im having problems with this math question?

Question

How do you change f(x)=3cos(2x)+4sin(2x), into a single function?

Answer 1

f(x) = 3cos(2x)+4sin(2x)
√(3² + 4²) = 5
Divide and mutlipy by 5
So f(x) = 5*(3/5 cos(2x) + 4/5 sin(2x))
Now because (3/5)² + (4/5)² = 1 there exists an angle such that cos ε = 3/5 and sin ε = 4/5 (OR vice versa if you prefer)

So f(x) = 5*(cosεcos(2x) + sinεsin(2x)))
= 5 cos(2x - ε) where ε ≈ 53°07'48" ≈ 0.9273 rad

ie f(x) = 5cos(2x - 0.9273)

You could also have got to f(x) = 5 sin (2x + 0.6435) if you used sinε = 3/5 and cosε = 4/5 It is easy to show that these are the same as (2x + 0.6435) - (2x - 0.9273) = 1.5708 = π/2

and if x - y = x + (-y) = π/2
then x = π/2 - (-y)
So sin x = cos (-y) = cos y as the cosine function is an odd function (ie f(-x) = f(x))

In general:
if(x) = AsinY + BcosY
= √(A² + B²)(A/√(A² + B²) * sinY + B/√(A² + B²) * cosY
= √(A² + B²)(sinY cosε + cosY sinε)
= √(A² + B²)sin(Y + ε) where ε = arctan (B/A)
(also = √(A² + B²)cos(Y - ε) where ε = arctan (A/B)

and other variations can be more useful depending on the signs of A and B and a desire to have 0 < ε < π/2 (ie ε acute)

There are 4 possibilities f(x) = Rsin(Y + ε)
f(x) = Rsin(Y - ε)
f(x) = Rcos(Y + ε)
f(x) = Rcos(Y - ε), the latter two if A and B are opposite in sign.

Answer 2

When you are faced with a question like this, it is often helpful to write out all of the the implciations.

Do you remember your double angle formulae? How about expanding all of the double angles. Those double angle formulae will involve some squares of the single angles. Maybe these can be collected into something a lot simpler. Perhaps you will have a square of a cosine that you can rewrite in terms of the square of a sine or vice-versa. That will probably get you where you need to go.

Answer 3

Using trig identities: cos(2x) = (cosx)^2 - (sinx)^2 and sin2x = 2sinxcosx. Therefore, f(x) = 3(cosx)^2 - 3(sinx)^2 +8sinxcosx. ...Im sure that factors to something :P

Answer 4

knowing that cos(x)=sqrt(1-sin^2(x))
f(x)=3*sqrt(1-sin^2(2x)) + 4sin(2x)

Answer 5

cos(x)=sqrt(1-sin^2(x))
f(x)=3*sqrt(1-sin^2(2x)) + 4sin(2x)

Answer 6

Put it in your calculator.
3cos(2x)=2.732
4sin(2x)=1.651

1.651+2.732=4.383

Answer 7

why don't you do your own homework ideit