Express sin32+sin54 as sum or difference.  

Expert Answers
Borys Shumyatskiy eNotes educator| Certified Educator


This expression is already a sum of two numbers, `sin(32)` and `sin(54).` Probably you want or express it as a product, or as an expression involving trigonometric functions of sum or difference of the arguments, `32` and `54.`

For this, we need the well-known formula

`sin(x) + sin(y) = 2sin((x+y)/2)cos((x-y)/2),`

which is true for any real `x` and `y.`

For the given numbers it gives us

`sin(32) + sin(54) = 2sin((32+54)/2)cos((32-54)/2) =`

`= 2sin(43)cos(-11) = 2sin(43)cos(11).`


If you misspelled the expression, and it is actually `sin(32)*sin(54),` then we really can express it as a sum with the help of the formula

`sin(x)sin(y) = 1/2(cos(x-y)-cos(x+y)),`

for the given numbers it is

`sin(32)sin(54) = 1/2(cos(-22)-cos(86)) = 1/2(cos(22)-cos(86)).`