SOLUTION: if sin B=-1/5 with B in Quadrant 3, find cos B/2

Draw angle B in 3rd quadrant:

Since sine = , we make y=-1 and r=5, so that the sin(B)
will be .



But we want cos(B) which is 

Then we find x by the Pythagorean relation: 









Since x goes to the left, we know to take the negative
value 



So 

Now use the formula



We need to determine whether to use the + or the - sign.

Since B is in the 3rd quadrant,  
                                 

Therefore  is in quadrant 2, and the cosine will be
negative, so





Multiply numerator and denominator by 5 to simplify compound fraction:



That is the correct answer, but it has a square root within a square
root, and often such an expression can be simplified to the sum or
difference of two square roots of rational numbers.  Assume rational 
a and b exist so that



I will assume a difference since a difference occurs under the square
root.

Square both sides



Multiply both sides by 10



We set rational parts equal and irrational parts equal

,  
,     
                
                 


Solve  for b, 
Substitute in 



 Multiply through by 25a







, 

Using , substitute in




Using , substitute in




So we either have , or ,

But since  is negative we must choose
the second answer:

,

So



Edwin