Question 831818
<pre>
Given: {{{cos(A)=-7/25}}}, {{{pi < A < 3pi/2}}} 
       {{{sin(B)=-3/5}}}, {{{3pi/2 <B<2pi}}}

Find: {{{sin(A-B)}}}
      {{{tan(A-B)}}}

{{{cos(A)=-7/25}}}, {{{pi < A < 3pi/2}}}. That's Q3 

Draw the picture of angle A in the third quadrant Q3. 
Since the cosine is adjacent/hypotenuse or x/r, and since we are given 
{{{cos(A)=-7/25}}}, make x=-7 and r={{{25}}}, and calculate y using the 
Pythagorean theorem:

    x²+y² = r²
 (-7)²+y² = 25²
    49+y² = 625
       y² = 576
        y = ±&#8730;<span style="text-decoration: overline">576</span>
        y = ±24, we take y = -24 

We take y negative because it goes down from the x-axis, 
and we ALWAYS take r positive because it begins on the
right side of the x-axis and doesn't change its sign when
it swings around into the other quadrants. 

{{{drawing(300,300,-2,2,-2,2,line(-30,0,30,0),line(0,-30,0,30),
green(line(0,0,-.583,-2.083)),line(-.583,0,-.583,-2.083),locate(-.6,.3,x=-7),locate(-1.3,-.9,y=-24),
locate(.1,.3,A), red(arc(0,0,.8,-.8,0,250)),
locate(-.2,-.9,r=25) )}}}

{{{sin(A)=y/r=(-24)/25=-7/25}}}
{{{cos(A)=x/r=(-7)/25=-7/25}}}
{{{tan(A)=y/x=(-24)/(-7)=7/24}}}


--------------------------------

{{{sin(B)=-3/5}}}, {{{3pi/2 <B < 2pi}}}. That's Q4 
 

Draw the picture of angle B in the fourth quadrant Q4. Since the sine is
opposite/hypotenuse or y/r, and since we are given  {{{sin(B)=-3/5}}},
make y=-3 and r=5, and calculate x using the Pythagorean theorem:

    x²+y² = r²
 x²+(-3)² = 5²
     x²+9 = 25
       x² = 25
        y = ±&#8730;<span style="text-decoration: overline">25</span>
        y = ±5, we take y = -5 

We take y negative because it goes down from the x-axis, 
and we ALWAYS take r positive because it begins on the
right side of the x-axis and doesn't change its sign when
it swings around into the other quadrants. 

{{{drawing(300,300,-1,3,-2,2,line(-30,0,30,0),line(0,-30,0,30),
green(line(0,0,1.6,-1.2)),line(1.6,0,1.6,-1.2),locate(.8,.3,x=4),locate(1.7,-.5,y=-3),
locate(.1,.3,B), red(arc(0,0,.8,-.8,0,323)),
locate(.3,-.5,r=5) )}}}

-----------------------------------------------
{{{sin(A)=y/r=(-24)/25=-24/25}}}
{{{cos(A)=x/r=(-7)/25=-7/25}}}
{{{tan(A)=y/x=(-24)/(-7)=24/7}}}

--------------------------------------

{{{sin(B)=y/r=(-3)/5=-3/5}}}
{{{cos(B)=x/r=4/5}}}
{{{tan(B)=y/x=(-3)/-4=-3/4}}}

---------------------------------------

{{{sin(A-B) = sin(A)cos(B)-cos(A)sin(B) = (-24/25)(4/5)-(-7/25)(-3/5) = -96/125-21/125 = -117/125}}} 

{{{tan(A-B)=(tan(A)-tan(B))/(1+tan(A)tan(B))=(24/7-(-3/4))/(1+(24/7)(-3/4))=(24/7+3/4)/(1+(-72/28))=(96/28+21/28)/(28/28-72/28)=(117/28)/(-44/28)=(117/28)*(-28/44)=-177/44}}} 

Edwin</pre>