Some Characteristics of Trigonometric Functions

Trigonometric functions have two structural properties that make them exceptionally useful for modelling repeating phenomena: they repeat their values at regular intervals (periodicity), and they have clean odd/even symmetry.

Periodicity. A function $f$ is called periodic with period $p$ if $p$ is the smallest positive number such that $f(x+p)=f(x)$ for every $x$ in the domain. Since adding $2\pi$ to an angle is a full rotation that returns the terminal side to exactly the same place,

so $\sin x$ is periodic with period $2\pi$; likewise $\cos x,\ \operatorname{cosec}x,\ \sec x$ are all periodic with period $2\pi$. By contrast, $\tan x$ and $\cot x$ repeat twice as often — they are periodic with the smaller period $\pi$ (a half rotation already brings the ratio $y/x$ back to itself, since both $y$ and $x$ flip sign together).

The graphs of $y=\sin x$ and $y=\cos x$. Plotting $y=\sin x$ against $x$ (horizontal axis = the angle $x$ in radians, vertical axis = $\sin x$) gives a smooth wave that:

• stays within one unit of the $x$-axis at all times, since $-1\le\sin x\le1$;
• repeats exactly every $2\pi$ — sliding the whole curve left or right by $2\pi$ maps it onto itself;
• rises from $-1$ to $1$ as $x$ runs from $-\pi/2$ to $\pi/2$, then falls back from $1$ to $-1$ as $x$ runs from $\pi/2$ to $3\pi/2$, and so on, alternately rising and falling in $\pi$-wide stretches;
• passes through $0$ at every integer multiple of $\pi$ (matching the zero rule from §3.4.1, $\sin\theta=0\iff\theta=n\pi$).

The graph of $y=\cos x$ has exactly the same shape and amplitude as $y=\sin x$, only slid to the left by $\dfrac{\pi}{2}$ — because $\cos x=\sin\!\left(x+\dfrac{\pi}{2}\right)$ (an immediate consequence of the allied-angle table's $\dfrac{\pi}{2}+\theta$ column, read in reverse). Both curves are called sinusoidal waves, and together they model an enormous range of periodic phenomena in nature and physics — the rising and setting of the sun, a spring oscillating up and down, ocean tides — precisely because any regular periodic behaviour can be built out of combinations of sine and cosine.

Odd and even functions. A real-valued function $f$ is:

• even if $f(-x)=f(x)$ for every real $x$ in the domain (its graph is symmetric about the $y$-axis);
• odd if $f(-x)=-f(x)$ for every real $x$ in the domain (its graph is symmetric about the origin).

From the negative-angle identities established in §3.4.3, $\cos(-x)=\cos x$ and $\sin(-x)=-\sin x$ for every $x$, so:

The same negative-angle reasoning extends to the other four: $\sec x$ (built from $\cos x$) is also even, while $\tan x,\ \operatorname{cosec}x,\ \cot x$ (each built using a $\sin x$) are all odd.

Not every combination of trigonometric functions is odd or even, though — e.g. $f(t)=t-\cos t$ is neither, since $f(-t)=-t-\cos t$, which matches neither $f(t)=t-\cos t$ nor $-f(t)=-t+\cos t$ in general.

Testing whether a combination is even, odd or neither. Given $f(x)$, compute $f(-x)$ using the negative-angle identities ($\sin(-x)=-\sin x$, $\cos(-x)=\cos x$) throughout, then compare the result with $f(x)$ and with $-f(x)$:

• if $f(-x)=f(x)$ exactly, $f$ is even;
• if $f(-x)=-f(x)$ exactly, $f$ is odd;
• if $f(-x)$ matches neither $f(x)$ nor $-f(x)$, $f$ is neither. …