Domain, range and graphs of trigonometric functions

Once each trigonometric function is defined for every real argument, we can draw its graph. The shapes are some of the most important curves in mathematics; you must be able to sketch each one from memory.

The six standard graphs

$\sin x$

• Domain: $\mathbb{R}$. Range: $[-1, 1]$.
• Period $2\pi$. Zeros at $x = n\pi$.
• Maximum $1$ at $x = \dfrac{\pi}{2} + 2n\pi$; minimum $-1$ at $x = \dfrac{3\pi}{2} + 2n\pi$.
• Odd function , graph symmetric about the origin.

$\cos x$

• Domain: $\mathbb{R}$. Range: $[-1, 1]$.
• Period $2\pi$. Zeros at $x = \dfrac{\pi}{2} + n\pi$.
• Maximum $1$ at $x = 2n\pi$; minimum $-1$ at $x = (2n+1)\pi$.
• Even function , graph symmetric about the $y$-axis.
• The cosine graph is the sine graph shifted left by $\pi/2$: $\cos x = \sin\!\left(x + \dfrac{\pi}{2}\right)$.

$\tan x$

• Domain: $\mathbb{R} \setminus \left\{\dfrac{\pi}{2} + n\pi\right\}$. Range: $\mathbb{R}$.
• Period $\pi$.
• Vertical asymptotes at $x = \dfrac{\pi}{2} + n\pi$.
• Increasing on each interval $\left(-\dfrac{\pi}{2} + n\pi,\ \dfrac{\pi}{2} + n\pi\right)$.
• Odd function.

$\cot x$

• Domain: $\mathbb{R} \setminus \{n\pi\}$. Range: $\mathbb{R}$. Period $\pi$.
• Vertical asymptotes at $x = n\pi$.
• Decreasing on each $(n\pi, (n+1)\pi)$. Odd.

$\sec x$ and $\csc x$

• $\sec x$: range $(-\infty, -1] \cup [1, \infty)$. Period $2\pi$. Vertical asymptotes where $\cos x = 0$. Even.
• $\csc x$: range $(-\infty, -1] \cup [1, \infty)$. Period $2\pi$. Vertical asymptotes where $\sin x = 0$. Odd.

Transformations

Given a base function $y = f(x)$, the transformed function $y = a\,f(b(x - c)) + d$ has:

• Amplitude $|a|$ , vertical stretch (relevant for $\sin, \cos$).
• Period $\dfrac{2\pi}{|b|}$ for $\sin, \cos$ (or $\dfrac{\pi}{|b|}$ for $\tan, \cot$).
• Phase shift $c$ , horizontal translation (positive $c$ shifts right).
• Vertical shift $d$ , moves the midline.

So $y = 3 \sin\!\left(2x - \dfrac{\pi}{3}\right) + 1$ has amplitude $3$, period $\dfrac{2\pi}{2} = \pi$, phase shift $\dfrac{\pi}{6}$ (rewriting $2(x - \pi/6)$), midline $y = 1$.

Even and odd

• $\sin(-x) = -\sin x$, $\cos(-x) = \cos x$, $\tan(-x) = -\tan x$.
• $\cot, \csc$ are odd; $\sec$ is even.

Combining trigonometric functions

The graph of $y = \sin x + \cos x$ can be rewritten as $\sin x + \cos x = \sqrt{2}\,\sin\!\left(x + \dfrac{\pi}{4}\right),$ revealing amplitude $\sqrt{2}$ and a phase shift of $-\dfrac{\pi}{4}$. More generally: $a \sin x + b \cos x = R\sin(x + \phi), \quad R = \sqrt{a^2 + b^2}, \quad \tan\phi = \dfrac{b}{a}.$

This is a workhorse trick , used in solving trigonometric equations, modelling, and Class-XII calculus.

Worked examples

Example 1. Find the maximum and minimum of $f(x) = 3 \cos 2x - 1$.

$\cos 2x \in [-1, 1]$, so $3\cos 2x \in [-3, 3]$, hence $f \in [-4, 2]$. Max $= 2$, min $= -4$.

Example 2. Period of $f(x) = \sin\dfrac{x}{3}$?

$\dfrac{2\pi}{1/3} = 6\pi$.

Example 3. Sketch $y = 2\sin(x - \pi/2)$ on $[0, 2\pi]$.

Amplitude $2$, period $2\pi$, phase shift $\pi/2$ right. Note $\sin(x - \pi/2) = -\cos x$, so this is $y = -2\cos x$: starts at $-2$, hits $0$ at $\pi/2$, reaches $2$ at $\pi$, $0$ at $3\pi/2$, $-2$ at $2\pi$.

Example 4. Find the range of $f(x) = 3\sin x + 4\cos x$.

By the $R$-formula: $R = \sqrt{9 + 16} = 5$. So $f \in [-5, 5]$.

Example 5 (harder). Sketch and describe the graph of $f(x) = \sin x + \sin 2x$ on $[0, 2\pi]$.

This is a non-elementary curve. By sampling:

• $f(0) = 0$.
• $f(\pi/4) = \dfrac{\sqrt{2}}{2} + 1 \approx 1.71$.
• $f(\pi/2) = 1 + 0 = 1$.
• $f(3\pi/4) = \dfrac{\sqrt{2}}{2} - 1 \approx -0.29$.
• $f(\pi) = 0$.

The graph has one tall hump near $\pi/4$, a smaller dip near $3\pi/4$, then mirrors in the second half. Calculus tools (Chapter 12) make the extrema explicit.

Try it yourself

1. Period of $\sin 3x$, $\cos 5x$, $\tan(2x + 1)$.
2. Amplitude and period of $y = -4\cos\dfrac{x}{2}$.
3. Sketch $y = |\cos x|$ on $[-\pi, \pi]$ and find its period.
4. Find max and min of $f(x) = 5\sin x - 12\cos x$.
5. Express $\sin x + \sqrt{3}\cos x$ in the form $R\sin(x + \phi)$.
6. Sketch $y = \tan(x - \pi/4)$ on $(-\pi/4, 3\pi/4)$.
7. Find the range of $f(x) = 2 + 3\sin x$.
8. Show $\sin(x + 2\pi) = \sin x$ from the unit circle definition.
9. Sketch $y = \cos 2x$ and $y = 2\cos x$ on the same axes; describe how they differ.
10. Find the values of $x$ in $[0, 2\pi]$ at which $\sec x$ is undefined.
11. Sketch $y = x + \sin x$ on $[-2\pi, 2\pi]$.
12. Prove that $\sin x$ is not a polynomial in $x$ by considering its zeros.

Pitfalls / Tricks

• A coefficient inside the argument divides the period: $\sin 3x$ has period $\dfrac{2\pi}{3}$, not $2\pi/3$ "stretched".
• $\sin x$ does not equal $\cos x$ except at isolated points: do not "swap" them mid-calculation.
• $\sec, \csc$ have gaps in their ranges , they never take values in $(-1, 1)$.
• Insight. Every linear combination $a \sin x + b \cos x$ is itself a sine wave with amplitude $\sqrt{a^2 + b^2}$. This is a complete description of all "single-frequency" oscillations.