Periodic function
A function whose values repeat after a fixed input interval $T$, so $f(x+T) = f(x)$ for all $x$ in its domain. The smallest positive such $T$ is the period.
In this topic you build the sine and cosine graphs, describe their key features (amplitude, period, midline), and learn how the equation changes when you translate, reflect, or dilate (stretch) the curve.
The graphs of $y = \sin x$ and $y = \cos x$ (with $x$ measured in degrees) have the same smooth wave shape.
For $0^\circ \le x \le 360^\circ$:
| $x$ | $0^\circ$ | $90^\circ$ | $180^\circ$ | $270^\circ$ | $360^\circ$ |
|---|---|---|---|---|---|
| $\sin x$ | 0 | 1 | 0 | -1 | 0 |
| $\cos x$ | 1 | 0 | -1 | 0 | 1 |
The sine and cosine curves repeat every $360^\circ$.
Period
The horizontal length of one complete cycle of a periodic graph.
So for $y = \sin x$ and $y = \cos x$ (degrees), the period is $360^\circ$.
Frequency
The number of complete cycles per unit interval. In this course, the coefficient $b$ is often referred to as the frequency, representing the number of cycles in $360^\circ$ (distinct from the physics definition of cycles per unit time).
For the basic curves, the frequency is 1 cycle per $360^\circ$.
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