Definition

Noun: A trigonometric function is a function of an angle, typically defined as a ratio of two side lengths of a right-angled triangle that contains that angle. These functions are fundamental in relating the angles of a triangle to the lengths of its sides and form the basis of trigonometry.

Usage

The term "trigonometric function" is used to refer to one of the core functions in mathematics: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). It is a standard term in academic, scientific, and engineering contexts. - The sine and cosine are the most basic trigonometric functions. - To solve this geometry problem, you must apply the correct trigonometric function.

Examples

• In a right triangle, the trigonometric function sine of angle θ is defined as the ratio of the length of the side opposite θ to the length of the hypotenuse.
• Calculus often involves finding the derivatives and integrals of various trigonometric functions.
• The values of the six primary trigonometric functions for common angles should be memorized.

Advanced Usage

• Periodic Nature: Trigonometric functions are periodic, meaning their values repeat at regular intervals. For example, the sine and cosine functions have a period of 2π.
• Unit Circle Definition: Beyond right triangles, trigonometric functions are defined for all real numbers using the coordinates of points on a unit circle. For an angle θ, cos θ is the x-coordinate and sin θ is the y-coordinate of the corresponding point.
• Inverse Trigonometric Functions: These functions (e.g., arcsin, arccos) are used to find an angle when the value of a trigonometric function is known.

Variants and Related Words

• Trigonometry (n): The branch of mathematics that deals with the relationships between the sides and angles of triangles, and with the properties and applications of trigonometric functions.
• Trigonometric (adj): Of or relating to trigonometry or trigonometric functions. (e.g., ).
• Sine (sin), Cosine (cos), Tangent (tan), Cosecant (csc), Secant (sec), Cotangent (cot) (n): The names of the individual trigonometric functions.

Synonyms

• Circular function: A synonym often used in higher mathematics, emphasizing the unit circle definition.
• Angle function: A less common, descriptive synonym.

Related Phrases and Concepts

• Trigonometric identity: An equation involving trigonometric functions that is true for all values of the variables for which both sides are defined (e.g., sin²θ + cos²θ = 1).
• Graph of a trigonometric function: The visual representation, typically a wave-like curve, showing how the function's value changes with the angle.
• Law of Sines/Cosines: Rules relating the sides and angles of any triangle, derived from trigonometric functions.

Noun

1. function of an angle expressed as a ratio of the length of the sides of right-angled triangle containing the angle