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Trigonometric simplifications and matrices

asked 2019-11-08 15:53:07 +0100

Jean-Sébastien gravatar image

updated 2019-11-08 15:53:53 +0100

I want to simplify trigonometric identities in a matrix. For example, say I want to show that the composition of two rotation matrices is a rotation, I can do with sage something like

  show(produit.apply_map(lambda x: x.trig_reduce()))

Note that simplify_trig or trig_reduce don't work on matrices and that you need to use apply_map to use it entry by entry, as detailed in Mike Hansen's answer in this question .

However when I get to 3 matrices, sage can't simplify with the above procedure:

show(produit.apply_map(lambda x: x.trig_reduce()))

For example, the 1-1 entry in this matrix is returned as cos(theta1 + theta2)*cos(theta3) - sin(theta1 + theta2)*sin(theta3) . The weird thing is that using (cos(theta1 + theta2)*cos(theta3) - sin(theta1 + theta2)*sin(theta3)).trig_reduce() produces the correct simplification cos(theta1 + theta2 + theta3) .

What's happening here? Any other way to force the simplification?

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answered 2019-11-08 16:50:23 +0100

tmonteil gravatar image

updated 2019-11-08 16:51:11 +0100

Apparently trig_reduce is not idempotent and you have to apply it twice to get the correct result:

sage: produit.apply_map(lambda x: x.trig_reduce().trig_reduce())
[ cos(theta1 + theta2 + theta3) -sin(theta1 + theta2 + theta3)]
[ sin(theta1 + theta2 + theta3)  cos(theta1 + theta2 + theta3)]
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Wow, what is even more weird is that it seems that if you go with an even number of matrices, sage can simplify with one trig-reduce, but needs two for an odd number of matrices

Jean-Sébastien gravatar imageJean-Sébastien ( 2019-11-08 17:49:54 +0100 )edit

It might depend on how "complex" are the expressions.

tmonteil gravatar imagetmonteil ( 2019-11-08 21:18:29 +0100 )edit

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Asked: 2019-11-08 15:53:07 +0100

Seen: 589 times

Last updated: Nov 08 '19