Graphing Sine & Cosine

Sine and cosine graphs are waves that repeat forever.

Basic y = sin x: starts at 0, peaks at 1, returns to 0, dips to −1, back to 0. One full cycle takes 2π.
y = cos x is the same wave starting at its peak instead of at zero.

For y = a·sin(b(x − h)) + k:

• |a| is the amplitude — how tall from the midline
• 2π/b is the period — how long one cycle takes
• h shifts left/right (a phase shift)
• k moves the midline up or down

For y = 3sin(2x): amplitude 3, period 2π/2 = π. It is three times taller and twice as fast as the basic wave.

Sine and cosine are periodic with period 2π, amplitude 1 and midline y = 0. The general form y = a·sin(b(x − h)) + k transforms these: |a| is the amplitude (with a < 0 reflecting across the midline), the period is 2π/|b|, h is the phase shift and k the vertical shift, with midline y = k and range [k − |a|, k + |a|].

The parameter b compresses horizontally by 1/b, which is the same inverted horizontal behaviour seen in all function transformations. As there, the argument must be factored — sin(2x − π) is sin(2(x − π/2)), so the phase shift is π/2 rather than π.

Cosine is sine shifted left by π/2, so cos x = sin(x + π/2) and either function can model any sinusoid; the choice is made by whichever starting point matches the data.

Tangent behaves differently: period π, no amplitude, and vertical asymptotes where cos x = 0, that is at x = π/2 + nπ.

Sinusoidal models describe anything cyclical — tides, daylight hours, alternating current, sound — where amplitude, period, phase and midline each carry a physical meaning.