Not every angle can be plugged into every trig function. Tangent blows up at $90^\circ$, cosecant is undefined at $0^\circ$, and cosine never exceeds 1. In this lesson, you will systematically determine the domain and range of all six trigonometric functions — an essential foundation for graphing, solving equations, and working with inverse trig functions.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
The function $y = \sin x$ has a maximum value of 1 and a minimum value of $-1$. But $y = \tan x$ has no maximum or minimum — it extends to $+\infty$ and $-\infty$. Why do you think $\sin x$ is bounded while $\tan x$ is not? And why is $\tan x$ undefined at $90^\circ$?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Wrong: (a + b)² = a² + b².
Right: (a + b)² = a² + 2ab + b²; the middle term 2ab is essential and commonly forgotten.
📚 Core Content
On the unit circle, $\sin \theta = y$ and $\cos \theta = x$. Since every point on the unit circle has $x^2 + y^2 = 1$, both $x$ and $y$ must lie between $-1$ and $1$ inclusive. The angle $\theta$ can be any real number — we can rotate infinitely many times around the circle.
| Function | Domain | Range |
|---|---|---|
| $y = \sin x$ | All real $x$ | $-1 \leq y \leq 1$ |
| $y = \cos x$ | All real $x$ | $-1 \leq y \leq 1$ |
$\tan x = \frac{\sin x}{\cos x}$. This is undefined whenever $\cos x = 0$, which occurs at $x = \frac{\pi}{2} + n\pi$ for any integer $n$. As $x$ approaches these values, $\tan x$ approaches $+\infty$ or $-\infty$. Therefore:
The domain restrictions of the reciprocal functions come from where their denominators are zero. Their ranges are determined by the fact that $\sin x$ and $\cos x$ are bounded between $-1$ and $1$.
| Function | Domain | Range |
|---|---|---|
| $y = \csc x = \frac{1}{\sin x}$ | All real $x$ except $x = n\pi$, $n \in \mathbb{Z}$ | $y \leq -1$ or $y \geq 1$ |
| $y = \sec x = \frac{1}{\cos x}$ | All real $x$ except $x = \frac{\pi}{2} + n\pi$, $n \in \mathbb{Z}$ | $y \leq -1$ or $y \geq 1$ |
| $y = \cot x = \frac{\cos x}{\sin x}$ | All real $x$ except $x = n\pi$, $n \in \mathbb{Z}$ | All real $y$ |
🧮 Worked Examples
🧪 Activities
1 $y = \cos x$
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2 $y = \sec x$
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3 $y = \cot x$
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4 $y = 2\sin x - 3$
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1 $y = \tan x$
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2 $y = \csc x$
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3 $y = \frac{1}{1 - \sin x}$
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Earlier you were asked: Why is $\sin x$ bounded while $\tan x$ is not? And why is $\tan x$ undefined at $90^\circ$?
$\sin x$ represents the $y$-coordinate on the unit circle, which can never be larger than 1 or smaller than $-1$. So sine is bounded. But $\tan x = \frac{\sin x}{\cos x}$ is a ratio. As $x$ approaches $90^\circ$, $\cos x$ approaches 0, making the ratio blow up to $+\infty$ or $-\infty$. At exactly $90^\circ$, $\cos x = 0$, so division by zero makes $\tan x$ undefined.
Now revisit your initial response. What did you get right? What has changed in your thinking?
Look back at your initial response in your book. Annotate it with what you now understand differently.
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
5 random questions from a replayable lesson bank — feedback shown immediately
✍️ Short Answer
8. (a) State the domain of $y = \tan 2x$. (b) Find all values of $x$ in $[0, \pi]$ where $\tan 2x$ is undefined. 3 MARKS
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9. Find the range of $y = 4 - 2\cos x$. Show your reasoning. 2 MARKS
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10. Explain why the range of $y = \sec x$ is $y \leq -1$ or $y \geq 1$. Your explanation should reference the range of $\cos x$. 3 MARKS
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Answer in your workbook.
1. Domain: all real $x$. Range: $-1 \leq y \leq 1$.
2. Domain: all real $x$ except $x = \frac{\pi}{2} + n\pi$. Range: $y \leq -1$ or $y \geq 1$.
3. Domain: all real $x$ except $x = n\pi$. Range: all real $y$.
4. Domain: all real $x$. Range: $-5 \leq y \leq -1$.
1. $x = \frac{\pi}{2}, \frac{3\pi}{2}$
2. $x = 0, \pi, 2\pi$ (or $0, \pi$ within $[0, 2\pi]$)
3. $1 - \sin x = 0 \Rightarrow \sin x = 1 \Rightarrow x = \frac{\pi}{2}$
1. A — Sine has domain all real $x$.
2. A — Tangent undefined when $\cos x = 0$, at odd multiples of $\frac{\pi}{2}$.
3. A — Cosecant range is $y \leq -1$ or $y \geq 1$.
4. A — $\sec x$ undefined where $\cos x = 0$.
5. A — $2\sin x + 1$ has range $[-1, 3]$.
Q8 (3 marks): (a) $2x \neq \frac{\pi}{2} + n\pi \Rightarrow x \neq \frac{\pi}{4} + \frac{n\pi}{2}$ [1]. (b) In $[0, \pi]$: $x = \frac{\pi}{4}, \frac{3\pi}{4}$ [2].
Q9 (2 marks): $-1 \leq \cos x \leq 1$ [0.5]. Multiply by $-2$: $-2 \leq -2\cos x \leq 2$ [0.5]. Add 4: $2 \leq 4 - 2\cos x \leq 6$ [1]. Range: $[2, 6]$.
Q10 (3 marks): $\sec x = \frac{1}{\cos x}$ [1]. Since $-1 \leq \cos x \leq 1$ and $\cos x \neq 0$, the reciprocal $|\sec x| \geq 1$ [1]. When $\cos x$ is positive, $\sec x \geq 1$; when negative, $\sec x \leq -1$ [1].
Domains and Ranges of Trigonometric Functions
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