Secant Calculator

The secant is the reciprocal of cosine:

sec(θ) = 1 / cos(θ)

It is undefined wherever cos(θ) = 0 — at 90°, 270°, 450°, and every odd multiple of 90° (every odd multiple of π/2 in radians). At those angles the calculator shows “undefined.”

Key values:

• sec(0°) = 1
• sec(30°) = 2/√3 ≈ 1.155
• sec(45°) = √2 ≈ 1.414
• sec(60°) = 2
• sec(90°) = undefined

Range: like cosecant, secant never falls strictly between −1 and +1. It lives in (−∞, −1] ∪ [1, +∞). The graph has vertical asymptotes at ±90°, ±270°, etc., and arcs between them that look like upward and downward parabolas.

Pythagorean identity: the most useful secant identity is:

1 + tan²(θ) = sec²(θ)

This comes directly from dividing the standard identity sin²(θ) + cos²(θ) = 1 through by cos²(θ). You use it constantly in integration — whenever you see 1 + tan²(θ) in an integral, replace it with sec²(θ) and the integral usually simplifies.

The name comes from the geometric construction on the unit circle: the secant line extended from the center to the tangent, and the length of that segment from the center to where it crosses the tangent is sec(θ). It is a literal line segment, not just an abstract ratio.

Integration: ∫sec(θ) dθ = ln|sec(θ) + tan(θ)| + C. This one is harder to remember and comes up in arc length calculations and in physics problems involving relativistic motion.