The Sine Function

Definition For a real number $x$, let $P(x)$ be the point where the terminal side of the angle of $x$ radians in standard position meets the unit circle. The sine of $x$ is the vertical coordinate of that point,

This defines a function $\sin$ whose domain is all real numbers.

Definition The input of $\sin$ is a real number read as an angle in radians. On the unit circle the input $x$ is the signed length of the arc from $(1, 0)$ to $P(x)$, counterclockwise for $x > 0$ and clockwise for $x < 0$. If an angle is given in degrees, it must first be converted using $180^\circ = \pi$ radians. The calculus rules for sine, such as the derivative of $\sin x$ being $\cos x$, hold only with radian input.

Definition The range of $\sin$ is the closed interval $[-1, 1]$. Every real $x$ satisfies $-1 \le \sin x \le 1$, and every number $y$ in $[-1, 1]$ equals $\sin x$ for some real $x$. The largest value $1$ is reached exactly at $x = \frac{\pi}{2} + 2\pi k$ and the smallest value $-1$ exactly at $x = \frac{3\pi}{2} + 2\pi k$, for integers $k$.

Definition A function $f$ is periodic with period $p > 0$ if $x + p$ is in the domain whenever $x$ is, and $f(x + p) = f(x)$ for every $x$ in the domain. For sine, adding $2\pi k$ to the input adds $k$ full turns, so for every real $x$ and every integer $k$

The least positive period of sine is $2\pi$. To see that nothing smaller works, suppose $p > 0$ is a period. Then

By the range facts above, sine equals $1$ only at the inputs $\frac{\pi}{2} + 2\pi k$, so $p = 2\pi k$ for a positive integer $k$, and therefore $p \ge 2\pi$.

Definition A function $f$ is odd if $-x$ is in its domain whenever $x$ is and $f(-x) = -f(x)$ for all such $x$. Sine is odd,

The reason is that $P(-x)$ is the reflection of $P(x)$ across the horizontal axis, which keeps the horizontal coordinate and flips the sign of the vertical one. The graph of an odd function is symmetric about the origin.

Definition The zeros of sine are the real numbers $x$ with $\sin x = 0$. They are exactly

These are the inputs whose point is $(1, 0)$ or $(-1, 0)$, the two places where the unit circle crosses the horizontal axis.

Definition The graph of sine is the set of points $(x, \sin x)$ for all real $x$. One full cycle runs over $[0, 2\pi]$, and the rest of the graph is copies of that cycle shifted by multiples of $2\pi$. The five landmarks of the cycle are the zeros, the peak and the trough,

Sine increases on $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and decreases on $\left[\frac{\pi}{2}, \frac{3\pi}{2}\right]$. A finite table of values only illustrates the graph. The smooth shape between the points comes from the point moving continuously around the circle, not from the table.

Definition For real numbers $A \ne 0$ and $B \ne 0$, the function $y = A \sin(Bx)$ has amplitude $\lvert A \rvert$, range $[-\lvert A \rvert, \lvert A \rvert]$ and least positive period

The amplitude is the distance from the midline $y = 0$ to a peak. Each landmark $(x, y)$ of $\sin x$ moves to $\left(\frac{x}{B}, Ay\right)$.