SYN projection equations in equatorial coordinates

If ( $\alpha$ , $\delta$ ) and ( $\alpha_{0}^{}$ , $\delta_{0}^{}$ ) are the J2000.0 right ascension and declination of e and e₀ then

cos $\displaystyle \theta$ sin $\displaystyle \phi$	=	cos $\displaystyle \delta$ sin( $\displaystyle \alpha$ - $\displaystyle \alpha_{0}^{}$ )	(9)
cos $\displaystyle \theta$ cos $\displaystyle \phi$	= -	sin $\displaystyle \delta$ cos $\displaystyle \delta_{0}^{}$ + cos $\displaystyle \delta$ sin $\displaystyle \delta_{0}^{}$ cos( $\displaystyle \alpha$ - $\displaystyle \alpha_{0}^{}$ )
sin $\displaystyle \theta$	=	sin $\displaystyle \delta$ sin $\displaystyle \delta_{0}^{}$ + cos $\displaystyle \delta$ cos $\displaystyle \delta_{0}^{}$ cos( $\displaystyle \alpha$ - $\displaystyle \alpha_{0}^{}$ )

These may be substituted into equations (7) to obtain the SYN projection equations in J2000.0 equatorial coordinates.