Secant (sec)
Calculate secant, see it on the unit circle and graph.
What is secant
Secant is the reciprocal of cosine: sec α = 1 / cos α. It is used less often but appears in integrals and some physics formulas.
Domain and properties
The domain is all real numbers except 90° + k·180°. The range is (−∞, −1] ∪ [1, +∞). The period is 360° and the function is even.
Notable values
sec 0° = 1, sec 60° = 2, sec 90° = undefined
Where secant is used in practice
- Higher mathematics – integral calculus and differential equations.
- Physics – some formulas in optics and mechanics.
- Historically in navigation and astronomical tables.
The unit circle and graphs
The best way to understand trigonometric functions is the unit circle – a circle of radius 1 centred at the origin. Moving along it by an angle α, the coordinates of the point are exactly (cos α, sin α). This gives the fundamental identity sin²α + cos²α = 1 (the Pythagorean theorem for that point). The graph of the function is obtained by plotting the function value against the angle – producing the characteristic wave.
Relationships between the functions
tan α = sin α / cos α • cot α = 1 / tan α • sin²α + cos²α = 1
Tip: switch between degrees and radians. 180° = π rad ≈ 3.14159 rad.