The Secant Function

Secant is the reciprocal of the cosine, sec θ = 1 ÷ cos θ, undefined wherever cos θ = 0 and never strictly between −1 and 1.

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Secant is the reciprocal of the cosine. Everything about it — where it breaks, how high and low it goes, how often it repeats — follows from that one relationship.

Concept Definition

\sec\theta = \frac{1}{\cos\theta}

In a right triangle that is \dfrac{\text{hypotenuse}}{\text{adjacent}} — the cosine ratio turned upside down.

It is undefined wherever \cos\theta = 0 , that is at 90° + 180°n . Since |\cos\theta| \le 1 , the reciprocal is never smaller than 1 in size, so no value of sec θ lies strictly between −1 and 1.

Graph sec θ against cos θ
180° 90° 270° solid: sec θ  ·  dotted: cos θ  ·  dashed verticals: asymptotes

Each branch touches the cosine curve where the cosine is at ±1 — its own closest approach to the axis — then bends away to ±∞ as the cosine drops toward 0.

Reference Special values
θ cos θ sec θ
0° 1 1
30° √3 ÷ 2 2√3 ÷ 3 ≈ 1.155
45° √2 ÷ 2 √2 ≈ 1.414
60° 0.5 2
90° 0 undefined
120° −0.5 −2
180° −1 −1
240° −0.5 −2
Example Four short problems
Find \sec 240° . The angle is in the third quadrant, where \cos 240° = -0.5 .
⟹ sec 240° = 1 ÷ (−0.5) = −2
Solve \sec\theta = 2 on [0°,\,360°) . Then \cos\theta = 0.5 .
⟹ θ = 60° or θ = 300°
Simplify \sec x \times \cos x = \dfrac{1}{\cos x} \times \cos x
⟹ = 1
A right triangle has hypotenuse 10 cm and adjacent side 6 cm, so \sec\theta = \dfrac{10}{6} and \cos\theta = 0.6 .
⟹ sec θ ≈ 1.667 and θ = cos⁻¹(0.6) ≈ 53.13°
Note Properties and identity

Domain: all angles except 90° + 180°n .  Range: \sec\theta \le -1 or \sec\theta \ge 1 .  Period: 360°.

Secant is an even function: \sec(-\theta) = \sec\theta , like the cosine it comes from.

It also carries its own Pythagorean identity: 1 + \tan^2\theta = \sec^2\theta .

Summary
  1. sec θ = 1 ÷ cos θ, or hypotenuse ÷ adjacent in a right triangle.
  2. It is undefined wherever cos θ = 0, giving asymptotes at 90° + 180°n.
  3. No values between −1 and 1; the period is 360° and the function is even.
  4. Identity to remember: 1 + tan²θ = sec²θ.
231. Arithmetic Sequences and Series
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