Given cos theta=4/9 and csc theta 0 find sin theta and tan theta

The correct question is
Given cos theta=4/9 and csc theta < 0 find sin theta and tan theta

we know that
csc theta=1/sin theta
if csc theta < 0
then 
sin theta < 0

we have that
cos theta=4/9

we know that
sin² theta+cos² theta=1
so
sin² theta=1-cos² theta> 1-(4/9)²> 1-(16/81)> 65/81
sin theta=-√(65/81)>-√65/9

the answer Part a) is 
sin theta=-√65/9

Part b) find tan theta

tan theta=sin theta/cos theta
tan theta=(-√65/9)/(4/9)> tan theta=-√65/4

the answer part b) is
tan theta=-√65/4

The answer would be a