Question

If ABC is a triangle prove the following:

Answer 1

\[c \sin \frac{ A-B }{ 2 } = (a-b) \cos \frac{ C }{ 2 }\]

Answer 2

I'm not quite sure where to start, do you guys have any tips?

Thanks in advance!

Answer 3

In geometry, drawing a diagram (if not already provided) is the first step to solving a problem.
For the given problem, using the traditional conventions, the diagram looks like the following.

where A,B,C are angles, and a,b,c are side lengths opposite the corresponding lettered angles.
Does that correspond to your own diagram provided with the question?

Answer 4

They did not provide a picture of a triangle, (it's a section on trigonometric identities.) but I solved a previous "proof" using a right angled triangle. So, I suppose I should start there? Unless using a right-angled triangle is incorrect and I nee to use a triangle similar to the one you have drawn.

Answer 5

I'm giving this question a shot right now. would it be fair to say that \[\sin \frac{ A-B}{ 2} = Sin (\frac{ A }{ 2 } - \frac{ B }{ 2 }) = Sin \frac{ A }{2 }Cos \frac{ B }{2 }-Sin \frac{ B }{2 }Cos \frac{ A }{2 }\]

Answer 6

hence:

\[Sin \frac{ A }{2 }Sin \frac{ A +C}{2 } - Sin \frac{ B }{2 }Sin \frac{ B+C }{2 }\]

But how do I continue from this point?

Answer 7

you probably have to use the fact that
A+B+C= 180 deg
so for example, B+C= 180- A

Answer 8

Let,
\(\Large\frac{a}{sin~A}=\frac{b}{sin~B}=\frac{c}{sin~C}=k\)

\(\large(a-b)cos{\frac{C}{2}}=(k\cdot sin~A-k\cdot sin~B)sin(\frac{A+B}{2})\\\qquad\qquad\qquad\large=k\cdot [2sin(\frac{A+B}{2})cos(\frac{A+B}{2})]\cdot sin(\frac{A-B}{2})\\\qquad\qquad\qquad\large=k\cdot sin(A+B)\cdot sin(\large\frac{A-B}{2})\\\qquad\qquad\qquad\large=k\cdot sin~C\cdot sin(\large\frac{A-B}{2})\\\qquad\qquad\qquad\large=c\cdot sin(\large\frac{A-B}{2})\)

Answer 9

@Sachintha

Sorry for the delay! Thank you for the detailed working! I feel like the little trick of X/Sin X = k will prove useful. I'll study it in detail, thank you again!

Answer 10

@sachintha Bravo!

Answer 11

It was the way I learned to solve them from my teacher when I went school. :3