The tangent is negative in the 2nd and 4th quadrants.  Its referent angle
is the first quadrant angle that has +1 for its tangent.  That's  or .  
Its sine is  and its cosine is also 

So to get the smallest 2nd quadrant positive solution, we either

subtract 45° from 180° to get it into the 2nd quadrant, and get 
180°-45° = 135°.  Its sine is  and it's cosine is .

or we

subtract  from  to get it into the 2nd quadrant, 
and get 
, and its sine is  and its cosine is 

That's the 2nd quadrant solutions worked out in both degrees and radians.

------

To get the smallest 4th quadrant positive solution, we either

subtract 45° from 360° to get it into the 4th quadrant, and get 
360°-45° = 315°,

or we

subtract  from  to get it into the 4th quadrant, 
and get 


.

The sine is , and the cosine is 

That's the 4th quadrant solutions worked out in both degrees and radians.  

So if theta is in the 2nd quadrant its sine is  and its 
cosine is , and

if theta is in the 4th quadrant its sine is  and its cosine 
is .

Edwin