I am following a tutorial in Given rotations. Suppose I have the quantum state $|100\rangle$ and would like to use two Givens $G$ with angle $\theta, \phi$ rotation to transform it into a superposition of $\alpha|100\rangle+\beta|010\rangle+\gamma|001\rangle$. I denote $\cos(\frac\cdot 2), \sin(\frac\cdot 2)$ as $c(\cdot), s(\cdot)$
Consider $G(1,4,\phi)$ which maps $|100\rangle$ to $|001\rangle$ $$G(1,4,\phi)=\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & c(\phi) & 0 & 0 & -s(\phi)& 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0& 0 & 0\\ 0 & s(\phi)& 0 & 0 & c(\phi) & 0& 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1& 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0& 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0& 0 & 1 \end{bmatrix}$$ and $G(2,4,\theta)$ which maps $|100\rangle$ to $|010\rangle$ $$G(2,4,\theta)=\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & c(\theta)& 0 & -s(\theta)& 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0& 0 & 0\\ 0 & 0 & s(\theta)& 0 & c(\theta)& 0& 0 &0\\ 0 & 0 & 0 & 0 & 0 & 1& 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0& 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0& 0 & 1 \end{bmatrix}$$
$G(2,4,\phi)G(1,2,\theta)$ is the mapping that I needs, $$G:=G(2,4,\phi)G(1,4,\theta)=\begin{bmatrix} 1&0&0&0&0&0&0&0\\ 0&c(\phi)&0&0&-s(\phi)&0&0&0\\ 0&-s(θ)s(\phi)&c(θ)&0&-cθs(\phi)&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&c(θ)s(\phi)&s(θ)&0&c(θ)c(\phi)&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1\end{bmatrix}$$
Now $G|100\rangle=-s(\phi)|001\rangle-s(\phi) c(\theta)|010\rangle +c(\theta) c(\phi) |100\rangle$. However the supposed solution is $c(\theta)s(\phi)|001\rangle - s(\theta)|010\rangle + c(\theta)c(\phi)|100\rangle$. I don't understand how the 1st and 2nd coefficient is that way
