Mathematics · Ch 3 — Trigonometric Functions

Signs, Domain, Range and Graphs of Trigonometric Functions

Signs in each quadrant. Since $\cos\theta$ is the x-coordinate and

$\sin\theta$ the y-coordinate of the point $P$ on the unit circle (Section 3), the sign of each

function is exactly the sign of $x$ or $y$ in that quadrant, and the sign of a ratio like

$\tan\theta=y/x$ follows from the signs of its two parts.

Quadrant	$\theta$ range	$\sin\theta$	$\cos\theta$	$\tan\theta$	Functions positive
I	$0<\theta<\pi/2$	$+$	$+$	$+$	all six
II	$\pi/2<\theta<\pi$	$+$	$-$	$-$	$\sin,\ \text{cosec}$
III	$\pi<\theta<3\pi/2$	$-$	$-$	$+$	$\tan,\ \cot$
IV	$3\pi/2<\theta<2\pi$	$-$	$+$	$-$	$\cos,\ \sec$

This pattern is often remembered by the mnemonic "All Silver Tea Cups" (or "All Students Take

Calculus"): reading the quadrants in order I, II, III, IV, the family of functions that is

positive is All, Sin (and cosec), Tan (and cot), Cos (and sec).

Domain, range and period. The domain of each function is all real numbers except where

its defining ratio has a zero denominator; the range and period follow from the unit-circle

definition directly.

Function	Domain	Range	Period
$\sin\theta$	$\mathbb{R}$	$[-1,1]$	$2\pi$
$\cos\theta$	$\mathbb{R}$	$[-1,1]$	$2\pi$
$\tan\theta$	$\mathbb{R} - \{(2n+1)\tfrac{\pi}{2} : n\in\mathbb{Z}\}$	$\mathbb{R}$	$\pi$
$\cot\theta$	$\mathbb{R} - \{n\pi : n\in\mathbb{Z}\}$	$\mathbb{R}$	$\pi$
$\sec\theta$	$\mathbb{R} - \{(2n+1)\tfrac{\pi}{2} : n\in\mathbb{Z}\}$	$\mathbb{R}-(-1,1)$	$2\pi$
$\text{cosec}\,\theta$	$\mathbb{R} - \{n\pi : n\in\mathbb{Z}\}$	$\mathbb{R}-(-1,1)$	$2\pi$

Sine and cosine are defined for every real number (the coordinates of a point on the unit

circle always exist) and confined to $[-1,1]$ (no coordinate of a point on a circle of radius

$1$ can exceed $1$ in size). Tangent and secant are undefined exactly where $\cos\theta=0$

(the odd multiples of $\pi/2$ ); cotangent and cosecant are undefined exactly where

$\sin\theta=0$ (the integer multiples of $\pi$ ). Because $\tan\theta=\sin\theta/\cos\theta$

repeats its value every time both $\sin\theta$ and $\cos\theta$ flip sign together, which

happens after only half a revolution, tangent (and cotangent) have period $\pi$ , half that of

sine and cosine.

Graphs. The three figures attached to this section show the graphs of $\sin x$ , $\cos x$

and $\tan x$ . All three curves repeat the pattern described above visually: the sine and cosine …

Figure 2Graph of $y=\sin x$

What this figure shows. Shows the graph of $y=\sin x$ plotted against $x$ over roughly two periods, from $-2\pi$ to $2\pi$ on the horizontal axis and from $-1$ to $1$ on the vertical axis. The curve is a smooth wave passing through the origin, rising to a maximum of $+1$ at $x=\pi/2$ (and at $x=\pi/2-2\pi$ , etc.), falling back through zero at $x=\pi$ , reaching a minimum of $-1$ at $x=3\pi/2$ , and returning to zero at $x=2\pi$ , repeating identically every $2\pi$ . The curve is symmetric about the origin (an odd function: $\sin(-x)=-\sin x$ ), has x-intercepts at every integer multiple of $\pi$ , has no asymptotes, and stays within the horizonta …

Figure 3Graph of $y=\cos x$

What this figure shows. Shows the graph of $y=\cos x$ plotted against $x$ over roughly two periods, from $-2\pi$ to $2\pi$ on the horizontal axis and from $-1$ to $1$ on the vertical axis. The curve starts at its maximum value $+1$ at $x=0$ , falls through zero at $x=\pi/2$ , reaches a minimum of $-1$ at $x=\pi$ , rises back through zero at $x=3\pi/2$ , and returns to $+1$ at $x=2\pi$ , repeating every $2\pi$ -- it is the sine graph shifted left by $\pi/2$ . The curve is symmetric about the vertical axis (an even function: $\cos(-x)=\cos x$ ), has x-intercepts at every odd multiple of $\pi/2$ , has no asymptotes, and stays within t …

Figure 4Graph of $y=\tan x$

What this figure shows. Shows the graph of $y=\tan x$ plotted against $x$ over roughly two periods, from just past $-3\pi/2$ to just past $3\pi/2$ on the horizontal axis, with the vertical axis unbounded. Vertical dashed lines mark asymptotes at every odd multiple of $\pi/2$ ( $x=\dots,-\pi/2,\pi/2,3\pi/2,\dots$ ), and between each pair of consecutive asymptotes the curve rises steadily from $-\infty$ to $+\infty$ , passing through zero at every integer multiple of $\pi$ (the x-intercepts), repeating identically every $\pi$ rather than every $2\pi$ . The curve is symmetric about the origin (an odd function: $\tan(-x)=-\tan x$ ) -- domain …

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