# How to obtain the Cosine symbolic Matrix of EulerAngles

Here in X-convention

Extrinsic

RotationMatrix3DExtrinsic[\[Phi]_,\[Theta]_,\[Psi]_]:=RotationMatrix[\[Pi]-\[Psi],{0,0,1}].RotationMatrix[\[Theta],{1,0,0}].RotationMatrix[\[Pi]-\[Phi],{0,0,1}]


Intrinsic

RotationMatrix3DIntrinsic[\[Phi]_,\[Theta]_,\[Psi]_]:=RotationMatrix[\[Pi]-\[Phi],RotationMatrix[\[Theta],RotationMatrix[\[Pi]-\[Psi],{0,0,1}].{1,0,0}].{0,0,1}].RotationMatrix[\[Theta],RotationMatrix[\[Pi]-\[Psi],{0,0,1}].{1,0,0}].RotationMatrix[\[Pi]-\[Psi],{0,0,1}]


In Numeric values, they got the same results.

RotationMatrix3DExtrinsic[\[Pi]/3, \[Pi]/4, \[Pi]/5]//N
RotationMatrix3DIntrinsic[\[Pi]/3,\[Pi]/4,\[Pi]/5]//N


But RotationMatrix3DIntrinsic does not work well in symbolic value, alpha,beta,gamma. After a long time calculation, I see some complex result with Conjugate Abs... So how to develop it.

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By default, Mathematica assumes your angles are complex. You thus need to use assumptions; look up the second argument of Simplify[], or look up Assuming[]. –  Ｊ. Ｍ. May 11 at 3:07
@J.M. ok, Let me try that, does any simpler way to Obtain the matrix just like Extrinsic way? –  HyperGroups May 11 at 3:09
My eyes ! ..... –  belisarius May 11 at 3:16
@bel, your eyes are sensitive to pink? –  Ｊ. Ｍ. May 11 at 3:19
–  belisarius May 11 at 3:34
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