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Pure · Trigonometry

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Reciprocal trigonometric functions

The three reciprocal functions secant, cosecant and cotangent — their definitions, the shape and key features of their graphs, exact values, and solving equations that involve them.

Pure · Trigonometry

Reciprocal trigonometric functions

The three reciprocal functions secant, cosecant and cotangent — their definitions, the shape and key features of their graphs, exact values, and solving equations that involve them.

Why it works

Three new functions are just the reciprocals of the ones you know: sec⁡θ=1cos⁡θ,cosec⁡θ=1sin⁡θ,cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ.\sec\theta = \frac{1}{\cos\theta}, \qquad \operatorname{cosec}\theta = \frac{1}{\sin\theta}, \qquad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}.

Pairing them up (the trap). The "co" doesn't match the obvious partner: sec goes with cos, and cosec goes with sin. A reliable rule — look at the third letter: sec →\to 1/cos, cosec →\to ... use the pairing sec⁡↔cos⁡\sec\leftrightarrow\cos, cosec⁡↔sin⁡\operatorname{cosec}\leftrightarrow\sin, cot⁡↔tan⁡\cot\leftrightarrow\tan and don't overthink it.

Exact values come straight from the reciprocal. For example sec⁡π3=1cos⁡π3=11/2=2\sec\frac{\pi}{3} = \dfrac{1}{\cos\frac{\pi}{3}} = \dfrac{1}{1/2} = 2, and cot⁡π4=cos⁡(π/4)sin⁡(π/4)=1\cot\frac{\pi}{4} = \dfrac{\cos(\pi/4)}{\sin(\pi/4)} = 1.

The graphs. Each reciprocal blows up wherever the original function is zero (you can't divide by 00), giving vertical asymptotes:
  • y=sec⁡θy = \sec\theta: asymptotes where cos⁡θ=0\cos\theta = 0 (i.e. $\theta = \frac{\pi}{2}
+ n\pi);period); period 2\pi;itneverlandsbetween; it never lands between -1and and 1,soits∗∗rangeis, so its **range is \sec\theta \le -1or or \sec\theta \ge 1$**.
  • y=cosec⁡θy = \operatorname{cosec}\theta: asymptotes where sin⁡θ=0\sin\theta = 0 (i.e.
θ=nπ\theta = n\pi); period 2π2\pi; range cosec⁡θ≤−1\operatorname{cosec}\theta \le -1 or ≥1\ge 1.
  • y=cot⁡θy = \cot\theta: asymptotes where sin⁡θ=0\sin\theta = 0 (θ=nπ\theta = n\pi); it
decreases across each branch; period π\pi (not 2π2\pi), and its range is all real numbers.

Solving equations. Take reciprocals to get back to sin⁡/cos⁡/tan⁡\sin/\cos/\tan, then solve as usual. The one slip to avoid: sec⁡x=2\sec x = 2 means cos⁡x=12\cos x = \frac12 (invert the 22), not cos⁡x=2\cos x = 2. And because ∣sec⁡θ∣,∣cosec⁡θ∣≥1|\sec\theta|, |\operatorname{cosec}\theta| \ge 1, an equation like cosec⁡x=0.5\operatorname{cosec} x = 0.5 has no solutions.