Understanding this proof for $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$ from Gelfand

Question

We would like to have a small question concerning this proof for $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$ from Gelfand's Trigonometry

Why is it true that $\frac{q}{d}=\frac{a}{d}$. Is it necessary to use properties of similar triangles?

score 4 · Accepted Answer · answered Jun 15 '18 at 07:41

4

Not at all. You simply use the fact that $q=a$.

answered Jun 15 '18 at 07:41

Is it possible to prove $q=a$ using elementary geometry? I think I will upload a fuller version from the book. – James Warthington Jun 15 '18 at 07:48
Ay, I also misunderstood the picture, thank you! So $PD=q$ and $q=QP$. – James Warthington Jun 15 '18 at 08:03
@JamesWarthington No. $q=PQ$ and $p=DP$. – José Carlos Santos Jun 15 '18 at 08:05
I can't help but mention my own picture-proof of this identity, whose geometry is essentially the same as Gelfand's, but without the visual and cognitive clutter of overlapping triangles. I like to think that my figure is less prone to be misunderstand. – Blue Jun 15 '18 at 10:28
1

Hi Blue! Recognize you! I like your visual proofs very much! This particular proof for the formula is very neat. So I know like 4 proofs already. :) – James Warthington Jun 15 '18 at 12:08

1 Answers1