Question

How do you prove `sin 5pi/12 = sin pi/6 cos pi/4 + cos pi/6 sin pi/4`?

Answer 1

Note that `pi/6+pi/4=(2pi)/12+(3pi)/12=(5pi)/12`.

Also, recognize the sine addition formula:

`sin(A+B)=sin(A)cos(B)+cos(A)sin(B)`

So, we can write `(5pi)/12` as `pi/6+pi/4` and then apply the sine addition formula.

`sin((5pi)/12)=sin(pi/6+pi/4)=sin(pi/6)cos(pi/4)+cos(pi/6)sin(pi/4)`