Graphs of Sine and Cosine

Definition For a real number $x$, let $P(x)$ be the point on the unit circle reached by starting at $(1, 0)$ and traveling an arc of length $\lvert x \rvert$, counterclockwise when $x \ge 0$ and clockwise when $x < 0$. As in the unit circle lesson, $P(x) = (\cos x, \sin x)$, so $\sin x$ is the second coordinate of $P(x)$. The sine wave is the graph of $y = \sin x$ with $x$ in radians. Its domain is all real numbers and its range is $[-1, 1]$. Its values repeat every $2\pi$ units, and it is odd, which means $\sin(-x) = -\sin x$. Its zeros are exactly the numbers $x = k\pi$ with $k$ an integer. It takes its maximum value $1$ exactly at $x = \frac{\pi}{2} + 2k\pi$ and its minimum value $-1$ exactly at $x = \frac{3\pi}{2} + 2k\pi$.

Definition The cosine wave is the graph of $y = \cos x$, where $\cos x$ is the first coordinate of $P(x)$ and $x$ is in radians. Its domain is all real numbers and its range is $[-1, 1]$. Its values repeat every $2\pi$ units, and it is even, which means $\cos(-x) = \cos x$. Its zeros are exactly the numbers $x = \frac{\pi}{2} + k\pi$ with $k$ an integer. It takes its maximum value $1$ exactly at $x = 2k\pi$ and its minimum value $-1$ exactly at $x = \pi + 2k\pi$. For every real $x$

so the cosine wave is the sine wave shifted $\frac{\pi}{2}$ units to the left.

Definition A function $f$ is periodic if there is a number $p > 0$ such that, whenever $x$ is in the domain of $f$, the number $x + p$ is also in the domain and

Each such number $p$ is called a period of $f$.

Definition If a periodic function $f$ has a smallest period, that is, a smallest number $p > 0$ with $f(x + p) = f(x)$ for every $x$ in its domain, that smallest number is the fundamental period of $f$, usually just called the period. The part of the graph over an interval of length $p$, such as $[0, p]$, is called one cycle. Sine and cosine both have period $2\pi$. Shifts by $4\pi$ or $6\pi$ also satisfy $f(x + p) = f(x)$, so they are periods too, but they are not the fundamental period, because they are not the smallest.

Definition Let $f$ be a periodic function with maximum value $M$ and minimum value $m$. The midline of its graph is the horizontal line

For $y = \sin x$ and $y = \cos x$ the midline is the $x$-axis, $y = 0$. Adding a constant $D$, as in $y = \sin x + D$, moves every point up by $D$ (down by $\lvert D \rvert$ when $D < 0$), so the midline becomes $y = D$.

Definition For a periodic function with maximum value $M$ and minimum value $m$, the amplitude is

which is the vertical distance from the midline to a highest point. For $y = A\sin x$ and $y = A\cos x$ with $A \ne 0$ the amplitude is $\lvert A \rvert$. Sine and cosine themselves have amplitude $1$.