How to make sin(pi) and cos(pi/2) zero?


I understand that in Python sin(pi) and cos(pi/2) won't produce 0, but I'm making calculations with matrices and I need to use those values.

I'm using SymPy and at first the values of sin(pi) and cos(pi/2) are a little annoying. After some multiplications they start to get in the way. Is there a way to make those values be equal to 0 in the entire module? How can I change it in the middle of expressions?

I'll use this matrix as an example:

A = Matrix([ [(-sin(theta1)*sin(theta2)*cos(alpha2) + cos(theta1)*cos(theta2))*cos(theta3) + (-sin(theta1)*cos(alpha2)*cos(theta2) - sin(theta2)*cos(theta1))*sin(theta3)*cos(alpha3) + sin(alpha2)*sin(alpha3)*sin(theta1)*sin(theta3), -(-sin(theta1)*sin(theta2)*cos(alpha2) + cos(theta1)*cos(theta2))*sin(theta3) + (-sin(theta1)*cos(alpha2)*cos(theta2) - sin(theta2)*cos(theta1))*cos(alpha3)*cos(theta3) + sin(alpha2)*sin(alpha3)*sin(theta1)*cos(theta3), -(-sin(theta1)*cos(alpha2)*cos(theta2) - sin(theta2)*cos(theta1))*sin(alpha3) + sin(alpha2)*sin(theta1)*cos(alpha3), a3*(-sin(theta1)*sin(theta2)*cos(alpha2) + cos(theta1)*cos(theta2)) + d2*sin(alpha2)*sin(theta1) - d3*(-sin(theta1)*cos(alpha2)*cos(theta2) - sin(theta2)*cos(theta1))*sin(alpha3) + d3*sin(alpha2)*sin(theta1)*cos(alpha3)], [(-sin(theta1)*sin(theta2) + cos(alpha2)*cos(theta1)*cos(theta2))*sin(theta3)*cos(alpha3) + (sin(theta1)*cos(theta2) + sin(theta2)*cos(alpha2)*cos(theta1))*cos(theta3) - sin(alpha2)*sin(alpha3)*sin(theta3)*cos(theta1), (-sin(theta1)*sin(theta2) + cos(alpha2)*cos(theta1)*cos(theta2))*cos(alpha3)*cos(theta3) - (sin(theta1)*cos(theta2) + sin(theta2)*cos(alpha2)*cos(theta1))*sin(theta3) - sin(alpha2)*sin(alpha3)*cos(theta1)*cos(theta3), -(-sin(theta1)*sin(theta2) + cos(alpha2)*cos(theta1)*cos(theta2))*sin(alpha3) - sin(alpha2)*cos(alpha3)*cos(theta1), a3*(sin(theta1)*cos(theta2) + sin(theta2)*cos(alpha2)*cos(theta1)) - d2*sin(alpha2)*cos(theta1) - d3*(-sin(theta1)*sin(theta2) + cos(alpha2)*cos(theta1)*cos(theta2))*sin(alpha3) - d3*sin(alpha2)*cos(alpha3)*cos(theta1)], [sin(alpha2)*sin(theta2)*cos(theta3) + sin(alpha2)*sin(theta3)*cos(alpha3)*cos(theta2) + sin(alpha3)*sin(theta3)*cos(alpha2), -sin(alpha2)*sin(theta2)*sin(theta3) + sin(alpha2)*cos(alpha3)*cos(theta2)*cos(theta3) + sin(alpha3)*cos(alpha2)*cos(theta3), -sin(alpha2)*sin(alpha3)*cos(theta2) + cos(alpha2)*cos(alpha3),a3*sin(alpha2)*sin(theta2) + d2*cos(alpha2) - d3*sin(alpha2)*sin(alpha3)*cos(theta2) + d3*cos(alpha2)*cos(alpha3)], [0,0,0,1]])

with SymPy I'll substitute the value

substitution = A.subs(alpha2, (-pi/2))

and I'll have a lot of 6.12323399573677e-17 in the middle of it.


Use the symbolic pi from SymPy, not the numeric pi from math or NumPy modules. This is what you are probably doing:

from sympy import sin, cos from math import pi print([sin(pi), cos(pi/2)]) # [1.22464679914735e-16, 6.12323399573677e-17]

And this is what you should do instead:

from sympy import sin, cos, pi print([sin(pi), cos(pi/2)]) # [0, 0]


You could always make a function! Something like

from math import sin as oldsin def sin(x): if x % pi == 0: #if x is an integer mult of pi, like pi, 2pi, -7pi, etc. return 0 else: return oldsin(x)


This is a more general answer than what you may be looking for, but you could try to round down the values to zero. In Numpy, you can do this quite easily: the below example will round any values in the array less than 1e-14 to zero.

threshold = 1e-14 array[array < threshold] = 0

Hope this helps.


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