# The roots of nonlinear equations don't exist?

i want to solve the nolinear equations and $\phi1\in(0,\frac{\pi}{2})$ and $\phi2\in(0,\frac{\pi}{2})$ and $\phi1<\phi2$

I calculated the maximum and minimum on the left of the equal sign, and the value on the right of the equal sign between them, indicating that the root should exist, but the result doesn't exist,is it because when I use FindRoot to solve the problem, I take $\phi1$ and $\phi2$ a point instead of a range? for example

{ϕ1, π/4}, {ϕ2, π/4}


rather than

0 < ϕ1 < π/2 && 0 < ϕ2 < π/2


How to calculate the root of this equation,

How to set a range of $\phi1$ and $\phi2$ rather a point of start

tried

h = 20;
l = 20*Sqrt[2];

FindRoot[{Csc[ϕ2] -
Csc[ϕ1] == -1, (Log[Tan[ϕ2/2]] -
Log[Tan[ϕ1/2]])/((Cot[ϕ1] - Cot[ϕ2])) ==
h/l}, {ϕ1, π/4}, {ϕ2, π/4}]
(*Power::infy: Infinite expression 1/0. encountered.*)
(*Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered.*)
(*FindRoot::nlnum: The function value {1.,Indeterminate} is not a list of numbers with dimensions {2} at {ϕ1,ϕ2} = {0.785398,0.785398}.*)


and

NMaximize[{Csc[ϕ2] - Csc[ϕ1],
0 < ϕ1 < π/2 && 0 < ϕ2 < π/2}, {ϕ1, ϕ2}]
(*{9.29958*10^11, {ϕ1 -> 1.05644, ϕ2 -> 1.07532*10^-12}}*)

NMinimize[{Csc[ϕ2] - Csc[ϕ1],
0 < ϕ1 < π/2 && 0 < ϕ2 < π/2}, {ϕ1, ϕ2}]
（*{-5.52822*10^10, {ϕ1 -> 1.8089*10^-11, ϕ2 -> 0.0239192}}*）

NMaximize[{(Log[Tan[ϕ2/2]] -
Log[Tan[ϕ1/2]])/((Cot[ϕ1] - Cot[ϕ2])),
0 < ϕ1 < π/2 && 0 < ϕ2 < π/2}, {ϕ1, ϕ2}]
（*{1., {ϕ1 -> 1.57079, ϕ2 -> 1.57079}}*）

NMinimize[{(Log[Tan[ϕ2/2]] -
Log[Tan[ϕ1/2]])/((Cot[ϕ1] - Cot[ϕ2])),
0 < ϕ1 < π/2 && 0 < ϕ2 < π/2}, {ϕ1, ϕ2}]
（*{3.14135*10^-18, {ϕ1 -> 0.799769, ϕ2 -> 7.15307*10^-20}}*）

• @HenrikSchumacher i think as i coded it,it is unnecessary to put it as formula again, and why it inequality? and is there really no roots and if exists,how to find all them? Commented May 25, 2018 at 8:49
• I aborted FindInstance[eq, {ϕ1, ϕ2},Reals]. May be you can wait longer?
– Acus
Commented May 25, 2018 at 9:18

To solve your equations numerically use NMinimize:

gl = {Csc[ϕ2] - Csc[ϕ1] == -1, (Log[Tan[ϕ2/2]] -Log[Tan[ϕ1/2]])/(Cot[ϕ1] - Cot[ϕ2]) == h/l}

NMinimize[{1, gl, 0 < ϕ1 < π/2, 0 < ϕ2 < π/2, ϕ1 < ϕ2}, {ϕ1, ϕ2}]

(* Out: {1., {ϕ1 -> 0.511714, ϕ2 -> 1.28531}} *)

• helps a lot and thanks a lot! Commented May 26, 2018 at 13:55
h = 20;
l = 20*Sqrt[2];


Use ContourPlot to find good starting values for use in FindRoot

ContourPlot[{Csc[ϕ2] -
Csc[ϕ1] == -1, (Log[Tan[ϕ2/2]] -
Log[Tan[ϕ1/2]])/((Cot[ϕ1] - Cot[ϕ2])) ==
h/l}, {ϕ1, 0, π/2}, {ϕ2, 0, π/2},
PlotLegends -> Placed["Expressions", {.6, .15}],
FrameLabel -> Automatic] // Quiet


FindRoot[{Csc[ϕ2] -
Csc[ϕ1] == -1, (Log[Tan[ϕ2/2]] -
Log[Tan[ϕ1/2]])/((Cot[ϕ1] - Cot[ϕ2])) ==
h/l}, {ϕ1, .5}, {ϕ2, 1.3}]

(* {ϕ1 -> 0.511714, ϕ2 -> 1.28531} *)

• great answer and thanks a lot! Commented May 26, 2018 at 13:56