Express sin 2x cos 4x as a sum of trigonometric functions
From the familiar
sin(A+B)=sin(A)cos(B)+cos(A)sin(B), and
sin(A-B)=sin(A)cos(B)-cos(A)sin(B)
Add both equations to get
sin(A+B)+sin(A-B)=2sin(A)cos(B) .....(3)
But
2x=3x-x
4x=3x+x,
so substitute A=3x, B=x into equation (3) and simplify to get your answer.
To express sin 2x cos 4x as a sum of trigonometric functions, we can utilize the trigonometric identity:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
Let's rewrite sin 2x cos 4x in a different form to match the identity:
sin 2x cos 4x = sin (2x + 4x)
Now, using the identity sin(A + B), we can expand this expression as:
sin (2x + 4x) = sin 2x cos 4x + cos 2x sin 4x
Hence, sin 2x cos 4x expressed as a sum of trigonometric functions is:
sin 2x cos 4x = sin 2x cos 4x + cos 2x sin 4x