Period and Frequency of a Sinusoid

Picture a point moving around a unit circle at steady speed. One full lap takes 2pi units of angle, and that lap is exactly what the sine wave traces out as one complete cycle. Period is the length of x it takes to finish one lap. Frequency just counts laps per unit of x instead of measuring the length of one lap, so the two describe the same motion from opposite directions: period is time per cycle, frequency is cycles per time. For y = sin(x), one full cycle happens over an x-interval of length 2pi, so the period is 2pi.

Now put a coefficient on x: y = sin(bx). The b scales how fast the input grows, so the wave reaches a full cycle (an angle of 2pi) after x moves through only 2pi/b instead of 2pi. That is why period = 2pi/b. A bigger b feeds angle into the sine function faster, so the wave repeats sooner, squeezing more cycles into the same stretch of x-axis. Frequency, the reciprocal b/(2pi), goes up exactly when period goes down.

This is why sin(4x) has period 2pi/4 = π/2, not a period stretched by 4. Two full cycles finish by x = π, each one squeezed into a quarter of the space sin(x) would use. Watching b reshape period and frequency here sets up the next move: once you can control how fast a sinusoid repeats, you are ready to also control its height, midline, and horizontal shift, building the full general sinusoidal form.