For calculating the identity for $\rm \large \sin(a + b + c)$, can I use the sum formula twice?

Like $\rm \large \sin((a + b) + c) = \sin(a + b)\cos(c) + \sin(c)\cos(a + b)$

Am I on the right track?

Question

For calculating the identity for $\rm \large \sin(a + b + c)$, can I use the sum formula twice? 

Like $\rm \large \sin((a + b) + c) = \sin(a + b)\cos(c) + \sin(c)\cos(a  + b)$

Am I on the right track?

baldymcgee6 · Answer

never heard of such an identity, but maybe they just don't teach it.

parthkohli · Answer

This is a whole lot complicated than it seems...$$\rm \large (\sin(a)\cos(b)+\sin(b)\cos(a))\cos(c) + \sin(c)(\cos(a)\cos(b)-\sin(a)\sin(b))$$

baldymcgee6 · Answer

just try plugging in values for a, b and c and see if the LHS = RHS

parthkohli · Answer

lol, I know, just trying to calculate stuff. :)

baldymcgee6 · Answer

We know that Sin(A+B) = SinA CosB + CosA SinB
Therefore Sin(A+B+C) = Sin(A+B)CosC + Cos(A+B)SinC

Now Sin(A+B) = SinACosB + CosASinC. and
Cos(A+B) = CosACosB -SinASinC

Substituting we get

Sin(A+B+C) = (SinACosB +CosASinB)CosC + (CosACosB -SinASinB)SinC

= SinACosBCosC + CosASinBCosC + CosACosBSinC - SinASinBSinC

parthkohli · Answer

Yes, that's what I am doing... and wow.