amplitude and phase
A pure oscillation has two things you can adjust independently: how big the swing is, and where in its cycle it starts. Amplitude is the size — the farthest the mass gets from the centre. Phase is the head start — whether the cosine begins at a peak, a trough, or somewhere in between. Knowing both pins down the wave completely, the way knowing height and starting position pins down a swing on a playground.
Solving the spring equation usually gives an answer like x(t) = C1 cos(omega t) + C2 sin(omega t), a mix of a cosine and a sine. That mix is awkward to read, so we rewrite it as a single shifted cosine: x(t) = A cos(omega t - phi). The two forms are the same motion. The amplitude is A = sqrt(C1^2 + C2^2) (think of C1 and C2 as the legs of a right triangle and A as its hypotenuse), and the phase angle phi satisfies tan(phi) = C2/C1. Now you can read the size and timing of the swing at a glance.
This amplitude-phase picture is how engineers talk about vibrations and signals: a speaker cone's loudness is its amplitude, two waves that cancel are out of phase by half a cycle, and a forced system lags its driver by a phase angle. The conversion from sine-plus-cosine to a single cosine is the everyday bridge between the algebra of solving and the physics of describing.
x(t) = 3 cos(2t) + 4 sin(2t) becomes x(t) = 5 cos(2t - phi) with amplitude A = sqrt(3^2 + 4^2) = 5 and tan(phi) = 4/3, so phi ≈ 0.927 radians.
The 3-and-4 mix is really one cosine of amplitude 5, shifted by phase phi.
When recovering phi from tan(phi) = C2/C1, you must place it in the correct quadrant using the signs of C1 and C2 — a calculator's arctangent alone can land you half a cycle off. Use atan2(C2, C1).