# How to find a value to make function positive and real? [closed]

Let

x=(((-3 + r) r^2 + a^2 (1 + r)) Csc[θ])/(a (-1 + r))

y=Sqrt[(a^2 (a^2 (1 + r)^2 + 2 r^2 (-3 + r^2)) +
a^4 (-1 + r)^2 Cos[θ]^2 - ((-3 + r) r^2 +
a^2 (1 + r))^2 Csc[θ]^2)/(a^2 (-1 + r)^2)]


where $0<a<1$, $0<\theta\leq \frac{\pi}{2}$

I am looking for a two values of $r$ where (1) both $x$ and $y$ are real and (2) both $x$ and $y$ are real and positive.

If we choose some value for $a$ and $\theta$, is there a way to find the two values of $r$ without solving the $x$ or $y$?

Thanks.

## closed as off-topic by Feyre, happy fish, MarcoB, march, ÖskåDec 10 '16 at 11:02

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question arises due to a simple mistake such as a trivial syntax error, incorrect capitalization, spelling mistake, or other typographical error and is unlikely to help any future visitors, or else it is easily found in the documentation." – Feyre, happy fish, MarcoB, march, Öskå
If this question can be reworded to fit the rules in the help center, please edit the question.

• You try Solve? Or NDSolve? – LCarvalho Dec 6 '16 at 8:33
• You can just use FindInstance[]? – Feyre Dec 6 '16 at 10:36
• fi[a_, \[Theta]_] := FindInstance[ x[a, \[Theta]] > 0 && y[a, \[Theta]] > 0 && x[a, \[Theta]] \[Element] Reals && y[a, \[Theta]] \[Element] Reals, r, 2],fi[0.5, 0.5] will just give you two values which satisfy (2), and therefore also (1). – Feyre Dec 6 '16 at 10:37
• @Feyre If you specify that sth > 0 it automatically is also Real; \[Element] Reals is unnecesary. – corey979 Dec 6 '16 at 12:42

## 1 Answer

The following, based on Freye's comment, will allow you to explore the solution space.

finder[a_, θ_] :=
Block[{x, y, r},
x[aa_, theta_] := ((-3 + r) r^2 + aa^2 (1 + r)) Csc[theta]/(aa (-1 + r));
y[aa_, theta_] :=
Sqrt[
(aa^2 (aa^2 (1 + r)^2 + 2 r^2 (-3 + r^2)) +
aa^4 (-1 + r)^2 Cos[theta]^2 -
((-3 + r) r^2 + aa^2 (1 + r))^2 Csc[theta]^2) /
(aa^2 (-1 + r)^2)];
Module[{sols},
sols = FindInstance[x[a, θ] > 0 && y[a, θ] > 0, r, 2];
{r, x[a, θ], y[a, θ]} /. sols]]

Manipulate[
Column[
Row[{"r = ", #[[1]], "; x = ", #[[2]], "; y = ", #[[3]]}] & /@ finder[a, θ]],
{{a, .5}, .01, 1., Appearance -> "Labeled"},
{{θ, .5}, .01, π/2., Appearance -> "Labeled"}]