Talk:Polarization correction

Angle		Adjacent	Opposite	Hypotenuse	Sine = O/H	Cosine = A/H	Tangent = O/A
$\chi$	Azimuth on detector (relative to $q_{z}$ axis)	$z$	$x$	$r={\sqrt {x^{2}+z^{2}}}$	$\sin \chi ={\frac {x}{r}}$	$\cos \chi ={\frac {z}{r}}$	$\tan \chi ={\frac {x}{z}}$
$2\theta$	Full scattering angle (between incident beam and scattering)	$y$	$r={\sqrt {x^{2}+z^{2}}}$	$R={\sqrt {x^{2}+y^{2}+z^{2}}}$	$\sin 2\theta ={\frac {r}{R}}$	$\cos 2\theta ={\frac {y}{R}}$	$\tan 2\theta ={\frac {r}{y}}$
$\gamma$	In-plane angle	$y$	$x$	$h={\sqrt {x^{2}+y^{2}}}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin \gamma = \frac{x}{h}}	$\cos \gamma ={\frac {y}{h}}$	$\tan \gamma ={\frac {x}{y}}$
$\delta$	Elevation angle	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h = \sqrt{x^2 + y^2}}	$z$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = \sqrt{x^2 + y^2 + z^2}}	$\sin \delta ={\frac {z}{R}}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos \delta = \frac{h}{R}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan \delta = \frac{z}{h}}

So:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \tan \gamma & = \frac{x}{y} \\ & = \frac{r \sin \chi}{r / \tan 2 \theta} \\ & = \tan 2 \theta \sin \chi \\ \gamma & = \tan^{-1} \left [ \tan 2 \theta \sin \chi \right ] \end{alignat} }

and:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \sin \delta & = \frac{z}{R} \\ & = \frac{r \cos \chi}{r / \sin 2 \theta} \\ & = \sin 2 \theta \cos \chi \\ \delta & = \sin^{-1} \left [ \sin 2 \theta \cos \chi \right ] \end{alignat} }

and so:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} P_h & = 1 - \cos^2 \delta \sin^2 \gamma \\ & = 1 - \left( 1 - \left[ \sin 2 \theta \cos \chi \right]^2 \right ) \frac{ \left [ \tan 2 \theta \sin \chi \right ]^2 }{ \left [ \tan 2 \theta \sin \chi \right ]^2 + 1 } \\ & = 1 - \sin^2(2 \theta) \sin^2(\chi) \end{alignat} }