Question 553615
If cot<font face = "symbol">q</font> = -3/4 and sin<font face = "symbol">q</font> < 0, then the value of cos<font face = "symbol">q</font> is:
<pre>

The cotangent, being {{{-3/4}}} is negative, and the cotangent is 
negative in quadrants II and IV

The sine, being < 0, is negative, and the sine is 
negative in quadrants III and IV.

Therefore <font face = "symbol">q</font> is in quadrant IV

The cotangent is {{{x/y}}} = {{{3/(-4)}}}

So we draw a triangle in quadrant IV with its hypotenuse as
the terminal side of <font face = "symbol">q</font>, its shorter leg x=+3 and its longer leg  
y=-4.  The shorter leg will be taken positive since it goes right,
and the longer leg will be taken negative because it goes down. 
The angle <font face = "symbol">q</font> is indicated by the red arc.

{{{drawing(400,400,-5,5,-5,5, graph(400,400,-5,5,-5,5),
locate(1.3,.5,x=3), locate(3.1,-1.7,y=-4), locate(1.1,-1.7,r),
red(arc(0,0,1.9,-1.9,0,306.8698976),locate(-.7,1.3,theta)), 

green(line(0,0,3,0), line(3,0,3,-4),line(0,0,3,-4)) )}}} 

Now we calculate r, the hypotenuse, by the Pythagorean theorem:

r² = x² + y²
r² = (3)² + (-4)²
r² = 9 + 16
r² = 25
 r = 5

{{{drawing(400,400,-5,5,-5,5, graph(400,400,-5,5,-5,5),
locate(1.3,.5,x=3), locate(3.1,-1.7,y=-4), locate(.6,-1.7,r=5),
red(arc(0,0,1.9,-1.9,0,306.8698976),locate(-.7,1.3,theta)), 

green(line(0,0,3,0), line(3,0,3,-4),line(0,0,3,-4)) )}}} 

Now since we want cos(<font face = "symbol">q</font>), we know that cos(<font face = "symbol">q</font>) = {{{x/r}}} = {{{(3)/5}}}.
It is positive as we would expect an angle's cosine to be in quadrant IV.

Edwin</pre>