How do you combine sin(4x) and cos(4x) into a single trigonometric function?

Answer 1

I suspect that what you want to do is make a linear combination

y = a cos4x + b sin4x

Now R Cos(u-v) = R cos u cos v + R sin u sin v

so you want to find R = sqrt( a^2 + b^2) and phase angle z such that
R cos z = a, R sin z = b
i.e. tan z = b/a or z = atan(b/a)

so a cos 4x + b sin 4x = R cos( 4x - atan(b/a) )

Answer 2

The question is not answerable in this simple form!!
For example from a very well known trig formula the following equation satisfies your question: 1 - squaredsin(4x) = squaredcos(4x) :-) . But I don't think you mean this???

Answer 3

does it not need to equal something?

If f(x) = sin(4x)cos(4x) then f(x) = 0.5sin(8x)

sin 4x = cos[4(90-x)]

Or square them both and add them:

(sin 4x)^2 + (cos 4x)^2 = 1

Dividing sin 4x by cos 4x yields tan 4x

Answer 4

Seems the question is unclear. Do you want to combine by adding sin4x and cos4x? subtract one from the other? some other algebraic combination? Do you want to graph the result?

Answer 5

sin(4x)/cos(4x)=tan(4x)