The Secant Function | Moosa Academy

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

In a right triangle that is — the cosine ratio turned upside down.

It is undefined wherever , that is at . Since , 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

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

. The angle is in the third quadrant, where

⟹ sec 240° = 1 ÷ (−0.5) = −2

Solve

. Then

⟹ θ = 60° or θ = 300°

Simplify

⟹ = 1

A right triangle has hypotenuse 10 cm and adjacent side 6 cm, so

and

⟹ sec θ ≈ 1.667 and θ = cos⁻¹(0.6) ≈ 53.13°

Note Properties and identity

Domain: all angles except . Range: or . Period: 360°.

Secant is an even function: , like the cosine it comes from.

It also carries its own Pythagorean identity: .

Summary

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