How to derive the bank angle of an aircraft from its roll angle and pitch angle?

Question

From Young (2017) (https://onlinelibrary.wiley.com/doi/book/10.1002/9781118534786) it is stated that we can define the bank angle ($\Phi$) of an aircraft as the angle between its Y body axis and the horizontal plane. He then states the following equivalence:

$\sin(\Phi)=\sin(\phi)\cos(\theta)$

where $\theta$ is the pitch angle of the aircraft and $\phi$ is its roll angle. I want to derive this equivalence and this is my attempt so far which yields an alternative expression:

I define a global axis system $E=(O,X_e,Y_e,Z_e)$ where $O=\begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix}$ is the origin of the system, and $X_e=\begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix}$, $Y_e=\begin{bmatrix} 0 \\ 1 \\ 0 \\ \end{bmatrix}$ and $Z_e=\begin{bmatrix} 0 \\ 0 \\ 1 \\ \end{bmatrix}$ are orthonormal unit vectors which define North, East and 'down' respectively.

I also define an aircraft body axis system, $B=(O,X_b,Y_b,Z_b)$, whose starting orientation and position is coincident with $E$.

I first rotate $B$ about the $Y_b$ axis an angle $\theta$, the pitch rotation, and then I rotate this rotated body axis system about its new $Xb$ axis an angle $\phi$, the roll rotation. The relevant rotation matrices are:

$R_Y(\theta)$=\begin{bmatrix} \cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0 \\ -\sin(\theta) & 0 & \cos(\theta) \end{bmatrix}

$R_X(\phi)$=\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\phi) & -\sin(\phi) \\ 0 & \sin(\phi) & \cos(\phi) \end{bmatrix}

which when applied in the correct order to specify the rotations outlined in the text above yields the composite matrix:

$R_Y(\theta)R_X(\phi)=R=$\begin{bmatrix} \cos(\theta) & \sin(\theta)\sin(\phi) & \sin(\theta)\cos(\phi) \\ 0 & \cos(\phi) & -\sin(\phi) \\ -\sin(\theta) & \cos(\theta)\sin(\phi) & \cos(\theta)\cos(\phi) \end{bmatrix}

I then have $B2=RB=R$ which is the body axis system after the rotations. $Y_{B2}= \begin{bmatrix} \sin(\theta)\sin(\phi) \\ \cos(\phi) \\ \cos(\theta)\sin(\phi) \end{bmatrix}$, the y axis of B2, and its projection onto the horizontal plane is $Y_{pB2}= \begin{bmatrix} \sin(\theta)\sin(\phi) \\ \cos(\phi) \\ 0 \end{bmatrix}$. I can then say that the cosine of the angle between them, the bank angle $\Phi$, is the normalised dot product of the two vectors:

\begin{align} \cos(\Phi)=\frac{Y_{B2}\cdot Y_{pB2}}{|Y_{B2}||Y_{pB2}|} \end{align}

Doing the computation I am left with:

\begin{align} \cos(\Phi)=\sin(\theta)\sin(\phi) + \cos(\phi) \end{align}

Whereas I want:

\begin{align} \sin(\Phi)=\sin(\phi)\cos(\theta) \end{align}

Any advice on where I have gone wrong would be much appreciated.

Answer 1

$$\vec{y'}=\left[ \begin {array}{c} \sin \left( \theta \right) \sin \left( \phi \right) \\\cos \left( \phi \right) \\ \cos \left( \theta \right) \sin \left( \phi \right) \end {array} \right] $$

$\Rightarrow$

$$\tan(\Phi)=\frac{\sin(\Phi)}{\cos(\Phi)}=\frac{\vec{y'}_x}{\vec{y'}_y}=\frac{\sin(\theta)\,\sin(\phi)}{\cos(\phi)}$$

thus:

$$\sin(\Phi)=\sin(\theta)\,\sin(\phi)$$

but this angle is not the angle between y‘ and the plane xy