Question

How do you simplify `sin 2x times sin7x`?

Answer 1

Use the exponential form of the trigonometric functions:

`sin(7x) = (e^(7ix) - e^(-7ix))/(2i)`

`sin(2x) = (e^(2ix) - e^(-2ix))/(2i)`

`sin(2x)sin(7x) = (e^(7ix) - e^(-7ix))/(2i)(e^(2ix) - e^(-2ix))/(2i) =`

`= (e^(9ix)-e^(5ix)-e^(-5ix) + e^(-9ix))/(2i)^2 = (e^(5ix)+e^(-5ix))/4-(e^(9ix)+e^(-9ix))/4`

`sin(2x)sin(7x) = frac (cos(5x)-cos(9x)) 2`