Secant Function — Definition, Formula & Examples
The secant function, written sec θ, is the reciprocal of the cosine function. For any angle θ, sec θ equals 1 divided by cos θ, and it is undefined wherever cosine equals zero.
For an angle θ in standard position on the unit circle with terminal point (x, y), the secant is defined as sec θ = 1/cos θ = r/x, where r is the radius (hypotenuse) and x ≠ 0. Equivalently, in a right triangle, sec θ equals the ratio of the hypotenuse to the side adjacent to θ.
Key Formula
Where:
- = The angle, measured in degrees or radians
- = The cosine of the angle, which must be nonzero
How It Works
To evaluate sec θ, first find cos θ, then take its reciprocal. Because division by zero is undefined, sec θ does not exist when cos θ = 0, which occurs at θ = 90°, 270°, and their coterminal angles (odd multiples of π/2). The range of secant is (−∞, −1] ∪ [1, ∞), meaning its output is never between −1 and 1. Its graph has vertical asymptotes at every odd multiple of π/2 and repeats with a period of 2π.
Worked Example
Problem: Find the exact value of sec 60°.
Step 1: Recall the cosine of 60°.
Step 2: Take the reciprocal to get secant.
Answer: sec 60° = 2
Why It Matters
Secant appears frequently in calculus, especially in integrals and derivatives involving trigonometric substitution. It also arises in physics and engineering when modeling quantities like the length of a line from a point to a circle, or the magnification factor in projections.
Common Mistakes
Mistake: Confusing sec θ with cos⁻¹ θ (arccos). Students sometimes think the reciprocal and the inverse function are the same thing.
Correction: sec θ = 1/cos θ is a reciprocal (division), while cos⁻¹ x (or arccos x) is the inverse function that returns an angle. They are completely different operations.