Question 1176584
<pre>
First you draw a picture of angle s in quadrant III, and draw a perpendicular
from the end of the terminal side up to the x-axis.

{{{drawing(300, 300,-7,7,-7,7, line(-8,0,8,0), line(0,-8,0,8),
line(0,0,-1,-sqrt(35)), line(-1,0,-1,-sqrt(35)), red(arc(0,0,1,-1,0,261)),
locate(-5,5,matrix(1,2,Angle, s))

 )}}}

You are given cos(s) = -1/6. You know that cosine = adjacent/hypotenuse,
which is x/r, so we label the x value as the numerator of -1/6, which
is -1, and we label the hypotenuse (the terminal side) as the denominator
of -1/6, which is 6. (The hypotenuse or terminal side is always taken
positive.

{{{drawing(300, 300,-7,7,-7,7, line(-8,0,8,0), line(0,-8,0,8),
line(0,0,-1,-sqrt(35)), line(-1,0,-1,-sqrt(35)), locate(-2,1,x=-1),
locate(-.3,-2.8,r=6), red(arc(0,0,1,-1,0,261)),
locate(-5,5,matrix(1,2,Angle, s))

 )}}}

We use the Pythagorean theorem to find the value of y:

{{{x^2+y^2=r^2}}}
{{{(-1)^2+y^2=6^2}}}
{{{1+y^2=36}}}
{{{y^2=35}}}
{{{y=""+-sqrt(35)}}}

Since y goes down from the x-axis we give it a negative sign.
We label the opposite side, which is y, as -√35

{{{drawing(300, 300,-7,7,-7,7, line(-8,0,8,0), line(0,-8,0,8),
line(0,0,-1,-sqrt(35)), line(-1,0,-1,-sqrt(35)), locate(-2,1,x=-1),
locate(-.3,-2.8,r=6), red(arc(0,0,1,-1,0,261)), locate(-3.8,-2.8,y=-sqrt(35)),
locate(-5,5,matrix(1,2,Angle, s)) 

 )}}}

We will need the sine and cosine of s,

{{{cos(s) = -1/6}}} which is given, and
{{{sin(s) = opposite/hypotenuse=y/r=-sqrt(35)/6}}}




The same way, draw the angle t, and get this graph:

{{{drawing(300, 300,-7,7,-7,7, line(-8,0,8,0), line(0,-8,0,8),
line(0,0,-3,-4), line(-3,0,-3,-4), locate(-2.7,.8,x=-3),
locate(-1.5,-2,r=5), red(arc(0,0,1,-1,0,234)), locate(-5,-1.7,y=-4),
locate(-5,5,matrix(1,2,Angle, t)) 

 )}}}

We will need the sine and cosine of t,

{{{cos(s) = -3/5}}} which is given, and
{{{sin(s) = opposite/hypotenuse=y/r=-4/5}}}

Now we use the formula for 

{{{cos(s-t) = cos(s)cos(t)+sin(s)sin(t)}}}

{{{cos(s-t) = (-1/6)(-3/5)+(-sqrt(35)/6)*(-4/5)}}}

{{{cos(s-t) = 3/30+(4sqrt(35)/30)}}}

{{{cos(s-t) = (3+4sqrt(35))/30}}}

Edwin</pre>