cotangent

/'kou'tændʤənt/

Thân thiện

A student calculates the cotangent of an angle in a right triangle.

Definition

1. Noun:
   • A trigonometric function: The cotangent is a trigonometric function defined for an angle in a right-angled triangle as the ratio of the length of the adjacent side to the length of the opposite side. It is the reciprocal of the tangent function.

Usage

• The cotangent is used in mathematics, particularly in trigonometry, geometry, and calculus.
• It is commonly abbreviated as cot.
• It is applied to solve problems involving angles and side lengths in triangles, wave functions, and oscillatory phenomena.

Examples

• Noun:
  • In a right triangle with an angle θ, the cotangent of θ is calculated as adjacent/opposite.
  • The value of the cotangent function decreases from infinity to zero as the angle increases from 0 to 90 degrees.
  • To find the unknown side, we used the cotangent of the known angle.

Advanced Usage

• Mathematical Identity: The cotangent can be expressed in terms of other trigonometric functions: cot(θ) = cos(θ) / sin(θ) = 1 / tan(θ).
• Inverse Function: The inverse cotangent function, written as arccot(x) or cot⁻¹(x), returns the angle whose cotangent is x.
• Calculus: The derivative of cot(x) is -csc²(x), and its integral is ln|sin(x)| + C.

Variants and Related Words

• Cot (abbr.): The standard abbreviation for cotangent.
  • Calculate cot(45°).
• Cotangential (adj.): Pertaining to or involving the cotangent.
  • The cotangential relationship was key to the proof.

Synonyms

• Reciprocal tangent: This phrase describes the fundamental relationship, as cotangent is the multiplicative inverse (reciprocal) of the tangent function.

Related Phrases

• Cotangent bundle (n.): A mathematical construct in differential geometry formed from a manifold and the cotangent spaces at each point.
  • The physics problem was analyzed using the cotangent bundle.
• Hyperbolic cotangent (n.): A related function, denoted coth, defined for hyperbolic trigonometry.
  • The hyperbolic cotangent is defined as coth(x) = cosh(x) / sinh(x).

Học thuật

Noun

1. ratio of the adjacent to the opposite side of a right-angled triangle

Từ đồng nghĩa

• cotan