Mathfolis

The Secant, Cosecant, and Cotangent Functions

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

The three reciprocal trig functions are defined as: sec⁡x=1/cos⁡x\sec x = 1/\cos x, csc⁡x=1/sin⁡x\csc x = 1/\sin x, cot⁡x=1/tan⁡x=cos⁡x/sin⁡x\cot x = 1/\tan x = \cos x / \sin x. Each is undefined where its denominator is zero — that's where the vertical asymptotes live. Secant and cosecant have ranges ∣y∣≥1|y| \geq 1; cotangent has range all reals.

Reciprocal definitions

Secant
sec⁡x=1cos⁡x\sec x = \frac{1}{\cos x}
Cosecant
csc⁡x=1sin⁡x\csc x = \frac{1}{\sin x}
Cotangent
cot⁡x=1tan⁡x=cos⁡xsin⁡x\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}

Asymptote locations

sec x
x=π2+nπ  (where cos x = 0)x = \tfrac{\pi}{2} + n\pi \;\text{(where cos x = 0)}
csc x
x=nπ  (where sin x = 0)x = n\pi \;\text{(where sin x = 0)}
cot x
x=nπ  (where sin x = 0)x = n\pi \;\text{(where sin x = 0)}
AP Tip: Range of sec and csc is (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty) — there is no value between −1-1 and 11 since cos and sin never exceed 1.
Type 1

Reciprocal definitions

Memorize: secant ↔ cosine, cosecant ↔ sine, cotangent ↔ tangent.

Example 1
Which equals sec⁡x\sec x? (A) sin⁡x\sin x (B) 1/cos⁡x1/\cos x (C) 1/sin⁡x1/\sin x (D) cos⁡x/sin⁡x\cos x / \sin x

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Type 2

Evaluate reciprocal trig at special angles

Compute the underlying sin/cos/tan value, then take its reciprocal.

Example 2
What is sec⁡(π/3)\sec(\pi/3)? (A) 1/21/2 (B) 3/2\sqrt 3/2 (C) 22 (D) 3\sqrt 3

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Type 3

Asymptotes and range

Asymptotes are at the zeros of the underlying sin or cos. Range of sec and csc is ∣y∣≥1|y| \geq 1.

Example 3
Where are the vertical asymptotes of y=csc⁡xy = \csc x? (A) x=nπx = n\pi (B) x=π/2+nπx = \pi/2 + n\pi (C) x=2nπx = 2n\pi (D) Nowhere — csc has no asymptotes

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