6.2PureCore

Graphs of Sec, Cosec and Cot

The reciprocal trig graphs are built directly from sin, cos and tan. Wherever the base function is zero, the reciprocal has a vertical asymptote; wherever the base reaches ±1, the reciprocal touches ±1.

25 min Trigonometric Functions

Edexcel A Level Maths: 6.2 Graphs of Sec, Cosec and CotCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Sketch y = sec x, y = cosec x, y = cot x
Locate vertical asymptotes
State periods and ranges
Relate each graph to its base function

From base to reciprocal

Asymptotes of the reciprocal occur where the base function is $0$ . The reciprocal touches $\pm 1$ where the base equals $\pm 1$ . So sketch the base lightly first, then build the reciprocal around it.

y = \sec x \text{ has asymptotes where } \cos x = 0

Asymptotes at zeros of the base.

Periods and ranges

$\sec x$ and $\operatorname{cosec}x$ have period $2\pi$ and range $y \le -1$ or $y \ge 1$ . $\cot x$ has period $\pi$ and range all real numbers, with asymptotes where $\sin x = 0$ .

Tip — sec/cosec never take values between −1 and 1 — there is a gap around the x-axis.

Formula recap

\sec x, \operatorname{cosec} x: \text{period } 2\pi

Range |y| ≥ 1.

\cot x: \text{period } \pi

Range all reals.

Common mistakes to avoid

Drawing sec/cosec passing through the x-axis.

They never enter −1 < y < 1; there is a gap.

Giving cot the same period as cosec.

cot has period π, not 2π.

Key takeaways

Asymptotes of the reciprocal sit at the zeros of the base.
sec, cosec: period 2π, |y| ≥ 1.
cot: period π, all real values.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

$y = \sec x$ has vertical asymptotes where:
- $\cos x = 0$
- $\sin x = 0$
- $\tan x = 0$
- $x = 0$
Show answer and explanation
Answer: $\cos x = 0$
Reciprocal of cos.
The range of $y = \operatorname{cosec}x$ is:
- $y \le -1$ or $y \ge 1$
- all reals
- $-1 \le y \le 1$
- $y \ge 0$
Show answer and explanation
Answer: $y \le -1$ or $y \ge 1$
Gap around the x-axis.
The period of $y = \cot x$ is:
- $\pi$
- $2\pi$
- $\tfrac{\pi}{2}$
- $4\pi$
Show answer and explanation
Answer: $\pi$
Same as tan.
$y = \cot x$ has asymptotes where:
- $\sin x = 0$
- $\cos x = 0$
- $\tan x = 1$
- nowhere
Show answer and explanation
Answer: $\sin x = 0$
cot = cos/sin.
The period of $y = \sec x$ is:
- $2\pi$
- $\pi$
- $\tfrac{\pi}{2}$
- $\tfrac{2\pi}{3}$
Show answer and explanation
Answer: $2\pi$
Same as cos.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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