Expert: Paul Klarreich - 11/20/2010

Question
If triangle ABC is right angled at c prove that
b=c-c(sinA).tan(A/2)

Answer
Questioner:   Dhananjay  
Country:  India
Category:  Advanced Math
Private:  No
 
Subject:  Maths-Trignometry

If triangle ABC is right angled at c (you mean 'C', don't you?) prove that b = c - c sinA  tan(A/2)

Your picture is:
     A
    /|
   / |
c /  | b
 /   |
/    |
B_____|C
  a


And basic trigonometry shows that  b = c sin B.

Now you can easily look up or prove that:

         1 - cos A
tan A/2 = ----------  << standard half-angle formula
           sin A

You have:

b = c - c sinA tan(A/2)

              1 - cos A
b = c - c sinA ----------  << use that formula
                sin A


b = c - c (1 - cos A)   << cancel

b = c cos A    << simplify

But cos A = sin B, since A = 90 - B

b = c sin B

Done.