y] - 2 \[Pi] C[1] +
2 \[Pi] C[2])) || (u ==
ArcCos[x] +
2 \[Pi] C[1] && (v == -ArcCos[x] - ArcCos[y] - 2 \[Pi] C[1] +
2 \[Pi] C[2] ||
v == -ArcCos[x] + ArcCos[y] - 2 \[Pi] C[1] + 2 \[Pi] C[2])))
On Apr 27, 2014, at 9:44 PM, Narasimham <mathma18 at gmail.com> wrote:

> Solve[ {x == Cos[u], y == Cos[u + v] }, {u, v} ]
>
> Its closed/analytic solution is not possible, even numerically.
>
> The known solutions are ellipses from sine waves with a phase difference, having x^2, x y and y^2 terms, as also sketched in Lissajous curves:
>
> ParametricPlot[{Cos[u], Cos[u + v]}, {u, -Pi, Pi}, {v, -Pi, Pi}]
>
> Can there be a work around?
>
> Narasimham
>

Murray Eisenberg                                murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 240 246-7240 (H)
University of Massachusetts
710 North Pleasant Street
Amherst, MA 01003-9305

```

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