Question

Express f(x) in the form f(x) = A Sin (bx+c) for the function: f(x) = sin (x) + sqrt3 cos (x)

Answer 1

\[disambiguation: f(x) = \sin(x) + \sqrt{3}\cos (x) \]

Answer 2

alright, i really have no way to explain it, it was a guess and check on my part, but I got:
\[f(x) = 2\sin(x+\frac{\pi}{3})\]
expanding this using sum of two angle formulas gives:\[2\sin(x+\frac{\pi}{3})=2(\sin(x)(\frac{1}{2})+\cos(x)(\frac{\sqrt3}{2}))\]
which simplifies back to what you started with.

Answer 3

\[A \sin(x + \phi) = A \sin x \cos \phi + A \cos x \sin \phi = \sqrt{3} \cos x\] for all x.
Hence,
\[A \cos \phi = 1\]\[ A \sin \phi = \sqrt{3}.\]We have, then that
\[ \tan \phi = \sqrt{3},\]so
\[\phi = \pi /3.\]If we square both sides of both equations and then add, we get
\[(A \cos \phi) ^{2} + (A \sin \phi)^{2} = 4,\]so A = 2, and the answer is
\[2 \sin(x + \pi/3).\]

Answer 4

\[f(x) = 2 \sin (x+\pi /3)\] is the answer given in the key. I very much appreciate the work! I will work to decipher the steps.

Answer 5

Right on:^)