While sine and cosine trace gentle waves, tangent and cotangent produce a very different pattern: a series of repeating curves separated by vertical asymptotes. In this lesson, you will learn how to sketch these graphs, identify their asymptotes, and understand why their period is only $\pi$ instead of $2\pi$.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
The tangent function is defined as $\tan x = \frac{\sin x}{\cos x}$. As $x$ gets closer to $90^\circ$ from below, $\cos x$ gets closer to 0 while $\sin x$ stays close to 1. What do you think happens to the value of $\tan x$? And what does this mean for the graph of $y = \tan x$ near $x = 90^\circ$?
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: The amplitude of y = 2sin(x) is 2 and the period is 2π.
Right: The amplitude of y = 2sin(x) is 2, but the period remains 2π. The coefficient in front of x (not the constant) changes the period: y = sin(2x) has period π.
📚 Core Content
Because $\tan x = \frac{\sin x}{\cos x}$, the tangent function is undefined wherever $\cos x = 0$. This occurs at:
$$x = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}$$
At these values, the graph has vertical asymptotes. Between each pair of asymptotes, the tangent graph forms a smooth, increasing curve that passes through the $x$-axis.
Because $\cot x = \frac{\cos x}{\sin x}$, the cotangent function is undefined wherever $\sin x = 0$. This occurs at:
$$x = n\pi, \quad n \in \mathbb{Z}$$
Note that $y = \cot x$ is related to $y = \tan x$ by a reflection and shift. Specifically:
$$\cot x = \tan\left(\frac{\pi}{2} - x\right)$$
🧮 Worked Examples
🧪 Activities
1 $y = \tan x$ for $-\pi < x < \pi$
Describe key features:
Sketch in your workbook.
2 $y = \cot x$ for $0 < x < 2\pi$
Describe key features:
Sketch in your workbook.
3 $y = \tan(2x)$ for $0 \leq x \leq \pi$
Describe key features:
Sketch in your workbook.
1 $y = \tan(3x)$
Type your answer:
Answer in your workbook.
2 $y = \cot\left(\frac{x}{2}\right)$
Type your answer:
Answer in your workbook.
3 $y = \tan\left(x + \frac{\pi}{4}\right)$
Type your answer:
Answer in your workbook.
Earlier you were asked what happens to $\tan x$ as $x$ approaches $90^\circ$.
As $x \to 90^\circ$ from below, $\cos x \to 0^+$ and $\sin x \to 1$, so $\tan x = \frac{\sin x}{\cos x} \to +\infty$. This means the graph of $y = \tan x$ has a vertical asymptote at $x = 90^\circ$ ($\frac{\pi}{2}$). The curve rises steeply and never crosses this line.
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. Sketch $y = \tan x$ for $-\frac{\pi}{2} < x < \frac{3\pi}{2}$. Label all asymptotes and $x$-intercepts. 3 MARKS
Describe your sketch below:
Sketch in your workbook.
9. Find the period and the equations of the vertical asymptotes of $y = \cot(2x)$. 3 MARKS
Type your answer below:
Answer in your workbook.
10. Explain why $\tan(x + \pi) = \tan x$ for all values of $x$ where $\tan x$ is defined. Use this result to explain why the period of $y = \tan x$ is $\pi$. 3 MARKS
Type your answer below:
Answer in your workbook.
1. Asymptotes at $x = -\frac{\pi}{2}, \frac{\pi}{2}$; intercept at $(0, 0)$. Increasing branches.
2. Asymptotes at $x = 0, \pi, 2\pi$; intercepts at $\frac{\pi}{2}, \frac{3\pi}{2}$. Decreasing branches.
3. Period = $\frac{\pi}{2}$. Asymptotes at $x = \frac{\pi}{4}, \frac{3\pi}{4}$. Intercepts at $0, \frac{\pi}{2}, \pi$.
1. Period = $\frac{\pi}{3}$. Asymptotes: $3x = \frac{\pi}{2} + n\pi \Rightarrow x = \frac{\pi}{6} + \frac{n\pi}{3}$.
2. Period = $2\pi$. Asymptotes: $\frac{x}{2} = n\pi \Rightarrow x = 2n\pi$.
3. Period = $\pi$. Asymptotes: $x + \frac{\pi}{4} = \frac{\pi}{2} + n\pi \Rightarrow x = \frac{\pi}{4} + n\pi$.
1. A — Tangent period is $\pi$.
2. A — Asymptotes at odd multiples of $\frac{\pi}{2}$.
3. A — Cotangent asymptotes at $n\pi$.
4. A — Period of $\tan(2x)$ is $\frac{\pi}{2}$.
5. A — Tangent range is all real numbers.
Q8 (3 marks): Asymptotes at $x = -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}$ [1]. $x$-intercepts at $x = 0, \pi$ [1]. Smooth increasing branches between asymptotes [1].
Q9 (3 marks): Period = $\frac{\pi}{2}$ [1]. $2x = n\pi \Rightarrow x = \frac{n\pi}{2}$ [2].
Q10 (3 marks): $\tan(x + \pi) = \frac{\sin(x + \pi)}{\cos(x + \pi)} = \frac{-\sin x}{-\cos x} = \tan x$ [2]. This shows the function repeats every $\pi$, so the period is $\pi$ [1].
Climb platforms using your knowledge of tangent and cotangent graphs. Pool: lessons 1–11.
Tick when you've finished all activities and checked your answers.