3.5 Sinusoidal Functions

A sinusoidal function is any function built from sin θ by additive and multiplicative transformations: shifting, stretching, shrinking and reflecting.

Since cos θ = sin(θ + π/2), cosine is just a shifted sine, so cosine and all its transformations are sinusoidal too. Any wave you can write with sine you can also write with cosine.

The period is the length of one full cycle. For sin θ and cos θ, it's 2π.

The frequency is the number of cycles per unit of input. Period and frequency are reciprocals: frequency = 1/period. For sin θ, the frequency is 1/(2π) cycles per radian.

In a context, if a wave repeats every 4 seconds, its period is 4 seconds and its frequency is 1/4 cycle per second.

The midline is the horizontal line halfway between the maximum and minimum: y = (max + min)/2. The graph oscillates around it.

The amplitude is half the distance between the maximum and minimum: (max − min)/2. It's how far the graph rises above or falls below the midline. The amplitude is always positive.

For sin θ and cos θ, max = 1 and min = −1, so the midline is y = 0 and the amplitude is 1.

Moving left to right, a sinusoidal graph keeps trading concave down for concave up and back again. It is concave down where it's above the midline (around each peak) and concave up where it's below (around each valley). The points of inflection are where the graph crosses its midline.

sin θ is an odd function: sin(−θ) = −sin θ, and its graph has rotational symmetry about the origin. For example, sin(−π/6) = −1/2 = −sin(π/6).

cos θ is an even function: cos(−θ) = cos θ, and its graph is symmetric over the y-axis. For example, cos(−π/3) = 1/2 = cos(π/3).

A sinusoidal function changes fastest where it crosses its midline (at the points of inflection) and slowest near its peaks and valleys. Right at a maximum or minimum, it stops rising and starts falling, or the reverse.

In a context, the midline is the average level, the amplitude is how far the quantity swings above and below that average, and the period is how long one full cycle takes.

For a tide that rises to 9 feet and falls to 1 foot every 12.4 hours, the water level averages 5 feet (the midline), swings 4 feet either way (the amplitude), and has a period of 12.4 hours, so its frequency is 1/12.4 cycle per hour.