Versine Calculator - versin(x) and aversin(x)

What is the Versine Function?

The versine function, denoted as versin(x) or vers(x), is a trigonometric function that represents the versed sine of an angle. It is defined as the complement of the cosine function, measuring the vertical distance from the center of a unit circle to the point where a line from the center at angle x intersects the circle.

The versine function is one of the lesser-known trigonometric functions but has important applications in navigation, astronomy, and spherical geometry. It was historically used in navigation tables before the advent of modern calculators and computers.

The mathematical definition of versine is:

Key properties of the versine function include:

• Range: The versine function has a range of [0, 2], reaching its minimum value of 0 when x = 0 and maximum value of 2 when x = π.
• Periodicity: Like cosine, versine is periodic with period 2π.
• Symmetry: versin(x) = versin(-x), making it an even function.
• Derivative: The derivative of versin(x) is sin(x).
• Integration: The integral of versin(x) is x - sin(x) + C.

The versine function is particularly useful in spherical trigonometry and navigation, where it helps calculate distances and angles on the Earth's surface. It's also used in signal processing and in the analysis of periodic functions.

What is Inverse Versine (Aversine)?

The inverse versine function, also known as aversine or arcversine, is the inverse function of the versine. It answers the question: 'What angle has a versine of y?' The inverse versine function is denoted as aversin(y) or arcversin(y).

The mathematical definition of inverse versine is:

Properties of the inverse versine function:

• Domain: The inverse versine is defined for y in the interval [0, 2].
• Range: The output range is [0, π].
• Monotonicity: aversin(y) is strictly increasing on its domain.
• Special values: aversin(0) = 0, aversin(1) = π/2, aversin(2) = π.

The inverse versine function is particularly useful in navigation and geodesy, where it's used to calculate angles from versine values obtained from measurements or calculations.

Common Versine Values

Here are some important versine values for common angles:

• versin(0°) = 0
• versin(30°) = 1 - √3/2 ≈ 0.134
• versin(45°) = 1 - √2/2 ≈ 0.293
• versin(60°) = 1 - 1/2 = 0.5
• versin(90°) = 1 - 0 = 1
• versin(120°) = 1 - (-1/2) = 1.5
• versin(180°) = 1 - (-1) = 2