Secant

The secant function (secθ) is a fundamental trigonometric function, defined as the reciprocal of the cosine function (i.e., secθ = 1/cosθ). In a right triangle, it represents the ratio of the hypotenuse to the adjacent side of an acute angle. Its graph exhibits periodicity, symmetry, and specific domain characteristics, serving as a key tool in solving problems in geometry, physics, and other fields.

1. Secant calculator

The secant value calculator is a practical tool for quickly obtaining secant values of any angle (in degrees or radians), providing efficient support for learning, scientific research, and engineering calculations.

Calculation Principle: Based on the reciprocal relationship between secant and cosine (secθ = 1/cosθ), after inputting an angle, the system first calculates the corresponding cosine value and then takes its reciprocal (results are undefined when the cosine value is 0).

Secant Calculator

Calculate secant of angles or radians

Degrees (°)
Radians (rad)
°
sec(0 °) = 1

Common secant values reference:

sec(0°) = 1
sec(30°) ≈ 1.1547
sec(45°) ≈ 1.4142
sec(60°) = 2
sec(90°) = NAN
sec(180°) = -1
sec(0) = 1
sec(π/6) ≈ 1.1547
sec(π/4) ≈ 1.4142
sec(π/3) = 2
sec(π/2) = NAN
sec(π) = -1

2. Secant definition

Right Triangle Definition

Before getting stuck into the functions, it helps to give a name to each side of a right triangle:

triangle showing Opposite, Adjacent and Hypotenuse
  • "Opposite" is opposite to the angle θ
  • "Adjacent" is adjacent to (next to) the angle θ
  • "Hypotenuse" is the long one
sec {\displaystyle \theta }=\frac{Hypotenuse}{Adjacent}=(\frac{1}{cos {\displaystyle \theta }})

General Angle Definition

In the Cartesian coordinate system, for any angle θ (whose terminal side does not coincide with the y-axis), if a point P(x, y) lies on its terminal side with a distance r = √(x² + y²) from the origin, then secθ = r/x (x ≠ 0). This definition breaks through the limitation of acute angles and applies to all angles (including obtuse and negative angles), serving as a general extension of the right triangle definition.

Feature

The graph of the secant function is derived from the cosine function, with the following core characteristics:

  • Periodicity: It has a period of 2π (360°), meaning sec(θ + 2π) = secθ, consistent with the period of the cosine function;
  • Asymptotes: The secant function is undefined when cosθ = 0 (i.e., θ = π/2 + kπ, where k is an integer), and there are vertical asymptotes at these points (e.g., θ = 90°, 270°, etc.);
  • Symmetry: As an even function, its graph is symmetric about the y-axis, satisfying sec(-θ) = secθ;
  • Range: y ≤ -1 or y ≥ 1. Since the range of the cosine function is [-1, 1] (excluding 0), the absolute value of the secant function value is always at least 1.

3. Table of Secant

The table lists secant values from 1° to 360° (rounded to four decimal places), intuitively presenting the numerical rules of the function. Note the following when using it:

  • Sign Rules: The sign of the secant value is the same as that of the cosine value (since secθ = 1/cosθ). That is, secθ is positive when θ is in the first or fourth quadrant, and negative when θ is in the second or third quadrant (e.g., sec(120°) = -2.0000, corresponding to 120° in the second quadrant where the cosine value is negative).
  • Undefined Cases: Secant values are undefined at angles such as 90° and 270°, where cosθ = 0, meaning the secant function does not exist at these points.
  • Value Change Trends: From 0° to 90°, the secant value increases from 1 to infinity (as the cosine value decreases from 1 to 0); from 90° to 180°, the secant value increases from negative infinity to -1 (as the cosine value decreases from 0 to -1), reflecting an inverse relationship with the cosine function.
Secant Values Table (1-360 Degrees)
The table presents secant values from 1° to 360°, formatted as sec (degree°)=value (rounded to 4 decimal places)

4. Origin of the Secant Function

The development of the secant function is closely linked to the evolution of the trigonometric system:

  • Early Embryonic Stage: Ancient Greek astronomers (such as Ptolemy) laid the foundation for the cosine function by describing the relationship between angles and circles through the concept of "chord length" in their studies of celestial movements, with the reciprocal property of secant implicitly contained within.
  • Arab Period: Medieval Arab mathematicians (such as al-Battani) systematically sorted out the proportional relationships of trigonometric functions while translating and organizing Greek classics, clarifying the reciprocal relationship between "cosine" and "secant," and applying them to astronomical calendar calculations (e.g., predicting the angles of solar and lunar eclipses).
  • Standardization Stage: In the 16th–17th centuries, European mathematicians (such as Euler) standardized trigonometric definitions through the unit circle and coordinate system. The term "secant" derives from the Latin word "secare" (meaning "to cut"), as it is geometrically related to the secant of a circle, eventually becoming an internationally recognized symbol.

5. Applications of the Secant Function

The secant function has important application value in multiple fields:

  • Mathematics: It is a core component of trigonometric identities, such as "1 + tan²θ = sec²θ" (derived from sin²θ + cos²θ = 1), which can simplify complex expressions. In calculus, the integral of the secant function (e.g., ∫secθdθ = ln|secθ + tanθ| + C) is a key tool for solving periodic problems.
  • Physics and Engineering: In wave theory (e.g., the propagation of light waves and sound waves), the secant function can describe the relationship between amplitude and angle. In mechanical design, when calculating the "hypotenuse-adjacent ratio" of inclined structures (such as the angle of bridge stay cables and the angle of robotic arm joints), the secant value directly reflects the proportion of force decomposition.
  • Navigation and Surveying: In azimuth and distance conversion, the secant function helps convert horizontal distance to hypotenuse distance (e.g., in satellite positioning, calculating the slant distance from a known adjacent side length and angle).
  • Electromagnetism: In the analysis of AC circuits, the secant function can be used to calculate the relationship between impedance and phase angle, aiding in optimizing circuit design.