Definition

1. Noun:
   • A trigonometric function: In mathematics, the secant (abbreviated as sec) is a trigonometric function defined as the ratio of the length of the hypotenuse to the length of the adjacent side in a right-angled triangle. It is the reciprocal of the cosine function.
   • A line intersecting a curve: In geometry, a secant is a straight line that intersects a curve, typically a circle or other conic section, at two or more distinct points.

Examples of Usage

• Noun (Trigonometry):
  • The secant of angle θ is defined as sec(θ) = hypotenuse / adjacent side.
  • To solve the equation, you need to find the value of the secant.
• Noun (Geometry):
  • Draw a secant through the circle to demonstrate the intersection points.
  • The line segment connecting two points on a curve is a chord; the extended line is a secant.

Advanced Usage

• "Secant line": A specific term in calculus and geometry for a line that cuts through a curve at two points, often used to approximate the slope (derivative) of the curve.
  • The slope of the secant line between two points gives the average rate of change of the function.
• "Secant method": A numerical analysis technique used to find roots of equations, which uses secant lines to approximate solutions.
  • The secant method is often faster than the bisection method for finding roots.

Variants and Related Words

• Sec (abbreviation): The standard abbreviation for the secant function in trigonometry.
  • Calculate sec(45°) using the reciprocal of cos(45°).
• Cosecant (noun): The reciprocal of the sine function.
• Tangent (noun): A trigonometric function (ratio of opposite to adjacent) and a line that touches a curve at a single point.

Synonyms

• Intersecting line (for the geometric meaning).
• Reciprocal of cosine (for the trigonometric function meaning).

Related Phrases

• Secant squared identity: Refers to the trigonometric identity: sec²(θ) = 1 + tan²(θ).
  • The secant squared identity is derived from the Pythagorean identity.

Noun

1. ratio of the hypotenuse to the adjacent side of a right-angled triangle
2. a straight line that intersects a curve at two or more points