Use the sumtoproduct formula sinu sinv 2 sinu v2 cosu v2

Use the sum-to-product formula sin(u) + sin(v) = 2 sin(u + v/2) cos(u - v/2) to simplify the expression: If sin(48 degree) + sin(12 degree) = sin(A degree), with 0

Solution

21. sin 48 + sin 12

we have to use the following formula

sin A + sin B= 2 sin (A+B)/2 cos (A-B)/2

sin 48 + sin 12= 2 sin (48+12)/2 cos (48-12)/2

                         = 2 sin 30 cos 18= cos 18= cos(90-72) sin 72

Therefore A=72

22. 2 sin 52.5 sin 97.5

       We have to use the following formula

       2sin u sin v= cos(u-v) - cos(u+v)

      2sin 52.5 sin 97.5 = cos(97.5- 52.5) - cos(97.5 +52.2)

                                      =   cos 45   - cos 150

                                      = (sqrt2)/2 - (-sqrt3/2)

                                       = ((sqrt2) + (sqrt3))/2

Therefore A=3