# How to find the zeros of a Complex trigonometric function

I tried this

ContourPlot[(-100 -
50 Cos[γ]) Sin[β] (50 Cos[γ] Sin[β] - (
50 Cos[β] Cos[γ] (-200 +
Cos[β] (50 + 50 Sin[γ])))/Sqrt[
6400 - (200 - Cos[β] (50 + 50 Sin[γ]))^2]) +
50 Cos[β] Sin[γ] (Cos[β] (50 +
50 Sin[γ]) + (
Sin[β] (50 + 50 Sin[γ]) (-200 +
Cos[β] (50 + 50 Sin[γ])))/Sqrt[
6400 - (200 -
Cos[β] (50 + 50 Sin[γ]))^2]), {β, 0,
2 Pi}, {γ, 0, 2 Pi}]


but it does not produce anything.

Could someone explain why and how to fix it?

-
Your function returns Complex results almost everywhere. Contour plotting needs real valued functions – Dr. belisarius Mar 3 '14 at 3:00
Many thanks, the function is the determinant of a matrix, I want ot know where the function equals to zero, what should I do? – wei ye Mar 3 '14 at 3:15
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In case you didn't know, you can format your code better by putting four spaces at the front of every code block (or click on the curly-brace button above the question editing area). Further, wrap short inline code snippets in a pair of backticks . This will make your post easier to read. – Dr. belisarius Mar 3 '14 at 17:04

Your determinant will be zero when the Real and Imaginary parts are zero. We can plot the contours independently for both and your answer is the intersection:

f[β_, γ_] := (-100 - 50 Cos[γ]) Sin[β] (50 Cos[γ] Sin[β] - (50 Cos[β] Cos[γ] (-200 +
Cos[β] (50 + 50 Sin[γ])))/ Sqrt[6400 - (200 - Cos[β] (50 + 50
Sin[γ]))^2]) + 50 Cos[β] Sin[γ] (Cos[β] (50 + 50 Sin[γ]) + (Sin[β] (50 +
50 Sin[γ]) (-200 + Cos[β] (50 + 50 Sin[γ])))/ Sqrt[6400 - (200 -
Cos[β] (50 + 50 Sin[γ]))^2])

ContourPlot[{Re@f[β, γ], Im@f[β, γ]}, {β, 0, 2 Pi}, {γ, 0, 2 Pi},
Contours -> {{0}},
Epilog -> {Red, PointSize[Large], Thickness[.01],
Line[{{0, 3/2 Pi}, {2 Pi, 3/2 Pi}}],
Point[Pi {{1, 1}, {1, 0}, {1/2, 1/2}, {3/2, 1/2}}]}]


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Seems to be missing solutions, e.g., beta=Pi, gamma=2Pi. – ciao Mar 3 '14 at 4:24
Thank you for the answer. – wei ye Mar 3 '14 at 4:29
But I don't know what do "Point[Pi {{1, 1}, {1, 0}, {1/2, 1/2}, {3/2, 1/2}, {1, 3/2}, {1/2, 3/2}, {3/2, 3/2}}]" means, could you explain it please. – wei ye Mar 3 '14 at 4:34
@weiye I'm drawing red points by hand at the intersections – Dr. belisarius Mar 3 '14 at 4:35
@weiye And the red line too – Dr. belisarius Mar 3 '14 at 4:38