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 Video by Zeeshan Zamurred Trigonometric Functions

Edexcel A Level Maths: 6.2 Graphs of Sec, Cosec and CotWatch the full walkthrough before the notes below.

Open on YouTube

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 has asymptotes where cos x = 0

Asymptotes at zeros of the base.

Periods and ranges

$sec x$ and $cosec x$ have period $2 π$ and range $y \leq - 1$ or $y \geq 1$ . $cot x$ has period $π$ 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, cosec x : period 2 π

Range |y| ≥ 1.

cot x : period π

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

Ready to lock in Graphs of Sec, Cosec and Cot? Pick a mode and earn XP & Dobloons.