Sinusoidal functions

Sine's graph is a smooth wave: it rises from its midline to a crest, falls through the midline to a trough, and returns, over and over, forever. A function is sinusoidal when its graph is that same wave after any combination of vertical or horizontal stretching, shifting, or reflecting. Cosine is already sinusoidal: it is sine slid a quarter cycle sideways.

Not every repeating graph qualifies. A sawtooth repeats but has corners; a damped vibration repeats its rhythm but shrinks each cycle; a square wave jumps. Sinusoidal means the specific smooth, symmetric wave shape, not merely "periodic."

Four numbers pin down a sinusoid, and each is read from the graph the same way every time:

1. Midline: the horizontal center, at the average of maximum and minimum: (max + min)/2.
2. Amplitude: the vertical distance from the midline to a crest, (max − min)/2. Half the total swing, and always positive.
3. Period: the horizontal length of one full cycle, measured crest to crest, trough to trough, or upward midline crossing to the next upward midline crossing.
4. Range: every output between min and max, which is midline ± amplitude.

The reliable order is midline first, then amplitude. A wave running between −2 and 6 has midline (6 + (−2))/2 = 2 and amplitude (6 − (−2))/2 = 4. Quoting 8 (the full swing) or 6 (the maximum) as "the amplitude" are the two most common wrong answers on this entire unit.

As in Topic 3.1, the period is input per cycle and the frequency is cycles per unit input, and they are reciprocals. A sinusoid with period 0.2 seconds has frequency 5 cycles per second; doubling the frequency halves the period.

Keep amplitude out of it: how tall a wave is and how often it repeats are independent dials. Two waves can share a period while one towers over the other, and a taller wave repeats no faster than a short one.

Sine and cosine respond differently to a sign flip on the input. Sine is odd: $\sin(-x) = -\sin(x)$, so its graph has point symmetry through the origin. Cosine is even: $\cos(-x) = \cos(x)$, so its graph is mirror-symmetric over the y-axis. One picture makes both obvious: sine starts at its midline (flipping x flips the direction of travel), while cosine starts at its crest (flipping x runs into the same crest).

These are exact identities, usable numerically: if $\cos(1.2) \approx 0.362$, then $\cos(-1.2) \approx 0.362$ with no further work, and if $\sin(0.8) \approx 0.717$, then $\sin(-0.8) \approx -0.717$.