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Pure · Trigonometry

Chapter 1 · 4

The idea

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.

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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

The three definitions

Three new functions are just the reciprocals of the ones you know:

sec⁡θ=1cos⁡θ,cosec⁡θ=1sin⁡θ,cot⁡θ=1tan⁡θ\sec\theta = \frac{1}{\cos\theta}, \qquad \operatorname{cosec}\theta = \frac{1}{\sin\theta}, \qquad \cot\theta = \frac{1}{\tan\theta}

Exact values come straight from the reciprocal: 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 has asymptotes where cos⁡θ=0\cos\theta = 0 (θ=π2+nπ\theta = \frac{\pi}{2} + n\pi), period 2π2\pi; y=cosec⁡θy = \operatorname{cosec}\theta where sin⁡θ=0\sin\theta = 0 (θ=nπ\theta = n\pi), period 2π2\pi. Both inherit a forbidden band: since ∣sin⁡∣,∣cos⁡∣≤1|\sin|, |\cos| \le 1, their reciprocals never land strictly between −1-1 and 11 — the range is ≤−1\le -1 or ≥1\ge 1.

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