Can someone please explain solving for the exact angle using the sum or difference formula

Question

In my Pre-Calc, class we are solving for the exact angle using the sum or difference formula, I understand how to apply the formula, I just don't know how to get the "expanded" version of the question. For instance, in the example our teacher showed us, Sin5pi/12, he said it is equal to 3pi/12 + 2pi/12, which I understand to be our "nice" angles that we can get from the unit circle, my problem comes when we have to solve for Sin11pi/12, there are no 2 angles that add or subtract to give us 11pi/12, so can anyone pls explain to me how to solve using this formula, and how to get the angles that add or subtract to give us the question. If I could also get an explanation on how to solve for the exact angle of degrees, for example, sin105degrees and sin 195degreesThank you very much!

Answer 1

Try sin(11π /12) = sin(3π /12 + 8π /12) = sin(π /4 + 2π /3)

= sin(π /4)cos(2π /3) + cos(π /4)sin(2π /3)

Answer 2

sin(11π/12)=sin(π/4+2π/3)=sin(π/4)cos(2π/3)+sin(2π/3)cos(π/4)

=((√2)/2)*(-(1/2)+(√3)/2),,,right?, and exact?

isn't 105=90+15

so sin(105)=1*cos(15)+0

and cos(15)=+/-√((1+cos(30))/2),,,,choose plus since 15 degrees in first quadrant

and cos(30)=(√3)/2,,,exact when finding sin(105)

need to know all standard values along with half and double angle formulas