Use the identity for sin(A+B) to find the exact value of sin 75.

From reading the question, we need to use the addition formula for sin(A+B)= sinAcosB+cosAsinB. So we need to think what are two common values for sine and cosine that we know and have exact answers for that also add up to 75? Our 'common angles' for sin and cosine are 30, 45 and 60, and we know that 30+45=75 so we can set A=30 and B=45. Plugging this into our formula we get:

sin(75)=sin(30+45)=sin(30)cos(45)+cos(30)sin(45)

Using our knowledge that sin(30)=1/2, cos(45)=sin(45)=sqrt(2)/2 and cos(30)=sqrt(3)/2, we can substitute them into the above formula:

sin(75)=1/2sqrt(2)/2+sqrt(3)/2sqrt(2)/2=(sqrt(2)+sqrt(2)*sqrt(3))/4

We can see that sqrt(2) is a common factor of the numerator so we can factorise giving us our answer:

sin(75)=sqrt(2)*(1+sqrt(3))/4