Mathematics Advanced • Year 11 • Module 2 • Lesson 9
Domains and Ranges of Trigonometric Functions
Build fluency in stating the domain and range of all six trig functions, finding undefined points, and writing general solutions with $n \in \mathbb{Z}$.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 Complete the table.
| Function | Domain | Range |
|---|---|---|
| $\sin x$ | ||
| $\tan x$ | ||
| $\sec x$ |
Q1.2 Why is $\tan x$ undefined where $\cos x = 0$? Answer in one sentence.
Q1.3 Define "bounded" and give one bounded and one unbounded trig function.
2. Worked example — domain of $f(x) = \frac{\sin x}{1 + \cos x}$
Every line annotated. Use as the template for the faded version.
Problem. State the domain of $f(x) = \frac{\sin x}{1 + \cos x}$.
Step 1 — Identify what can break the function.
Division by zero is the only risk — $\sin x$ itself is defined for all real $x$.
Reason: numerator (sin) has domain $\mathbb{R}$; the denominator is the only constraint.
Step 2 — Set the denominator equal to zero.
1 + cos x = 0 ⇒ cos x = −1
Reason: forbidden inputs are those that make the denominator vanish.
Step 3 — Solve $\cos x = -1$ over all reals.
cos x = −1 at x = π, then every $2\pi$ apart: x = π + 2nπ, $n \in \mathbb{Z}$.
Reason: cosine has period $2\pi$ and equals $-1$ once per cycle (Trap 3: must use general solution).
Step 4 — Write the domain.
Domain: all real x except x = π + 2nπ, $n \in \mathbb{Z}$.
Conclusion. Domain = $\mathbb{R} \setminus \{\pi + 2n\pi : n \in \mathbb{Z}\}$.
3. Faded example — fill in the missing steps
Find the range of $y = 3 \sin x + 1$. Fill in each blank line. 4 marks
Step 1 — Start with the range of $\sin x$.
________ ≤ sin x ≤ ________
Step 2 — Multiply through by 3 (positive, so inequality direction unchanged).
________ ≤ 3 sin x ≤ ________
Step 3 — Add 1 to every part.
________ ≤ 3 sin x + 1 ≤ ________
Step 4 — Write the range as an interval.
Range = [________, ________]
Conclusion. Range of $3 \sin x + 1$ is ____________.
4. Graduated practice
State the domain and range. Use general-solution notation ($n \in \mathbb{Z}$) where infinitely many points are excluded.
Foundation — basic six (4 questions)
| Q | Function | Domain | Range |
|---|---|---|---|
| 4.1 1 | $y = \cos x$ | ||
| 4.2 1 | $y = \sec x$ | ||
| 4.3 1 | $y = \cot x$ | ||
| 4.4 1 | $y = \csc x$ |
Standard — transformed and combined (6 questions)
4.5 State the range of $y = 2 \sin x - 3$. Show the inequality manipulation. 2 marks
4.6 State the range of $y = 4 - 2\cos x$. (Careful with the sign on the cos coefficient.) 2 marks
4.7 Find all $x \in [0, 2\pi]$ where $\csc x$ is undefined. 2 marks
4.8 State the domain of $y = \tan 2x$ in general-solution form (use $n \in \mathbb{Z}$). Then list every undefined value on $[0, \pi]$. 2 marks
4.9 State the domain of $g(x) = \frac{1}{\sin x - 1}$ in general-solution form. 2 marks
4.10 A student writes "the range of $\csc x$ is $[-1, 1]$". State the trap (referencing the lesson by number) and give the correct range. 2 marks
Extension — combine concepts (2 questions)
4.11 Find the domain and range of $y = \sec\left(x - \frac{\pi}{4}\right) + 2$. Use general-solution form for the domain. 3 marks
4.12 State the domain of $h(x) = \frac{\cos x}{\sin^2 x - \sin x}$. Hint: factorise the denominator first. 3 marks
5. Self-check the easy 3
Tick the first three once you've verified your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Domains and ranges
$\sin x$: domain all real $x$, range $[-1, 1]$. $\tan x$: domain all real $x$ except $x = \frac{\pi}{2} + n\pi$, range all real $y$. $\sec x$: domain all real $x$ except $x = \frac{\pi}{2} + n\pi$, range $(-\infty, -1] \cup [1, \infty)$.
Q1.2 — Why $\tan x$ is undefined at zeros of cos
$\tan x = \frac{\sin x}{\cos x}$; when $\cos x = 0$ the denominator is zero, so the ratio is undefined (division by zero).
Q1.3 — Bounded/unbounded
A function is bounded if its outputs stay between two fixed values. $\sin x$ is bounded ($[-1, 1]$). $\tan x$ is unbounded (extends to $\pm \infty$).
Q3 — Faded example: range of $3 \sin x + 1$
Step 1: $-1 \leq \sin x \leq 1$.
Step 2: $-3 \leq 3 \sin x \leq 3$.
Step 3: $-2 \leq 3 \sin x + 1 \leq 4$.
Step 4: Range = $[-2, 4]$.
Conclusion: Range = $\mathbf{[-2, 4]}$.
Q4.1 — $\cos x$
Domain: all real $x$. Range: $[-1, 1]$.
Q4.2 — $\sec x$
Domain: all real $x$ except $x = \frac{\pi}{2} + n\pi$, $n \in \mathbb{Z}$. Range: $(-\infty, -1] \cup [1, \infty)$.
Q4.3 — $\cot x$
Domain: all real $x$ except $x = n\pi$, $n \in \mathbb{Z}$. Range: all real $y$.
Q4.4 — $\csc x$
Domain: all real $x$ except $x = n\pi$, $n \in \mathbb{Z}$. Range: $(-\infty, -1] \cup [1, \infty)$.
Q4.5 — Range of $2 \sin x - 3$
$-1 \leq \sin x \leq 1 \Rightarrow -2 \leq 2\sin x \leq 2 \Rightarrow -5 \leq 2 \sin x - 3 \leq -1$. Range = $\mathbf{[-5, -1]}$.
Q4.6 — Range of $4 - 2\cos x$
$-1 \leq \cos x \leq 1 \Rightarrow -2 \leq -2\cos x \leq 2$ (multiplying by $-2$ reverses the inequalities, then re-ordering: $-2 \leq -2\cos x \leq 2$). Add 4: $2 \leq 4 - 2\cos x \leq 6$. Range = $\mathbf{[2, 6]}$.
Q4.7 — Where $\csc x$ is undefined on $[0, 2\pi]$
$\csc x = \frac{1}{\sin x}$, undefined when $\sin x = 0$. On $[0, 2\pi]$: $x = \mathbf{0, \pi, 2\pi}$.
Q4.8 — Domain of $\tan 2x$
$\tan 2x$ undefined when $2x = \frac{\pi}{2} + n\pi$, i.e. $x = \frac{\pi}{4} + \frac{n\pi}{2}$, $n \in \mathbb{Z}$. On $[0, \pi]$: $x = \mathbf{\frac{\pi}{4}, \frac{3\pi}{4}}$.
Q4.9 — Domain of $\frac{1}{\sin x - 1}$
Need $\sin x - 1 \neq 0$, i.e. $\sin x \neq 1$. Solutions of $\sin x = 1$: $x = \frac{\pi}{2} + 2n\pi$. Domain: all real $x$ except $x = \mathbf{\frac{\pi}{2} + 2n\pi}$, $n \in \mathbb{Z}$.
Q4.10 — Critique of "$\csc x \in [-1, 1]$"
This commits Trap 2: cosecant is the reciprocal of sine, so it can never take values in $(-1, 1)$. Correct range: $\mathbf{(-\infty, -1] \cup [1, \infty)}$.
Q4.11 — Domain and range of $\sec\left(x - \frac{\pi}{4}\right) + 2$
$\sec$ undefined when its argument equals $\frac{\pi}{2} + n\pi$: $x - \frac{\pi}{4} = \frac{\pi}{2} + n\pi \Rightarrow x = \frac{3\pi}{4} + n\pi$. Domain: all real $x$ except $\mathbf{x = \frac{3\pi}{4} + n\pi}$, $n \in \mathbb{Z}$.
Range of inner $\sec(\cdot)$: $(-\infty, -1] \cup [1, \infty)$. Add 2: $(-\infty, 1] \cup [3, \infty)$. Range = $\mathbf{(-\infty, 1] \cup [3, \infty)}$.
Q4.12 — Domain of $\frac{\cos x}{\sin^2 x - \sin x}$
Denominator: $\sin^2 x - \sin x = \sin x(\sin x - 1)$. Need both $\sin x \neq 0$ and $\sin x \neq 1$. $\sin x = 0$: $x = n\pi$. $\sin x = 1$: $x = \frac{\pi}{2} + 2n\pi$. Domain: all real $x$ except $x = n\pi$ and $x = \frac{\pi}{2} + 2n\pi$, $n \in \mathbb{Z}$.