Secant Calculator - Calculate sec(x) and arcsec(x)

What is the Secant Function?

The secant function, denoted as sec(x), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function, representing the ratio of the hypotenuse to the adjacent side in a right triangle.

The secant function is widely used in mathematics, physics, engineering, and various scientific applications. It's particularly important in calculus, where it appears in derivatives and integrals of trigonometric functions.

The mathematical definition of secant is:

Key properties of the secant function include:

• Domain: sec(x) is defined for all real numbers except x = (2n+1)π/2, where n is any integer.
• Range: The secant function has a range of (-∞, -1] ∪ [1, ∞).
• Periodicity: sec(x) is periodic with period 2π.
• Symmetry: sec(-x) = sec(x), making it an even function.
• Asymptotes: Vertical asymptotes occur at x = (2n+1)π/2.
• Derivative: The derivative of sec(x) is sec(x)tan(x).

The secant function is essential in solving trigonometric equations, analyzing periodic phenomena, and in applications involving right triangles and circular motion.

What is Inverse Secant (Arcsecant)?

The inverse secant function, denoted as arcsec(x) or sec⁻¹(x), is the inverse function of the secant. It answers the question: 'What angle has a secant of x?' The inverse secant function returns the angle whose secant is the given value.

The mathematical definition of inverse secant is:

Properties of the inverse secant function:

• Domain: The inverse secant is defined for |x| ≥ 1.
• Range: The principal value range is [0, π/2) ∪ (π/2, π].
• Monotonicity: arcsec(x) is strictly decreasing on its domain.
• Special values: arcsec(1) = 0, arcsec(2) = π/3, arcsec(√2) = π/4.
• Derivative: The derivative of arcsec(x) is 1/(|x|√(x²-1)).

The inverse secant function is particularly useful in solving trigonometric equations and in applications where you need to find angles from secant values.

Common Secant Values

Here are some important secant values for common angles:

• sec(0°) = 1
• sec(30°) = 2/√3 ≈ 1.155
• sec(45°) = √2 ≈ 1.414
• sec(60°) = 2
• sec(90°) = undefined
• sec(120°) = -2
• sec(135°) = -√2 ≈ -1.414
• sec(150°) = -2/√3 ≈ -1.155