3.9 The Secant, Cosecant, and Cotangent Functions

Reciprocal functions

Secant, cosecant, and cotangent are reciprocals of cosine, sine, and tangent: sec(θ) = 1/cos(θ), csc(θ) = 1/sin(θ), cot(θ) = 1/tan(θ) = cos(θ)/sin(θ). Each is undefined wherever its denominator equals zero, producing vertical asymptotes at those input values.

Domain, range, and periodicity

sec(θ) and csc(θ) have period 2π and range (-∞,-1]∪[1,∞), since |cos| and |sin| never exceed 1. cot(θ) has period π like tangent, with range all real numbers. sec is undefined where cos = 0 (θ = π/2 + πk); csc is undefined where sin = 0 (θ = πk); cot is undefined where sin = 0 too.

Graph behavior near asymptotes

Near a vertical asymptote, the reciprocal function shoots toward +∞ or -∞ depending on the sign of the original sine/cosine/tangent value nearby. Local minima of sec/csc occur where cos/sin have local maxima (value 1), and local maxima occur where cos/sin have local minima (value -1) — the reciprocal flips extrema into the nearest branch.

Key terms

sec(θ)

1/cos(θ); undefined where cos(θ) = 0; period 2π; range (-∞,-1]∪[1,∞).

csc(θ)

1/sin(θ); undefined where sin(θ) = 0; period 2π; range (-∞,-1]∪[1,∞).

cot(θ)

cos(θ)/sin(θ) = 1/tan(θ); undefined where sin(θ) = 0; period π; range all real numbers.

Vertical asymptote (reciprocal trig)

Occurs wherever the original sine/cosine/tangent value equals zero, since dividing by zero is undefined.

Extrema flip in reciprocal functions

Where the original function has a max (value 1), the reciprocal has a local min; where original has a min (value -1), reciprocal has a local max.

Try an AP-style question

sec(π/3) = ?

1. √3
2. 2/√3
3. 2
4. 1/2