Sign Issue, Side Issue

$sin(\theta)$ is still positive on the other side of the center $O$.  $\theta$ is always positive, when measured consistently.

From this reason, both $\small{v_{cir}}$ and $\small{v_{shm}}$ are negative at the same time.  Where, from the post "Twirl Plus SHM, Spinning Coin" dated 17 Jul 2015,

$v^{ 2 }_{ cir }=cos(\theta )(x+x_{ z })\left\{ -2{ c^{ 2 } }cos(\theta )ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } }  } x))+\cfrac { \psi _{ d } }{ m }  \right\}$

and

$v^{ 2 }_{ shm }=sin(\theta )(x+x_{ z })\left\{ -2{ c^{ 2 } }cos(\theta )ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } }  } x))+\cfrac { \psi _{ d } }{ m }  \right\}$

their signs depend on the common term,

$(x+x_{ z })\left\{ -2{ c^{ 2 } }cos(\theta )ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } }  } x))+\cfrac { \psi _{ d } }{ m }  \right\}$

When

$v_{cir}=v_{shm}=c$

$x_d=x_v$

circular motion is in a plane whose normal at the center of the circular path passes above $O$, the center of oscillation.  This occurs instantaneously and does not alter the analysis presented.

Have a nice day.