My Math Forum eigenvectors of matrix of eigenvectors entries

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 August 13th, 2016, 03:08 AM #1 Newbie   Joined: Oct 2014 From: China Posts: 12 Thanks: 2 eigenvectors of matrix of eigenvectors entries The standard method (Characteristic polynomial) for the eigenvectors of $$\begin{bmatrix} -\sin(y)\cos(z) & \sin(y)\cos(x) & \sin(x)\cos(z) \\ \sin(y)\cos(x) & -\sin(z)\cos(x) & \sin(z)\cos(y) \\ \sin(x)\cos(z) & \sin(z)\cos(y) & -\sin(x)\cos(y) \\ \end{bmatrix}$$ leads to a complicated polynomial (degree 3) that cannot be easily solved. Is there any other methods for the eigenvectors of this particular matrix (there are a lot of symmetry inside)? Thanks from 123qwerty
 August 17th, 2016, 07:55 AM #2 Member     Joined: Aug 2007 Posts: 42 Thanks: 4 Three positive eigenvalues (expected, as the matrix is symmetric) First: (cos(z)*sin(y))/3 - (cos(x)*sin(z))/3 - (cos(y)*sin(x))/3 + ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + \\(cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)/(((((cos(y)*sin(x) + cos(x)*sin(z) - \\ cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + \\(cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 - (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 \\ + (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 - (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^2 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - \\(cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)^3)^(1/2) - (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*\\(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) +\\ cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 + (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 - (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 + (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^(1/3) +\\ (((((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 +\\ (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 - (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 + (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 - \\(3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^2 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 -\\ (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)^3)^(1/2) - (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*\\(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + \\cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 + (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 - (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 + (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^(1/3) Second: (cos(z)*sin(y))/3 - (cos(x)*sin(z))/3 - (cos(y)*sin(x))/3 - (3^(1/2)*(((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)/(((((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 - (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 + (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 - (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^2 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)^3)^(1/2) - (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 + (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 - (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 + (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^(1/3) - (((((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 - (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 + (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 - (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^2 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)^3)^(1/2) - (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 + (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 - (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 + (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^(1/3))*i)/2 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)/(2*(((((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 - (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 + (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 - (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^2 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)^3)^(1/2) - (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 + (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 - (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 + (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^(1/3)) - (((((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 - (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 + (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 - (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^2 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)^3)^(1/2) - (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 + (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 - (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 + (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^(1/3)/2 Third: (cos(z)*sin(y))/3 - (cos(x)*sin(z))/3 - (cos(y)*sin(x))/3 + (3^(1/2)*(((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)/(((((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 - (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 + (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 - (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^2 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)^3)^(1/2) - (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 + (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 - (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 + (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^(1/3) - (((((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 - (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 + (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 - (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^2 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)^3)^(1/2) - (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 + (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 - (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 + (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^(1/3))*i)/2 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)/(2*(((((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 - (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 + (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 - (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^2 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)^3)^(1/2) - (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 + (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 - (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 + (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^(1/3)) - (((((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 - (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 + (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 - (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^2 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^2/9 + (cos(x)^2*sin(y)^2)/3 + (cos(z)^2*sin(x)^2)/3 + (cos(y)^2*sin(z)^2)/3 - (cos(x)*cos(y)*sin(x)*sin(z))/3 + (cos(y)*cos(z)*sin(x)*sin(y))/3 + (cos(x)*cos(z)*sin(y)*sin(z))/3)^3)^(1/2) - (cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))^3/27 - ((cos(y)*sin(x) + cos(x)*sin(z) - cos(z)*sin(y))*(cos(x)^2*sin(y)^2 + cos(z)^2*sin(x)^2 + cos(y)^2*sin(z)^2 - cos(x)*cos(y)*sin(x)*sin(z) + cos(y)*cos(z)*sin(x)*sin(y) + cos(x)*cos(z)*sin(y)*sin(z)))/6 + (cos(x)^2*cos(y)*sin(x)*sin(y)^2)/2 + (cos(x)*cos(z)^2*sin(x)^2*sin(z))/2 - (cos(y)^2*cos(z)*sin(y)*sin(z)^2)/2 + (3*cos(x)*cos(y)*cos(z)*sin(x)*sin(y)*sin(z))/2)^(1/3)/2 Last edited by Rebesques; August 17th, 2016 at 07:59 AM.

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