Derive the trigonometric addition formula for sine :

Question

anonymous · Answer

$$\sin(a+b)=sinacosb+cosasinb$$

anonymous · Answer

also, derive these identities using the addition or subtraction formulas for sine or cosine :
$$sinasinb=\frac{ 1 }{ 2 }(\cos(a-b)-\cos(a+b))$$

irishboy123 · Answer

easiest i know is:

$ e^{i \ a} =  (cos a + i \ sin a)$
$ e^{i \ b} = (cosb + i \ sin b)$

$ e^{i \ a} . e^{i \ b} = e^{i \  (a+b)} = $

then multiply out and equate the real and complex bits  

for the second bit, just churn the formula.  eg expand the RHS using the cos (a+b) and cos (a-b).

irishboy123 · Answer

if you post some stuff, i'm sure you will get all the help you need :p

anonymous · Answer

I've never seen that first formula before . I'm even more confused

freckles · Answer

you can derive cos(a-b)=cos(a)cos(b)+sin(a)sin(b) using the distance formula 
then you can use cos(a-b)=cos(a)cos(b)+sin(a)sin(b) along with cofunction identities 
to show sin(a+b)=sin(a)cos(b)+sin(b)cos(a)

anonymous · Answer

what are cofunction identities ?

anonymous · Answer

This is an algebra 2 class and they threw in a trig section

freckles · Answer

you are asking to derive the formula sin(a+b)=sin(a)cos(b)+sin(b)cos(a) right?

anonymous · Answer

yes

freckles · Answer

hmmm maybe you don't remember the name of the identity 
but you remember that cos(90-A)=sin(A) and sin(90-A=cos(A)?
If not you easily show these by drawing right triangles and comparing the above ratios.

anonymous · Answer

the notes didn't name the identities and I'm sure I learned that . Trigonometry is just really hard for me to understand

irishboy123 · Answer

hi @jxaf sorry for the confusion, i should have asked first what you knew. do you think you could follow this proof, with some guidance. https://gyazo.com/be936132eb2a67f24dbd6841f0cf94a3

anonymous · Answer

Um, that looks really complex. Do you want a picture of my notes to see what I learned ?

irishboy123 · Answer

yeah, go for it!

anonymous · Answer

okay it might take a bit cause my computer is really slow

irishboy123 · Answer

take your time.

irishboy123 · Answer

well i've drawn the shape for the proof anyways just in case it comes in useful.   i think it's more straightforward that it might appear.

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