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

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Answer 1 · 2016-12-01T15:56:44

Use the exponential form of the trigonometric functions:

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

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

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

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

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

How do you simplify sin 2x times sin7xsin 2x times sin7x?