SOLUTION: Given that sin x equals {{{-sqrt(2)/2}}} and that cos x is negative, find the other functions of x and the value of x.

There are two ways to do it, graphical and identity.

Graphical way (the easier way).  To avoid conflicting notation with
x being the angle and also the name of the horizontal axis, let's
change the problem to 

Given that sinq =  and that cosq is negative, find the other 
functions of q and the value of q. 

Since sinq and cosq are both negative numbers, we must be
in the third quadrant, and since we know that sin45° = 
we know that the reference angle is 45°.  Therefore we draw the
angle in the 3rd quadrant with reference angle 45°:



Now we just use the definitions of the trig functions:

sinq = y/r =  = 
cosq = x/r =  = 
tanq = y/x =  = 
cotq = x/y =  =  
secq = r/x =  = 
cscq = r/y =  = 

The hard way is by identities.  If your teacher requires
you to do it by identities such as sin²x + cos²x = 1, then post again
asking how.  But you'll just get the same answers as above.

In degrees the value of q is

q = 225° + 360°n where n is any integer

In radians the value of q is

q = 5p/4 + 2np, where n is any integer.

Edwin