cosecant
Definition
- Noun:
- A trigonometric function: The cosecant is a trigonometric function defined for an angle in a right triangle as the ratio of the length of the hypotenuse to the length of the side opposite that angle. It is the reciprocal of the sine function.
Usage
- The cosecant function is abbreviated as csc.
- It is used in mathematics, particularly in trigonometry, geometry, and calculus.
- The cosecant of an angle θ is defined as: csc(θ) = hypotenuse / opposite side.
- In a right triangle, if the side opposite angle θ has length 'a' and the hypotenuse has length 'h', then csc(θ) = h/a.
- The function is undefined for angles where the sine is zero (e.g., 0°, 180°), as this would involve division by zero.
Examples
- Noun:
- In a right triangle with an opposite side of 3 and a hypotenuse of 5, the cosecant of the angle is 5/3.
- The value of csc(30°) is 2, because sin(30°) is 1/2.
- To solve the equation, you need to understand the properties of the cosecant function.
Advanced Usage
- In the Unit Circle: On the unit circle, for an angle θ with coordinates (x, y), the cosecant is defined as csc(θ) = 1 / y (where y is the sine value), provided y ≠ 0.
- Graphical Representation: The graph of y = csc(x) is a series of disjoint curves with vertical asymptotes where sin(x) = 0.
- Identities: The cosecant function appears in various trigonometric identities, such as 1 + cot²(θ) = csc²(θ).
Variants and Related Words
- Cosecant Function (n): The full name for the mathematical function.
- Csc (abbr.): The standard abbreviation for cosecant.
- Reciprocal Trigonometric Functions: A category including cosecant (csc), secant (sec), and cotangent (cot).
Synonyms
- Reciprocal of sine: This is a descriptive synonym, as csc(θ) = 1 / sin(θ).
Related Terms (Not Phrasal Verbs or Idioms)
- Sine (sin): The trigonometric function whose reciprocal is the cosecant.
- Secant (sec): The reciprocal of the cosine function.
- Cotangent (cot): The reciprocal of the tangent function.
- Hypotenuse: The longest side of a right triangle, opposite the right angle.
- Opposite Side: The side of a triangle opposite a given angle.
Noun
-
ratio of the hypotenuse to the opposite side of a right-angled triangle
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