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As we all know in spherical coordinates a function phi = π/3 gives us a cone. The cone makes an angle of π/3 with the imagined z-axis. How do I type this formula into Mathematica?

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RevolutionPlot3D[3 z, {z, 0, -2}, ColorFunction -> "Rainbow", Mesh -> None] – Dr. belisarius Dec 14 '13 at 5:49
Related cones on spheres: Cone image refinement. – Artes Dec 14 '13 at 12:20

You can use CoordinateTransform to change coordinates to Cartesian and then use ParametricPlot3D to make the plot.

cone[r_, θ_] := 
  Evaluate[CoordinateTransform["Spherical" -> "Cartesian", {r, ϕ, θ}] /. ϕ -> π/4]
ParametricPlot3D[cone[r, θ], {r, 0, 1}, {θ, 0, 2. π}]

This gives


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